575 results on '"Hermitian matrix"'
Search Results
2. Topological complex-energy braiding of non-Hermitian bands
- Author
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Avik Dutt, Shanhui Fan, Kai Wang, and Charles C. Wojcik
- Subjects
Physics ,Ring (mathematics) ,Multidisciplinary ,Hopf link ,Braid group ,Braid ,Topology ,Unknot ,Mathematics::Geometric Topology ,Hermitian matrix ,Topology (chemistry) ,Unlink - Abstract
Effects connected with the mathematical theory of knots1 emerge in many areas of science, from physics2,3 to biology4. Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems5, where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹2. We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension6,7. Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems. Experiments using two coupled optical ring resonators and based on the concept of synthetic dimension reveal non-Hermitian energy band structures exhibiting topologically non-trivial knots and links.
- Published
- 2021
3. Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions
- Author
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Bin Han and Ran Lu
- Subjects
FOS: Computer and information sciences ,Pure mathematics ,Information Theory (cs.IT) ,Computer Science - Information Theory ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,42C40, 42C15, 41A25, 41A35, 65T60 ,010103 numerical & computational mathematics ,Spectral theorem ,Trigonometric polynomial ,01 natural sciences ,Hermitian matrix ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Spline (mathematics) ,Wavelet ,Factorization ,FOS: Mathematics ,0101 mathematics ,Vector-valued function ,Mathematics - Abstract
Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.
- Published
- 2021
4. The metrics of Hermitian holomorphic vector bundles and the similarity of Cowen-Douglas operators
- Author
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Kui Ji and Shanshan Ji
- Subjects
Pure mathematics ,Multiplication operator ,Similarity (network science) ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Holomorphic function ,Vector bundle ,Hermitian matrix ,Mathematics - Abstract
In this note, we investigate the similarity of Cowen-Douglas operators with index one in terms of the ratio of metrics of the corresponding holomorphic bundles. For the case of index two, we give some sufficient and necessary conditions for the similarity of $$M_{z}^{*}\oplus M_{z}^{*}$$ by using the ratio of determinants of the metrics, where $$M_{z}$$ is the multiplication operator of weighted Bergman spaces.
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- 2021
5. Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes
- Author
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Masaaki Harada
- Subjects
FOS: Computer and information sciences ,Computer Science - Information Theory ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Binary number ,94B05 ,Cryptography ,GeneralLiterature_MISCELLANEOUS ,law.invention ,law ,FOS: Mathematics ,Code (cryptography) ,Mathematics - Combinatorics ,Mathematics ,Discrete mathematics ,Liquid-crystal display ,business.industry ,Information Theory (cs.IT) ,Applied Mathematics ,Construct (python library) ,Hermitian matrix ,Computer Science::Other ,Computer Science Applications ,Computer Science::Computer Vision and Pattern Recognition ,Combinatorics (math.CO) ,business ,Ternary operation - Abstract
We give two methods for constructing many linear complementary dual (LCD for short) codes from a given LCD code, by modifying some known methods for constructing self-dual codes. Using the methods, we construct binary LCD codes and quaternary Hermitian LCD codes, which improve the previously known lower bound on the largest minimum weights., Comment: 25 pages
- Published
- 2021
6. CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data
- Author
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Hongtu Zhu, Jianfeng Yao, Zhidong Bai, Tingting Zou, and Shurong Zheng
- Subjects
Statistics and Probability ,Combinatorics ,Physics ,Matrix (mathematics) ,Series (mathematics) ,Multivariate random variable ,Dimension (graph theory) ,Positive-definite matrix ,Statistics, Probability and Uncertainty ,Covariance ,Hermitian matrix ,Central limit theorem - Abstract
This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $${\mathbf {B}}_n=n^{-1}\sum _{j=1}^{n}{\mathbf {Q}}{\mathbf {x}}_j{\mathbf {x}}_j^{*}{\mathbf {Q}}^{*}$$ under the assumption that $$p/n\rightarrow y>0$$ , where $${\mathbf {Q}}$$ is a $$p\times k$$ nonrandom matrix and $$\{{\mathbf {x}}_j\}_{j=1}^n$$ is a sequence of independent k-dimensional random vector with independent entries. A key novelty here is that the dimension $$k\ge p$$ can be arbitrary, possibly infinity. This new model of sample covariance matrix $${\mathbf {B}}_n$$ covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $$k=p$$ and $${\mathbf {Q}}={\mathbf {T}}_n^{1/2}$$ for some positive definite Hermitian matrix $${\mathbf {T}}_n$$ . Also with $$k=\infty $$ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (Ann Probab 32(1):553–605, 2004). Applications of this new CLT are proposed for testing the AR(1) or AR(2) structure for a causal process. Our proposed tests are then used to analyze a large fMRI data set.
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- 2021
7. The $$\partial \overline \partial $$-Bochner Formulas for Holomorphic Mappings between Hermitian Manifolds and Their Applications
- Author
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Kai Tang
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Schwarz lemma ,General Mathematics ,010102 general mathematics ,Holomorphic function ,General Physics and Astronomy ,Type (model theory) ,Curvature ,Mathematics::Geometric Topology ,01 natural sciences ,Hermitian matrix ,010101 applied mathematics ,Hermitian manifold ,Mathematics::Differential Geometry ,0101 mathematics ,Degeneracy (mathematics) ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics - Abstract
In this paper, we derive some $$\partial \overline \partial $$ -Bochner formulas for holomorphic maps between Hermitian manifolds. As applications, we prove some Schwarz lemma type estimates, and some rigidity and degeneracy theorems. For instance, we show that there is no non-constant holomorphic map from a compact Hermitian manifold with positive (resp. non-negative) l-second Ricci curvature to a Hermitian manifold with non-positive (resp. negative) real bisectional curvature. These theorems generalize the results [5, 6] proved recently by L. Ni on Kahler manifolds to Hermitian manifolds. We also derive an integral inequality for a holomorphic map between Hermitian manifolds.
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- 2021
8. Universality for 1d Random Band Matrices
- Author
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Mariya Shcherbina and Tatyana Shcherbina
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Physics ,Gaussian ,010102 general mathematics ,Spectrum (functional analysis) ,Block (permutation group theory) ,Statistical and Nonlinear Physics ,Correlation function (quantum field theory) ,Lambda ,01 natural sciences ,Hermitian matrix ,Transfer matrix ,Combinatorics ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Limit (mathematics) ,0101 mathematics ,Mathematical Physics - Abstract
We consider 1d random Hermitian $$N\times N$$ block band matrices consisting of $$W\times W$$ random Gaussian blocks (parametrized by $$j,k \in \Lambda =[1,n]\cap \mathbb {Z}$$ , $$N=nW$$ ) with a fixed entry’s variance $$J_{jk}=W^{-1}(\delta _{j,k}+\beta \Delta _{j,k})$$ in each block. Considering the limit $$W, n\rightarrow \infty $$ , we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as $$W\gg \sqrt{N}$$ , is determined by the Wigner–Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in Shcherbina and Shcherbina (J Stat Phys 172:627–664, 2018).
