Singular Spaces Research Articles

Singular geometrical optics for differential operators on surfaces

Inspired by the optical phenomenon of conical refraction, discovered by Hamilton in 1832, we study the existence of singular optical phenomena associated with linear differential operators acting on vector fields on a surface. We do this by studying the singularities of the Fresnel hyper-surface associated with the differential operator and show that the existence of these singularities can be accounted for from purely topological considerations. We associate a topological number (an integer) to the space of singularities of the Fresnel hyper-surface, which can be used to “count” the number points at which singular optical phenomena occur.

$d$-Minimal Surfaces in Three-Dimensional Singular Semi-Euclidean Space $\mathbb{R}^{0,2,1}$

In this paper, we investigate surfaces in singular semi-Euclidean space $\mathbb{R}^{0,2,1}$ endowed with a degenerate metric. We define $d$-minimal surfaces, and give a representation formula of Weierstrass type. Moreover, we prove that $d$-minimal surfaces in $\mathbb{R}^{0,2,1}$ and spacelike flat zero mean curvature (ZMC) surfaces in four-dimensional Minkowski space $\mathbb{R}^{4}_{1}$ are in one-to-one correspondence.

Poincar\xe9 duality for L^p cohomology on subanalytic singular spaces

We investigate the problem of Poincaré duality for L^p differential forms on bounded subanalytic submanifolds of mathbb {R}^n (not necessarily compact). We show that, when p is sufficiently close to 1 then the L^p cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that L^p cohomology is dual to intersection homology. As a consequence, we can deduce that the L^p cohomology is Poincaré dual to L^q cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large.

On the reduction of repetitive processes into singular and non-singular Roesser models

In this paper we present new methods for the reduction of a polynomial system matrix describing a discrete linear repetitive process, to equivalent singular and non-singular 2-D state space representations. Particularly, a zero coprime system equivalence transformation resulting in a singular Roesser state space model, preserving the core algebraic structure of the original system matrix, is proposed. As a second step utilizing the singular Roesser model introduced, we further reduce the system to a non-singular, zero coprime system equivalent Roesser model. Both models are constructed by inspection or by applying elementary matrix manipulations and have significantly smaller dimensions compared to similar reductions found in the literature.

Linear relations and their singular chains

Singular chain spaces for linear relations in linear spaces play a fundamental role in the decomposition of linear relations in finite dimensional spaces. In this paper singular chains and singular chain spaces are discussed in detail for not necessarily finite dimensional linear spaces. This leads to an identity that characterizes a singular chain space in terms of root spaces. The so-called proper eigenvalues of a linear relation play an important role in the finite dimensional case. <br><br> Сингулярні ланцюгові простори для лінійних відношень у лінійних просторах грають Фундаментальна роль у розкладанні лінійних відношень в скінчено вимірних просторах. У даній роботі надається детальний розгляд сингулярних ланцюгів і просторів сингулярних ланцюгів для не обов'язково скінченновимірних просторів. З цього отримується тотожність, що характеризує простір сингулярних ланцюгів в термінах кореневих просторів. У скінченновимірному випадкуважливу роль відіграють так звані правильні власні значення.

Arquile Varieties – Varieties Consisting of Power Series in a Single Variable

Abstract Spaces of power series solutions $y(\mathrm {t})$ in one variable $\mathrm {t}$ of systems of polynomial, algebraic, analytic or formal equations $f(\mathrm {t},\mathrm {y})=0$ can be viewed as ‘infinite-dimensional’ varieties over the ground field $\mathbf {k}$ as well as ‘finite-dimensional’ schemes over the power series ring $\mathbf {k}[[\mathrm {t}]]$ . We propose to call these solution spaces arquile varieties, as an enhancement of the concept of arc spaces. It will be proven that arquile varieties admit a natural stratification ${\mathcal Y}=\bigsqcup {\mathcal Y}_d$ , $d\in {\mathbb N}$ , such that each stratum ${\mathcal Y}_d$ is isomorphic to a Cartesian product ${\mathcal Z}_d\times \mathbb A^{\infty }_{\mathbf {k}}$ of a finite-dimensional, possibly singular variety ${\mathcal Z}_d$ over $\mathbf {k}$ with an affine space $\mathbb A^{\infty }_{\mathbf {k}}$ of infinite dimension. This shows that the singularities of the solution space of $f(\mathrm {t},\mathrm {y})=0$ are confined, up to the stratification, to the finite-dimensional part. Our results are established simultaneously for algebraic, convergent and formal power series, as well as convergent power series with prescribed radius of convergence. The key technical tool is a linearisation theorem, already used implicitly by Greenberg and Artin, showing that analytic maps between power series spaces can be essentially linearised by automorphisms of the source space. Instead of stratifying arquile varieties, one may alternatively consider formal neighbourhoods of their regular points and reprove with similar methods the Grinberg–Kazhdan–Drinfeld factorisation theorem for arc spaces in the classical setting and in the more general setting.