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- 2021
9. On spectrally convex ordinary algebras
- Author
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A. Ouhmidou and A. El Kinani
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Modulo ,Spectrum (functional analysis) ,Regular polygon ,Jacobson radical ,Algebraic number ,Element (category theory) ,Hermitian matrix ,Convexity ,Mathematics - Abstract
We prove that if A is an ordinary and advertibly complete $$l.m.c.a.\ $$ each element of which has a convex spectrum, then A modulo its Jacobson radical is isomorphic to $${\mathbb {C}}$$ . We obtain the same conclusion for l.A.c.a and l.u.A-c.a. A purely algebraic version is also given. In the involutive case, the same conclusion, for an involutive ordinary Arens-Michael algebra, is obtained only under the convexity hypothesis on the spectrum of each normal element. Finally, if the algebra is additionally hermitian, it suffices to assume that the spectrum of each unitary element is convex.
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- 2021
10. A partial ordering approach to characterize properties of a pair of orthogonal projectors
- Author
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Götz Trenkler and Oskar Maria Baksalary
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Binary number ,Star (graph theory) ,Hermitian matrix ,law.invention ,Matrix (mathematics) ,Projector ,law ,Product (mathematics) ,Idempotence ,Partially ordered set ,Mathematics - Abstract
It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns, inter alia, the star, minus, diamond, sharp, core, and Lowner orderings. The situation changes, though, when instead of two orthogonal projectors, various functions of the pair (being either no longer Hermitian or no longer idempotent) are compared. The present paper provides an extensive investigation of the matrix partial orderings of functions of two orthogonal projectors. In addition to the six orderings mentioned above, three further binary functions are covered by the analysis, one of which is the space preordering. A particular attention is paid to the requirements that either product, sum, or difference of two orthogonal projectors is itself an orthogonal projector, i.e., inherits both features, Hermitianness and idempotency. Links of the results obtained with the research areas of applied origin (e.g., physics and statistics) are pointed out as well.
- Published
- 2021
11. Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces
- Author
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Andrea Iannuzzi and Laura Geatti
- Subjects
Hermitian symmetric space ,Pure mathematics ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,General Mathematics ,Hermitian symmetric spaces ,Stein domains ,Plurisubharmonic functions ,Hermitian matrix ,Settore MAT/03 ,Domain (ring theory) ,FOS: Mathematics ,32M15, 31C10, 32T05 ,Complex Variables (math.CV) ,Invariant (mathematics) ,Mathematics - Abstract
Let $$\,G/K\,$$ G / K be an irreducible non-compact Hermitian symmetric space and let $$\,D\,$$ D be a $$\,K$$ K -invariant domain in $$\,G/K$$ G / K . In this paper we characterize several classes of $$\,K$$ K -invariant plurisubharmonic functions on $$\,D\,$$ D in terms of their restrictions to a slice intersecting all $$\,K$$ K -orbits. As applications we show that $$\,K$$ K -invariant plurisubharmonic functions on $$\,D\,$$ D are necessarily continuous and we reproduce the classification of Stein $$\,K$$ K -invariant domains in $$\,G/K\,$$ G / K obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).
- Published
- 2021
12. On Ovoids of the Generalized Quadrangle $$H(3,q^2)$$
- Author
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Bart De Bruyn
- Subjects
SETS ,Generalized quadrangle ,Ovoid ,010102 general mathematics ,SPREADS ,0102 computer and information sciences ,Polynomial ,01 natural sciences ,Hermitian matrix ,Combinatorics ,Mathematics and Statistics ,Indicator ,010201 computation theory & mathematics ,Discrete Mathematics and Combinatorics ,(Hermitian) generalized quadrangle ,Locally Hermitian ,0101 mathematics ,set ,Mathematics - Abstract
We construct examples and families of locally Hermitian ovoids of the generalized quadrangle $$H(3,q^2)$$ . We also obtain a computer classification of all locally Hermitian ovoids of $$H(3,q^2)$$ for $$q \le 4$$ , and compare the obtained classification for $$q=3$$ with the classification of all ovoids of H(3, 9) which is also obtained by computer.
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- 2021
13. Representations of the necklace braid group $${{\mathcal {N}}{\mathcal {B}}}_n$$ of dimension 4 ($$n=2,3,4$$)
- Author
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Taher I. Mayassi and Mohammad N. Abdulrahim
- Subjects
Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Braid group ,Necklace ,Positive-definite matrix ,01 natural sciences ,Hermitian matrix ,010101 applied mathematics ,Combinatorics ,Tensor product ,Irreducible representation ,0101 mathematics ,Mathematics - Abstract
We consider the irreducible representations each of dimension 2 of the necklace braid group $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ). We then consider the tensor product of the representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) and determine necessary and sufficient condition under which the constructed representations are irreducible. Finally, we determine conditions under which the irreducible representations of $${\mathcal {N}}{\mathcal {B}}_n$$ N B n ($$n=2,3,4$$ n = 2 , 3 , 4 ) of degree 2 are unitary relative to a hermitian positive definite matrix.
- Published
- 2021
14. Stable Solutions to the Abelian Yang–Mills–Higgs Equations on $$S^2$$ and $$T^2$$
- Author
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Da Rong Cheng
- Subjects
Reduction (recursion theory) ,010102 general mathematics ,01 natural sciences ,Hermitian matrix ,Vortex ,High Energy Physics::Theory ,symbols.namesake ,Differential geometry ,Fourier analysis ,0103 physical sciences ,Line (geometry) ,symbols ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Abelian group ,Yang–Mills–Higgs equations ,Mathematics ,Mathematical physics - Abstract
We show under natural assumptions that stable solutions to the abelian Yang–Mills–Higgs equations on Hermitian line bundles over the round 2-sphere actually satisfy the vortex equations, which are a first-order reduction of the (second-order) abelian Yang–Mills–Higgs equations. We also obtain a similar result for stable solutions on a flat 2-torus. Our method of proof comes from the work of Bourguignon–Lawson (Commun Math Phys 79(2):189–230, 1981) concerning stable SU(2) Yang–Mills connections on compact homogeneous 4-manifolds.
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- 2021
15. Quaternionic contact 4n + 3-manifolds and their 4n-quotients
- Author
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Yoshinobu Kamishima
- Subjects
010102 general mathematics ,Zero (complex analysis) ,Structure (category theory) ,Lie group ,Space (mathematics) ,01 natural sciences ,Hermitian matrix ,Combinatorics ,Differential geometry ,0103 physical sciences ,Domain (ring theory) ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Analysis ,Quotient ,Mathematics - Abstract
We study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .
- Published
- 2021
16. Minimal linear codes from Hermitian varieties and quadrics
- Author
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Stefano Lia, Marco Timpanella, and Matteo Bonini
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Algebra and Number Theory ,Applied Mathematics ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Hermitian matrix ,Algebra ,Linear codes ,Quadrics ,010201 computation theory & mathematics ,Weight distribution ,Theory of computation ,0202 electrical engineering, electronic engineering, information engineering ,Minimal codes ,Secret sharing schemes ,Hermitian varieties ,Mathematics - Abstract
In this note we investigate minimal linear codes arising from Hermitian varieties and quadrics. We study their parameters and formulate some open problems about their weight distribution.
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- 2021
17. The J-equation and the supercritical deformed Hermitian–Yang–Mills equation
- Author
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Gao Chen
- Subjects
General Mathematics ,010102 general mathematics ,Yang–Mills existence and mass gap ,01 natural sciences ,Hermitian matrix ,Omega ,Supercritical fluid ,0103 physical sciences ,Metric (mathematics) ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Constant (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics ,Mathematical physics ,Bar (unit) ,Scalar curvature - Abstract
In this paper, we prove that for any Kahler metrics $$\omega _0$$ and $$\chi $$ on M, there exists a Kahler metric $$\omega _\varphi =\omega _0+\sqrt{-1}\partial {\bar{\partial }}\varphi >0$$ satisfying the J-equation $${\mathrm {tr}}_{\omega _\varphi }\chi =c$$ if and only if $$(M,[\omega _0],[\chi ])$$ is uniformly J-stable. As a corollary, we find a sufficient condition for the existence of constant scalar curvature Kahler metrics with $$c_1
- Published
- 2021
18. On Transitive Ovoids of Finite Hermitian Polar Spaces
- Author
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Tao Feng and Weicong Li
- Subjects
51E20, 05B25, 51A50 ,Combinatorics ,Computational Mathematics ,Transitive relation ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Polar ,Combinatorics (math.CO) ,Hermitian matrix ,Mathematics - Abstract
In this paper, we complete the classification of transitive ovoids of finite Hermitian polar spaces., Comment: 17 pages. To appear in Combinatorica
- Published
- 2021
19. The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization
- Author
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Miroslav S. Pranić and Stefano Pozza
- Subjects
Generalization ,Applied Mathematics ,Numerical analysis ,Lanczos algorithm ,010103 numerical & computational mathematics ,Positive-definite matrix ,01 natural sciences ,Hermitian matrix ,010101 applied mathematics ,symbols.namesake ,Orthogonal polynomials ,symbols ,Applied mathematics ,Gaussian quadrature ,0101 mathematics ,Realization (systems) ,Mathematics - Abstract
The concept of Gauss quadrature can be generalized to approximate linear functionals with complex moments. Following the existing literature, this survey will revisit such generalization. It is well known that the (classical) Gauss quadrature for positive definite linear functionals is connected with orthogonal polynomials, and with the (Hermitian) Lanczos algorithm. Analogously, the Gauss quadrature for linear functionals is connected with formal orthogonal polynomials, and with the non-Hermitian Lanczos algorithm with look-ahead strategy; moreover, it is related to the minimal partial realization problem. We will review these connections pointing out the relationships between several results established independently in related contexts. Original proofs of the Mismatch Theorem and of the Matching Moment Property are given by using the properties of formal orthogonal polynomials and the Gauss quadrature for linear functionals.
- Published
- 2021
20. Constructions and bounds on quaternary linear codes with Hermitian hull dimension one
- Author
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Somphong Jitman and Todsapol Mankean
- Subjects
General Mathematics ,Dimension (graph theory) ,Minimum weight ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Hermitian matrix ,Combinatorics ,010201 computation theory & mathematics ,Hull ,0202 electrical engineering, electronic engineering, information engineering ,Focus (optics) ,Mathematics - Abstract
Due to their practical applications, hulls of linear codes have been of interest and extensively studied. In this paper, we focus on constructions and bounds on quaternary linear codes with Hermitian hull dimension one. Optimal $$[n,2]_4$$ [ n , 2 ] 4 codes with Hermitian hull dimension one are constructed for all lengths $$n\ge 3$$ n ≥ 3 , such that $$n \equiv 1,2,4 \ (\mathrm{mod}\ 5)$$ n ≡ 1 , 2 , 4 ( mod 5 ) . For positive integers $$n \equiv 0,3 \ (\mathrm{mod}\ 5)$$ n ≡ 0 , 3 ( mod 5 ) , good lower and upper bounds on the minimum weight of quaternary $$[n,2]_4$$ [ n , 2 ] 4 codes with Hermitian hull dimension one are given.
- Published
- 2021
21. Algebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
- Author
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Alessio Savini
- Subjects
010102 general mathematics ,Lie group ,Geometric Topology (math.GT) ,Algebraic geometry ,01 natural sciences ,Centralizer and normalizer ,Hermitian matrix ,Combinatorics ,Mathematics - Geometric Topology ,Bounded function ,0103 physical sciences ,Complex geodesic ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Invariant (mathematics) ,Algebraic number ,Mathematics - Abstract
Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$. The Toledo remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle $\sigma$ is reductive and the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$ is compact. If additionally $\sigma$ admits a boundary map, then $H$ is of tube type and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. In the particular case $G=\text{PU}(n,1)$ maximality is sufficient to prove that $\sigma$ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles., Comment: 29 pages, more general definition of pullback added, explicit example of $G=\text{PU}(n,1)$. To appear on Geometriae Dedicata
- Published
- 2020
22. Analytically stable Higgs bundles on some non-Kähler manifolds
- Author
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Chuanjing Zhang and Xi Zhang
- Subjects
Physics ,Applied Mathematics ,High Energy Physics::Phenomenology ,010102 general mathematics ,01 natural sciences ,Hermitian matrix ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Higgs boson ,High Energy Physics::Experiment ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematical physics - Abstract
In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kahler, we solve the Hermitian–Einstein equation on analytically stable Higgs bundles.