New construction of error-correcting pooling designs from singular linear spaces over finite fields

The group testing problem is that we are asked to identify all the defects with the minimum number of tests when given a set of n items with at most d defects. In this paper, as a generalization of Liu et al.’s construction in the paper (Liu and Gao in Discret Math 338:857–862, 2015), new pooling designs are constructed from singular linear spaces over finite fields. Then we make comparisons with Liu et al.’s construction in the aspects of parameters of pooling designs. By choosing appropriate parameters in our pooling designs, the performance of test efficiency in our pooling designs is better than that given by Liu et al. Finally, the analysis of parameters in our pooling designs is provided.

Statistical Inferences of Linear Forms for Noisy Matrix Completion

AbstractWe introduce a flexible framework for making inferences about general linear forms of a large matrix based on noisy observations of a subset of its entries. In particular, under mild regularity conditions, we develop a universal procedure to construct asymptotically normal estimators of its linear forms through double-sample debiasing and low-rank projection whenever an entry-wise consistent estimator of the matrix is available. These estimators allow us to subsequently construct confidence intervals for and test hypotheses about the linear forms. Our proposal was motivated by a careful perturbation analysis of the empirical singular spaces under the noisy matrix completion model which might be of independent interest. The practical merits of our proposed inference procedure are demonstrated on both simulated and real-world data examples.

Dark Energy, Dark Matter, and the Multiverse

Our third spatial dimension (3-D) is currently in the process of building the fourth spatial dimension. Dark Energy is being used to create higher dimensions of space from the first to the nth, and it is also responsible for the formation and the rotation of the fourth dimension from the point singularities of the vacuum space of the third dimension. To start the creation up to the nth dimension, n! zero dimensional singularities arise simultaneously from the n = 0 level of the void. Once the creation of an n dimensional ellipsoidal matter Universe has been completed, the outward Dark Energy force ends, and then the inward force of gravity takes over to crush the n dimensional Universe into the center of the nth dimensional singularity of the void. The system is now in a non-equilibrium state because the entropy of the system has attained its lowest possible value. To bring the system back to equilibrium there will arise out of the nth dimensional void an antimatter Universe which will reverse the process to deposit all the Dark Energy back into the original levels of the void so that the net change in entropy of the system becomes zero. Dark Matter is the matter that exists in the fourth spatial dimension as 4-D matter spewed there through the Black Hole singularities of our third dimension. Ellipsoidal matter and antimatter Multiverses rotating in opposite directions are a natural byproduct of this new theory of Physics which also solves certain particle Physics problems of the Standard Model.

Performance Analysis of a Two–Tile Reconfigurable Intelligent Surface Assisted 2 × 2 MIMO System

We consider a two-tile reconfigurable intelligent surface (RIS) assisted wireless network with a two-antenna transmitter and receiver over Rayleigh fading. We show that the average received signal-to-noise-ratio (SNR) optimal combining and transmission vectors are given by the left and right singular spaces of the RIS-receiver and transmit-RIS channel matrices, respectively. Moreover, the optimal phases at the two tiles of the RIS are determined by the phases of the elements of the latter spaces. To further study the effect of phase compensation, we statistically characterize the average SNR of all possible combinations of transmission and combining directions pertaining to the latter singular spaces by deriving novel expressions for the outage probability and throughput of each of those modes. Furthermore, for comparison, we derive the corresponding expressions in the absence of RIS. Our results show an approximate SNR improvement of 2 dB due to the phase compensation at the RIS.