- Published
- 2020
23. Eventual Positivity of Hermitian Algebraic Functions and Associated Integral Operators
- Author
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Colin Tan and Wing-Keung To
- Subjects
Polynomial ,Pure mathematics ,Mathematics::Complex Variables ,Sesquilinear form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,01 natural sciences ,Hermitian matrix ,010104 statistics & probability ,Operator (computer programming) ,Multiplication ,Algebraic function ,0101 mathematics ,Mathematics ,Bergman kernel - Abstract
Quillen proved that repeated multiplication of the standard sesquilinear form to a positive Hermitian bihomogeneous polynomial eventually results in a sum of Hermitian squares, which was the first Hermitian analogue of Hilbert’s seventeenth problem in the nondegenerate case. Later Catlin-D’Angelo generalized this positivstellensatz of Quillen to the case of Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds by proving the eventual positivity of an associated integral operator. The arguments of Catlin-D’Angelo involve subtle asymptotic estimates of the Bergman kernel. In this article, the authors give an elementary and geometric proof of the eventual positivity of this integral operator, thereby yielding another proof of the corresponding positivstellensatz.
- Published
- 2020
24. A Non-Hermitian Generalisation of the Marchenko–Pastur Distribution: From the Circular Law to Multi-criticality
- Author
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Sung-Soo Byun, Gernot Akemann, and Nam-Gyu Kang
- Subjects
Nuclear and High Energy Physics ,Pure mathematics ,FOS: Physical sciences ,Marchenko–Pastur distribution ,Type (model theory) ,01 natural sciences ,Matrix (mathematics) ,60B20, 33C45, 76D27 ,0103 physical sciences ,FOS: Mathematics ,Complex Variables (math.CV) ,0101 mathematics ,Mathematical Physics ,Eigenvalues and eigenvectors ,Physics ,Mathematics - Complex Variables ,Probability (math.PR) ,010102 general mathematics ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Hermitian matrix ,Circular law ,010307 mathematical physics ,Complex plane ,Random matrix ,Mathematics - Probability - Abstract
We consider the complex eigenvalues of a Wishart type random matrix model $X=X_1 X_2^*$, where two rectangular complex Ginibre matrices $X_{1,2}$ of size $N\times (N+\nu)$ are correlated through a non-Hermiticity parameter $\tau\in[0,1]$. For general $\nu=O(N)$ and $\tau$ we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation $X_1=X_2$ when $\tau=1$. The square root of the complex Wishart eigenvalues, corresponding to the non-zero complex eigenvalues of the Dirac matrix $\mathcal{D}=\begin{pmatrix} 0 & X_1 \\ X_2^* & 0 \end{pmatrix},$ are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value $\tau_c,$ where the interior of the spectrum splits into two connected components. At multi-criticality we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane., Comment: 30 pages, 4 figures, v2: references added, typos corrected
- Published
- 2020
25. The Generalized Modified Hermitian and Skew-Hermitian Splitting Method for the Generalized Lyapunov Equation
- Author
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Juan Zhang and Huihui Kang
- Subjects
Bilinear systems ,0209 industrial biotechnology ,business.industry ,Robotics ,02 engineering and technology ,Mechatronics ,Hermitian matrix ,Computer Science Applications ,Controllability ,symbols.namesake ,020901 industrial engineering & automation ,Skew-Hermitian matrix ,Control and Systems Engineering ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Convergence (routing) ,symbols ,Applied mathematics ,Lyapunov equation ,Artificial intelligence ,business ,Mathematics - Abstract
In this paper, we propose the generalized modified Hermitian and skew-Hermitian splitting (GMHSS) approach for computing the generalized Lyapunov equation. The GMHSS iteration is convergent to the unique solution of the generalized Lyapunov equation. Moreover, we discuss the convergence analysis of the GMHSS algorithm. Further, the inexact version of the GMHSS (IGMHSS) method is formulated to improve the GMHSS method. Finally, some numerical experiments are carried out to demonstrate the effectiveness and competitiveness of the derived methods
- Published
- 2020
26. A Natural Hermitian Line Bundle on the Moduli Space of Semistable Representations of a Quiver
- Author
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Pradeep Das
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Quiver ,Holomorphic function ,Type (model theory) ,Curvature ,01 natural sciences ,Hermitian matrix ,Moduli space ,Mathematics::Algebraic Geometry ,Line bundle ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Stratum ,Mathematics - Abstract
This paper describes the construction of a natural Hermitian holomorphic line bundle on the stratified moduli space of complex representations of a finite quiver, which are semistable with respect to a fixed rational weight and have a fixed type. It is shown that the curvature of this Hermitian line bundle on each stratum of the moduli space is essentially the Kahler form of that stratum.
- Published
- 2020
27. Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices
- Author
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Zlatko Drmač
- Subjects
Numerical Analysis ,Control and Optimization ,Applied Mathematics ,Computation ,MathematicsofComputing_NUMERICALANALYSIS ,Backward error ,Condition number ,Eigenvalues ,Hermitian matrices ,Jacobi method ,LAPACK ,Perturbation theory ,Rank revealing decomposition ,Singular value decomposition ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Vandermonde matrix ,Hermitian matrix ,010101 applied mathematics ,Singular value ,Factorization ,Modeling and Simulation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,65F15, 15A18, 15A42 ,0101 mathematics ,Frobenius matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. % Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the \textsf{LDU} factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.
- Published
- 2020
28. Hermitian curvature flow on complex locally homogeneous surfaces
- Author
-
Mattia Pujia and Francesco Pediconi
- Subjects
Mathematics - Differential Geometry ,Normalization (statistics) ,Physics ,Mathematics - Complex Variables ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Curvature ,01 natural sciences ,Hermitian matrix ,Singularity ,Differential Geometry (math.DG) ,Homogeneous ,0103 physical sciences ,FOS: Mathematics ,53C44 (Primary), 53C15, 53C30, 53C55 (Secondary) ,Mathematics::Differential Geometry ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Finite time ,Mathematics::Symplectic Geometry - Abstract
We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-K\"ahler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov-Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry., Comment: Some minor changes. A new appendix containing explicit tensor components. To appear on Ann. Mat. Pura Appl
- Published
- 2020
29. Codes with locality from cyclic extensions of Deligne–Lusztig curves
- Author
-
Gretchen L. Matthews and Fernando Piñero
- Subjects
Discrete mathematics ,Fiber (mathematics) ,business.industry ,Applied Mathematics ,Locality ,Cryptography ,Disjoint sets ,Hermitian matrix ,Computer Science Applications ,Set (abstract data type) ,Symbol (programming) ,Code (cryptography) ,business ,Mathematics - Abstract
Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverable codes allow for the recovery of a single symbol by accessing only a few others which form what is known as a recovery set. If every symbol has at least two disjoint recovery sets, the code is said to have availability. Three constructions are described, as each best fits a particular situation. The first employs the original construction of locally recoverable codes from curves by Tamo and Barg. The second yields codes with availability by appealing to the use of fiber products as described by Haymaker, Malmskog, and Matthews, while the third accomplishes availability by taking products of codes themselves. We see that cyclic extensions of the Deligne–Lusztig curves provide codes with smaller locality than those typically found in the literature.
- Published
- 2020
30. A Riccati-type algorithm for solving generalized Hermitian eigenvalue problems
- Author
-
Takafumi Miyata
- Subjects
Computer science ,Krylov subspace ,Hermitian matrix ,Theoretical Computer Science ,Hardware and Architecture ,Iterated function ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Convergence (routing) ,Riccati equation ,MATLAB ,computer ,Algorithm ,Software ,Subspace topology ,Eigenvalues and eigenvectors ,Information Systems ,computer.programming_language - Abstract
The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.
- Published
- 2020
31. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits
- Author
-
Stefan Imhof, Ronny Thomale, Martin Greiter, Tobias Kiessling, Alexander Szameit, Tobias Hofmann, Tobias Helbig, M. AbdelGhany, Ching Hua Lee, and Laurens W. Molenkamp
- Subjects
Physics ,General Physics and Astronomy ,Metamaterial ,Invariant (physics) ,01 natural sciences ,Hermitian matrix ,Open system (systems theory) ,010305 fluids & plasmas ,Classical mechanics ,Reciprocity (electromagnetism) ,0103 physical sciences ,Skin effect ,010306 general physics ,Eigenvalues and eigenvectors ,Electronic circuit - Abstract
The study of the laws of nature has traditionally been pursued in the limit of isolated systems, where energy is conserved. This is not always a valid approximation, however, as the inclusion of features such as gain and loss, or periodic driving, qualitatively amends these laws. A contemporary frontier of metamaterial research is the challenge open systems pose to the characterization of topological matter1,2. Here, one of the most relied upon principles is the bulk–boundary correspondence (BBC), which intimately relates the surface states to the topological classification of the bulk3,4. The presence of gain and loss, in combination with the violation of reciprocity, has been predicted to affect this principle dramatically5,6. Here, we report the experimental observation of BBC violation in a non-reciprocal topolectric circuit7, which is also referred to as the non-Hermitian skin effect. The circuit admittance spectrum exhibits an unprecedented sensitivity to the presence of a boundary, displaying an extensive admittance mode localization despite a translationally invariant bulk. Intriguingly, we measure a non-local voltage response due to broken BBC. Depending on the a.c. current feed frequency, the voltage signal accumulates at the left or right boundary, and increases as a function of nodal distance to the current feed. Boundary-localized bulk eigenstates given by the non-Hermitian skin effect are observed in a non-reciprocal topological circuit. A fundamental revision of the bulk–boundary correspondence in an open system is required to understand the underlying physics.
- Published
- 2020
32. Non-Hermitian linear response theory
- Author
-
Hui Zhai, Yu Chen, Xin Chen, and Lei Pan
- Subjects
Physics ,Strongly Correlated Electrons (cond-mat.str-el) ,Thermodynamic equilibrium ,Operator (physics) ,FOS: Physical sciences ,General Physics and Astronomy ,01 natural sciences ,Hermitian matrix ,010305 fluids & plasmas ,Condensed Matter - Strongly Correlated Electrons ,Classical mechanics ,Correlation function ,Quantum Gases (cond-mat.quant-gas) ,Luttinger liquid ,0103 physical sciences ,Dissipative system ,Quantum system ,Condensed Matter - Quantum Gases ,010306 general physics ,Critical exponent - Abstract
Linear response theory lies at the heart of quantum many-body physics because it builds up connections between the dynamical response to an external probe and correlation functions at equilibrium. Here we consider the dynamical response of a Hermitian system to a non-Hermitian probe, and we develop a non-Hermitian linear response theory that can also relate this dynamical response to equilibrium properties. As an application of our theory, we consider the real-time dynamics of momentum distribution induced by one-body and two-body dissipations. We find that, for many cases, the dynamics of momentum occupation and the width of momentum distribution obey the same universal function, governed by the single-particle spectral function. We also find that, for critical state with no well-defined quasi-particles, the dynamics are slower than normal state and our theory provides a model independent way to extract the critical exponent. We apply our results to analyze recent experiment on the Bose-Hubbard model and find surprising good agreement between theory and experiment. We also propose to further verify our theory by carrying out a similar experiment on a one-dimensional Luttinger liquid., Comment: 6+3 pages, 3 figures; v2: supplementary material added
- Published
- 2020
33. Hermitian manifolds with quasi-negative curvature
- Author
-
Man-Chun Lee
- Subjects
Mathematics - Differential Geometry ,General Mathematics ,010102 general mathematics ,Curvature ,01 natural sciences ,Hermitian matrix ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,Line bundle ,Flow (mathematics) ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,Point (geometry) ,Mathematics::Differential Geometry ,010307 mathematical physics ,Negative curvature ,0101 mathematics ,Ricci curvature ,Mathematics ,Mathematical physics - Abstract
In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasi-negative Chern-Ricci curvature is ample., final version to appear in Math. Ann. arXiv admin note: text overlap with arXiv:1812.04610
- Published
- 2020
34. Characteristic Polynomials for Random Band Matrices Near the Threshold
- Author
-
Tatyana Shcherbina
- Subjects
Physics ,Discrete mathematics ,Gaussian ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Covariance ,01 natural sciences ,Transfer matrix ,Hermitian matrix ,010305 fluids & plasmas ,symbols.namesake ,Matrix (mathematics) ,0103 physical sciences ,symbols ,010306 general physics ,Mathematical Physics - Abstract
The paper continues previous works which study the behavior of second correlation function of characteristic polynomials of the special case of $n\times n$ one-dimensional Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix $J=(-W^2\triangle+1)^{-1}$. Applying the transfer matrix approach, we study the case when the bandwidth $W$ is proportional to the threshold $\sqrt{n}$, 21p
- Published
- 2020
35. Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
- Author
-
László Erdős, Dominik Schröder, and Torben Krüger
- Subjects
Independent and identically distributed random variables ,Cusp (singularity) ,Physics ,60B20, 15B52 ,Spectral radius ,Probability (math.PR) ,010102 general mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,16. Peace & justice ,01 natural sciences ,Hermitian matrix ,Article ,Universality (dynamical systems) ,010104 statistics & probability ,Law ,FOS: Mathematics ,Gravitational singularity ,0101 mathematics ,Random matrix ,Mathematics - Probability ,Mathematical Physics ,Eigenvalues and eigenvectors - Abstract
For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also used in the companion paper [arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type matrices is proven., Comment: 58 pages, 2 figures. Updated introduction and references
- Published
- 2020
36. Passive Localization of Mixed Near-Field and Far-Field Sources Without Eigendecomposition via Uniform Circular Array
- Author
-
Xiaolong Su, Bo Peng, Tianpeng Liu, Zhen Liu, Xin Chen, and Xiang Li
- Subjects
0209 industrial biotechnology ,Noise power ,Covariance matrix ,Computer science ,Applied Mathematics ,Oblique projection ,02 engineering and technology ,Covariance ,Hermitian matrix ,Circular buffer ,Noise ,020901 industrial engineering & automation ,Signal Processing ,Algorithm ,Eigendecomposition of a matrix - Abstract
In this paper, we employ the geometry of uniform circular array to achieve classification and localization of mixed near-field and far-field sources. Considering that the eigendecomposition of the covariance matrix requires high computational cost, we develop the propagator method to obtain the noise subspace and reduce complexity. Firstly, since the direction parameters of far-field sources at centrosymmetry sensors hold a conjugate structure while the covariance matrix of near-field sources holds a Hermitian structure, we exploit the covariance differencing approach to extract the pure near-field sources from mixed sources. Then, we improve the ESPRIT-like method and one-dimensional MUSIC method to determine the 2-D direction-of-arrival (DOA) and range of near-field sources, respectively. Finally, by calculating the noise power of mixed sources, we utilize the oblique projection approach to extract the pure far-field sources and exploit the 2-D MUSIC method to determine the 2-D DOA of far-field sources. Simulations demonstrate that the proposed algorithm can avoid the pseudo-peaks in the 2-D DOA spatial spectrum of far-field sources and provide the satisfactory performance of mixed source localization.
- Published
- 2020
37. A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles
- Author
-
Takahiro Inayama and Genki Hosono
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,Generalization ,General Mathematics ,010102 general mathematics ,Vector bundle ,Extension (predicate logic) ,01 natural sciences ,Hermitian matrix ,Domain (mathematical analysis) ,0103 physical sciences ,Converse ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We study conditions of Hormander’s L2-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of the Hormander-type condition, we show a converse of Hormander L2-estimate under some regularity assumptions on an n-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson (1998). We also de_ne new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.
- Published
- 2020
38. Non-Hermitian bulk–boundary correspondence in quantum dynamics
- Author
-
Kunkun Wang, Zhong Wang, Wei Yi, Lei Xiao, Tianshu Deng, Gaoyan Zhu, and Peng Xue
- Subjects
Physics ,Photon ,Quantum dynamics ,General Physics and Astronomy ,Boundary (topology) ,01 natural sciences ,Hermitian matrix ,Measure (mathematics) ,010305 fluids & plasmas ,Brillouin zone ,Theoretical physics ,Scheme (mathematics) ,0103 physical sciences ,010306 general physics ,Eigenvalues and eigenvectors - Abstract
Bulk–boundary correspondence, a guiding principle in topological matter, relates robust edge states to bulk topological invariants. Its validity, however, has so far been established only in closed systems. Recent theoretical studies indicate that this principle requires fundamental revisions for a wide range of open systems with effective non-Hermitian Hamiltonians. Therein, the intriguing localization of nominal bulk states at boundaries, known as the non-Hermitian skin effect, suggests a non-Bloch band theory in which non-Bloch topological invariants are defined in generalized Brillouin zones, leading to a general bulk–boundary correspondence beyond the conventional framework. Here, we experimentally observe this fundamental non-Hermitian bulk–boundary correspondence in discrete-time non-unitary quantum-walk dynamics of single photons. We demonstrate pronounced photon localizations near boundaries even in the absence of topological edge states, thus confirming the non-Hermitian skin effect. Facilitated by our experimental scheme of edge-state reconstruction, we directly measure topological edge states, which are in excellent agreement with the non-Bloch topological invariants. Our work unequivocally establishes the non-Hermitian bulk–boundary correspondence as a general principle underlying non-Hermitian topological systems and paves the way for a complete understanding of topological matter in open systems. Measurements of non-Hermitian photon dynamics show boundary-localized bulk eigenstates given by the non-Hermitian skin effect. A fundamental revision of the bulk–boundary correspondence in open systems is required to understand the underlying physics.
- Published
- 2020
39. Orthogonality Spaces Arising from Infinite-Dimensional Complex Hilbert Spaces
- Author
-
Thomas Vetterlein
- Subjects
Pure mathematics ,Physics and Astronomy (miscellaneous) ,Binary relation ,General Mathematics ,010102 general mathematics ,Hilbert space ,02 engineering and technology ,Space (mathematics) ,Automorphism ,01 natural sciences ,Linear subspace ,Hermitian matrix ,symbols.namesake ,Orthogonality ,Reflexive relation ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,020201 artificial intelligence & image processing ,0101 mathematics ,Mathematics - Abstract
The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.