Semi-abelian Spectral Data for Singular Fibres of the 𝖲𝖫(2,ℂ)-Hitchin System

Abstract We describe spectral data for singular fibres of the $\textsf{SL}(2,{\mathbb{C}})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations, we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations, this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications, we study irreducible components of singular Hitchin fibres and give a description of $\textsf{SL}(2,{\mathbb{R}})$-Higgs bundles in terms of these semi-abelian spectral data.

Degenerate perturbation theory in thermoacoustics: high-order sensitivities and exceptional points

Abstract

Geometric conditions for the null-controllability of hypoelliptic quadratic parabolic equations with moving control supports

We study the null-controllability of some hypoelliptic quadratic parabolic equations posed on the whole Euclidean space with moving control supports, and provide necessary or sufficient geometric conditions on the moving control supports to ensure null-controllability. The first class of equations is the one associated to non-autonomous Ornstein–Uhlenbeck operators satisfying a generalized Kalman rank condition. In particular, when the moving control supports comply with the flow associated to the transport part of the Ornstein–Uhlenbeck operators, a necessary and sufficient condition for null-controllability on the moving control supports is established. The second class of equations is the class of accretive non-selfadjoint quadratic operators with zero singular spaces for which some sufficient geometric conditions on the moving control supports are also given to ensure null-controllability.

Trees of manifolds as boundaries of spaces and groups

We show that trees of manifolds, the topological spaces introduced by Jakobsche, appear as boundaries at infinity of various spaces and groups. In particular, they appear as Gromov boundaries of some hyperbolic groups, of arbitrary dimension, obtained by the procedure of strict hyperbolization. We also recognize these spaces as boundaries of arbitrary Coxeter groups with manifold nerves, and as Gromov boundaries of the fundamental groups of singular spaces obtained from some finite volume hyperbolic manifolds by cutting off their cusps and collapsing the resulting boundary tori to points.

Butterfly-Like Anisotropic Magnetoresistance and Angle-Dependent Berry Phase in a Type-II Weyl Semimetal WP2Supported by the National Natural Science Foundation of China (Grant Nos. 11974324, 11804326, U1832151, and 11674296), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDC07010000), the National Key Research and Development Program of China (Grant No. 2017YFA0403600), the Anhui Initiative in Quantum Information Technologies

The Weyl semimetal has emerged as a new topologically nontrivial phase of matter, hosting low-energy excitations of massless Weyl fermions. Here, we present a comprehensive study of a type-II Weyl semimetal WP2. Transport studies show a butterfly-like magnetoresistance at low temperature, reflecting the anisotropy of the electron Fermi surfaces. This four-lobed feature gradually evolves into a two-lobed variant with an increase in temperature, mainly due to the reduced relative contribution of electron Fermi surfaces compared to hole Fermi surfaces for magnetoresistance. Moreover, an angle-dependent Berry phase is also discovered, based on quantum oscillations, which is ascribed to the effective manipulation of extremal Fermi orbits by the magnetic field to feel nearby topological singularities in the momentum space. The revealed topological character and anisotropic Fermi surfaces of the WP2 substantially enrich the physical properties of Weyl semimetals, and show great promises in terms of potential topological electronic and Fermitronic device applications.

AS SINGULARIDADES DO ESPAÇO URBANO E SUAS IMPLICAÇÕES NAS PRÁTICAS TERRITORIAIS DO ENTORNO DE MAPUTO/MOZ

Buscamos apresentar neste artigo o relato de um recorte dos olhares obtidos durante experiência de pesquisa realizada em Maputo, capital de Moçambique, de abril a maio de 2015. No caso do presente olhar, destacamos a análise acerca das singularidades do espaço urbano e suas implicações nas práticas territoriais do Vale do Infulene e Distrito de Marracuene, no entorno de Maputo/MOZ; e do entorno de Xai-Xai, capital da Província de Gaza; além da praia de Chidenguele/Xai-Xai. No deslindar da pesquisa percebemos histórias singulares, com maior destaque às das mulheres. Histórias estas vividas no contexto sempre de acontecimentos marcantes da sociedade moçambicana, tal como o processo de independência nacional (1975) e, como uma de suas consequências, a incorporação do aproveitamento do solo / uso tradicional da terra; além das profundas marcas deixada pela Guerra Civil (1976-1992). As práticas produtivas alternativas efetivadas pelas comunidades, na construção dos territórios, podem ser entendidas como coexistência do “avanço” da modernização capitalista e manutenção de práticas tradicionais, que ressignificam os espaços rural e urbano e se exprimem numa paisagem urbana com fortíssimas imbricações rurais.