- Published
- 2020
40. Lie-algebraic curvature conditions preserved by the Hermitian curvature flow
- Author
-
Yury Ustinovskiy
- Subjects
Mathematics - Differential Geometry ,Mathematics - Complex Variables ,General Mathematics ,010102 general mathematics ,Convex set ,Holomorphic function ,Curvature ,01 natural sciences ,Hermitian matrix ,Canonical bundle ,Combinatorics ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Hermitian manifold ,Mathematics::Differential Geometry ,010307 mathematical physics ,Complex Variables (math.CV) ,0101 mathematics ,Invariant (mathematics) ,Scalar curvature ,Mathematics - Abstract
The purpose of this paper is to prove that the Hermitian Curvature Flow (HCF) on an Hermitian manifold $(M,g,J)$ preserves many natural curvature positivity conditions. Following Wilking, for an $Ad\,{GL(T^{1,0}M)}$-invariant subset $S\subset End(T^{1,0}M)$ and a ncie function $F\colon End(T^{1,0}M)\to\mathbb R$ we construct a convex set of curvature operators $C(S,F)$, which is invariant under the HCF. Varying $S$ and $F$, we prove that the HCF preserves Griffiths positivity, Dual-Nakano positivity, positivity of holomorphic orthogonal bisectional curvature, lower bounds on the second scalar curvature. As an application, we prove that periodic solutions to the HCF can exist only on manifolds $M$ with the trivial canonical bundle on the universal cover $\widetilde{M}$., 20 pages, 1 figure
- Published
- 2020
41. A factorisation theorem for the coinvariant algebra of a unitary reflection group
- Author
-
Gustav I. Lehrer
- Subjects
Finite group ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Hermitian matrix ,Combinatorics ,Reflection (mathematics) ,Factorization ,0103 physical sciences ,010307 mathematical physics ,Isomorphism ,0101 mathematics ,Reflection group ,Mathematics - Abstract
We prove the following theorem. Let G be a finite group generated by unitary reflections in a complex Hermitian space $$V={\mathbb {C}}^\ell $$ and let $$G'$$ be any reflection subgroup of G. Let $${\mathcal {H}}={\mathcal {H}}(G)$$ be the space of G-harmonic polynomials on V. There is a degree preserving isomorphism $$\mu :{\mathcal {H}}(G')\otimes {\mathcal {H}}(G)^{G'}\overset{\sim }{{\longrightarrow \;}}{\mathcal {H}}(G)$$ of graded $${\mathcal {N}}$$-modules, where $${\mathcal {N}}:=N_{{\text {GL}}(V)}(G)\cap N_{{\text {GL}}(V)}(G')$$ and $${\mathcal {H}}(G)^{G'}$$ is the space of $$G'$$-fixed points of $${\mathcal {H}}(G)$$. This generalises a result of Douglass and Dyer for parabolic subgroups of real reflection groups. An application is given to counting rational conjugates of reductive groups over $${\mathbb {F}}_q$$.
- Published
- 2020
42. Cryptosystem design based on Hermitian curves for IoT security
- Author
-
Omar A. Alzubi, Mohammad Alsayyed, Osama M. Dorgham, and Jafar A. Alzubi
- Subjects
020203 distributed computing ,business.industry ,Computer science ,Plaintext ,Cryptography ,02 engineering and technology ,Encryption ,Hermitian matrix ,Theoretical Computer Science ,Algebra ,Elliptic curve ,Algebraic geometric ,Hardware and Architecture ,Elliptic curve cryptosystem ,McEliece cryptosystem ,0202 electrical engineering, electronic engineering, information engineering ,Key (cryptography) ,Astrophysics::Solar and Stellar Astrophysics ,Cryptosystem ,business ,Error detection and correction ,Block size ,Software ,Computer Science::Cryptography and Security ,Information Systems - Abstract
The ultimate goal of modern cryptography is to protect the information resource and make it absolutely unbreakable and beyond compromise. However, throughout the history of cryptography, thousands of cryptosystems emerged and believed to be invincible and yet attackers were able to break and compromise their security. The main objective of this paper is to design a robust cryptosystem that will be suitable to be implemented in Internet of Things. The proposed cryptosystem is based on algebraic geometric curves, more specifically on Hermitian curves. The new cryptosystem design is called Hermitian-based cryptosystem (HBC). During the development of the HBC design, Kerckhoffs’s desideratum was the main guidance principle, which has been satisfied by choosing the Hermitian curves as the core of the proposed design. The proposed HBC inherits all the advantageous characteristics of Hermitian curve which are large number of points that satisfy the curve and high genus curves. The aforementioned characteristics play a crucial role in generating a large size encryption key for HBC and determine the block size of plaintext. Due to the fact that HBC used algebraic geometric codes over Hermitian curve, it has the ability to perform error correction in addition to data encryption. The error correction is another advantage of HBC compared with many existing cryptosystems such as McEliece cryptosystem. The number of errors that can be corrected by HBC is larger (high data rate) than other algebraic geometric codes such as elliptic and hyperelliptic curves. It also uses non-binary representation which increases its attack resistance. In this paper, the proposed HBC has been mathematically compared with elliptic curve cryptosystem. The results show that HBC has many advantages over the elliptic curves in terms of number of points and genus of the curve.
- Published
- 2020
43. On the $$\mathrm{PSU}(4,2)$$-Invariant Vertex-Transitive Strongly Regular (216, 40, 4, 8) Graph
- Author
-
Francesco Pavese, Andrea Švob, and Dean Crnković
- Subjects
Strongly regular graph ,Transitive relation ,0211 other engineering and technologies ,021107 urban & regional planning ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Hermitian matrix ,Graph ,Theoretical Computer Science ,Vertex (geometry) ,Combinatorics ,Hermitian surface ,Generalized quadrangle ,Projective geometry ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Invariant (mathematics) ,Mathematics - Abstract
In 2018 the first, Rukavina and the third author constructed with the aid of a computer the first example of a strongly regular graph $$\Gamma$$ with parameters (216, 40, 4, 8) and proved that it is the unique $$\mathrm{PSU}(4,2)$$-invariant vertex-transitive graph on 216 vertices. In this paper, using the geometry of the Hermitian surface of $$\mathrm{PG}(3,4)$$, we provide a computer-free proof of the existence of the graph $$\Gamma$$. The maximal cliques of $$\Gamma$$ are also determined.
- Published
- 2020
44. Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics
- Author
-
Lingling Chen, Haojie Chen, and Xiaolan Nie
- Subjects
Mathematics - Differential Geometry ,Pointwise ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Holomorphic function ,Conformal map ,Kähler manifold ,Curvature ,01 natural sciences ,Hermitian matrix ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,Sectional curvature ,Complex Variables (math.CV) ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Mathematical physics - Abstract
We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is K\"{a}hler. We also give examples of complete non-K\"{a}hler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-K\"{a}hler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor., Comment: 19 pages
- Published
- 2019
45. A simple finite element for the geometrically exact analysis of Bernoulli–Euler rods
- Author
-
Paulo de Mattos Pimenta, Cátia da Costa e Silva, Jörg Schröder, and Sascha Maassen
- Subjects
Applied Mathematics ,Mechanical Engineering ,Mathematical analysis ,Computational Mechanics ,Lagrange polynomial ,Torsion (mechanics) ,Ocean Engineering ,02 engineering and technology ,Rodrigues' rotation formula ,Rigid body ,01 natural sciences ,Hermitian matrix ,Finite element method ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Bernoulli's principle ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Computational Theory and Mathematics ,symbols ,Euler's formula ,0101 mathematics ,Bauwissenschaften ,Mathematics - Abstract
This work develops a simple finite element for the geometrically exact analysis of Bernoulli–Euler rods. Transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod. The cross-section undergoes a rigid body motion. A rotation tensor with the Rodrigues formula is used to describe the rotation, which makes the updating of the rotational variables very simple. A formula for the Rodrigues parameters in function of the displacements derivative and the torsion angle is for the first time settled down. The consistent connection between elements is thoroughly discussed, and an appropriate approach is developed. Cubic Hermitian interpolation for the displacements together with linear Lagrange interpolation for the torsion incremental angle were employed within the usual Finite Element Method, leading to adequate C1 continuity. A set of numerical benchmark examples illustrates the usefulness of the formulation and numerical implementation.