Codimension 1 foliations with numerically trivial canonical class on singular spaces

In this article, we describe the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira and Touzet to this context.

Verified inclusions for a nearest matrix of specified rank deficiency via a generalization of Wedin’s $$\sin (\theta )$$ theorem

For an $$m \times n$$ matrix A, the mathematical property that the rank of A is equal to r for $$0< r < \min (m,n)$$ is an ill-posed problem. In this note we show that, regardless of this circumstance, it is possible to solve the strongly related problem of computing a nearby matrix with at least rank deficiency k in a mathematically rigorous way and using only floating-point arithmetic. Given an integer k and a real or complex matrix A, square or rectangular, we first present a verification algorithm to compute a narrow interval matrix $$\varDelta $$ with the property that there exists a matrix within $$A-\varDelta $$ with at least rank deficiency k. Subsequently, we extend this algorithm for computing an inclusion for a specific perturbation E with that property but also a minimal distance with respect to any unitarily invariant norm. For this purpose, we generalize Wedin’s $$\sin (\theta )$$ theorem by removing its orthogonality assumption. The corresponding result is the singular vector space counterpart to Davis and Kahan’s generalized $$\sin (\theta )$$ theorem for eigenspaces. The presented verification methods use only standard floating-point operations and are completely rigorous including all possible rounding errors and/or data dependencies.

Secrecy precoding in MIMOME wireless communication system under partial CSI

In this study, a precoding scheme which is called singular value decomposition and null space (SVDNS) based scheme is proposed to enhance physical layer security of a multiple-input multiple-output multiantenna eavesdropper (MIMOME) system. Under the partial channel state information (CSI) of the eavesdropper, the precoding matrix for the information bearing signal is constructed firstly via SVDNS-based scheme to acquire a compromise between improving performance of the main channel and impairing that of the wiretap channel. Then the optimal power allocation over subchannels formed by precoding matrix is obtained through solving the secrecy rate maximise problem to further improve secrecy performance. After that, under the scenario that CSI of the wiretap channel is less knowable, precoding matrix for artificial noise (AN) is proposed via SVDNS-based scheme, and then the optimal power allocation between the information bearing signal and AN is obtained while optimal power allocation over subchannels is solved subsequently. Simulation results show that the SVDNS-based scheme with optimal power allocation achieves a higher secrecy rate than the existing schemes, and the SVDNS-based scheme with AN aiding could make up for the performance degradation of it without AN in the situation that CSI of the wiretap channel is more unknowable.

On the motive of O’Grady’s ten-dimensional hyper-Kähler varieties

We investigate how the motive of hyper-Kähler varieties is controlled by weight-2 (or surface-like) motives via tensor operations. In the first part, we study the Voevodsky motive of singular moduli spaces of semistable sheaves on K3 and abelian surfaces as well as the Chow motive of their crepant resolutions, when they exist. We show that these motives are in the tensor subcategory generated by the motive of the surface, provided that a crepant resolution exists. This extends a recent result of Bülles to the O’Grady-10 situation. In the non-commutative setting, similar results are proved for the Chow motive of moduli spaces of (semi-)stable objects of the K3 category of a cubic fourfold. As a consequence, we provide abundant examples of hyper-Kähler varieties of O’Grady-10 deformation type satisfying the standard conjectures. In the second part, we study the André motive of projective hyper-Kähler varieties. We attach to any such variety its defect group, an algebraic group which acts on the cohomology and measures the difference between the full motive and its weight-2 part. When the second Betti number is not 3, we show that the defect group is a natural complement of the Mumford–Tate group inside the motivic Galois group, and that it is deformation invariant. We prove the triviality of this group for all known examples of projective hyper-Kähler varieties, so that in each case the full motive is controlled by its weight-2 part. As applications, we show that for any variety motivated by a product of known hyper-Kähler varieties, all Hodge and Tate classes are motivated, the motivated Mumford–Tate conjecture 7.3 holds, and the André motive is abelian. This last point completes a recent work of Soldatenkov and provides a different proof for some of his results.