- Published
- 2019
46. On VT-harmonic maps
- Author
-
Jürgen Jost, Qun Chen, and Hongbing Qiu
- Subjects
Dirichlet problem ,Pure mathematics ,Computer Science::Information Retrieval ,010102 general mathematics ,Harmonic map ,Riemannian geometry ,01 natural sciences ,Hermitian matrix ,Connection (mathematics) ,symbols.namesake ,Maximum principle ,Differential geometry ,0103 physical sciences ,symbols ,010307 mathematical physics ,Geometry and Topology ,Affine transformation ,0101 mathematics ,Analysis ,Mathematics - Abstract
VT-harmonic maps generalize the standard harmonic maps, with respect to the structure of both domain and target. These can be manifolds with natural connections other than the Levi-Civita connection of Riemannian geometry, like Hermitian, affine or Weyl manifolds. The standard harmonic map semilinear elliptic system is augmented by a term coming from a vector field V on the domain and another term arising from a 2-tensor T on the target. In fact, this geometric structure then also includes other geometrically defined maps, for instance magnetic harmonic maps. In this paper, we treat VT-harmonic maps and their parabolic analogues with PDE tools. We establish a Jäger–Kaul type maximum principle for these maps. Using this maximum principle, we prove an existence theorem for the Dirichlet problem for VT-harmonic maps. As applications, we obtain results on Weyl/affine/Hermitian harmonic maps between Weyl/affine/Hermitian manifolds, as well as on magnetic harmonic maps from two-dimensional domains. We also derive gradient estimates and obtain existence results for such maps from noncompact complete manifolds.
- Published
- 2019
47. Quotients of the Hermitian curve from subgroups of $$\mathrm{PGU}(3,q)$$ without fixed points or triangles
- Author
-
Giovanni Zini and Maria Montanucci
- Subjects
Automorphism group ,Hermitian curve ,Maximal curves ,Quotient curves ,Unitary groups ,Algebra and Number Theory ,010102 general mathematics ,0102 computer and information sciences ,Fixed point ,01 natural sciences ,Hermitian matrix ,Combinatorics ,010201 computation theory & mathematics ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Invariant (mathematics) ,Quotient ,Mathematics - Abstract
In this paper, we deal with the problem of classifying the genera of quotient curves $${\mathcal {H}}_q/G$$ , where $${\mathcal {H}}_q$$ is the $${\mathbb {F}}_{q^2}$$ -maximal Hermitian curve and G is an automorphism group of $${\mathcal {H}}_q$$ . The groups G considered in the literature fix either a point or a triangle in the plane $$\mathrm{PG}(2,q^6)$$ . In this paper, we give a complete list of genera of quotients $${\mathcal {H}}_q/G$$ , when $$G \le \mathrm{Aut}({\mathcal {H}}_q) \cong \mathrm{PGU}(3,q)$$ does not leave invariant any point or triangle in the plane. Also, the classification of subgroups G of $$\mathrm{PGU}(3,q)$$ satisfying this property is given up to isomorphism.
- Published
- 2019
48. Biorthogonal quantum criticality in non-Hermitian many-body systems
- Author
-
Jia-Chen Tang, Gaoyong Sun, and Su-Peng Kou
- Subjects
Physics ,Strongly Correlated Electrons (cond-mat.str-el) ,Statistical Mechanics (cond-mat.stat-mech) ,Physics and Astronomy (miscellaneous) ,FOS: Physical sciences ,Hermitian matrix ,Many body ,Condensed Matter - Other Condensed Matter ,Condensed Matter - Strongly Correlated Electrons ,Criticality ,Biorthogonal system ,Quantum ,Condensed Matter - Statistical Mechanics ,Other Condensed Matter (cond-mat.other) ,Mathematical physics - Abstract
We develop the perturbation theory of the fidelity susceptibility in biorthogonal bases for arbitrary interacting non-Hermitian many-body systems with real eigenvalues. The quantum criticality in the non-Hermitian transverse field Ising chain is investigated by the second derivative of ground-state energy and the ground-state fidelity susceptibility. We show that the system undergoes a second-order phase transition with the Ising universal class by numerically computing the critical points and the critical exponents from the finite-size scaling theory. Interestingly, our results indicate that the biorthogonal quantum phase transitions are described by the biorthogonal fidelity susceptibility instead of the conventional fidelity susceptibility., 7 pages, 4 figures
- Published
- 2021
49. An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations
- Author
-
Manuel De la Sen, Reena Jain, and Hemant Kumar Nashine
- Subjects
Applied Mathematics ,Order (ring theory) ,Fixed point ,Positive-definite matrix ,Type (model theory) ,w-distance ,Hermitian matrix ,Combinatorics ,Nonlinear system ,Metric space ,Matrix (mathematics) ,Positive definite matrix ,Convergence analysis ,QA1-939 ,Discrete Mathematics and Combinatorics ,Mathematics ,Analysis ,Implicit relation ,Nonlinear matrix equation - Abstract
We propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form $\mathcal{U}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G}\mathcal{(U)}\mathcal{A}_{i}$ U = Q + ∑ i = 1 k A i ∗ G ( U ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.
- Published
- 2021
50. On decompositions and approximations of conjugate partial-symmetric tensors
- Author
-
Taoran Fu, Zhening Li, and Bo Jiang
- Subjects
Computational Mathematics ,Pure mathematics ,Algebra and Number Theory ,Optimization problem ,Generalization ,Numerical analysis ,Convex optimization ,Theory of computation ,Computer Science::Programming Languages ,Tensor ,Quadratic programming ,Hermitian matrix ,Mathematics - Abstract
Hermitian matrices have played an important role in matrix theory and complex quadratic optimization. The high-order generalization of Hermitian matrices, conjugate partial-symmetric (CPS) tensors, have shown growing interest recently in tensor theory and computation, particularly in application-driven complex polynomial optimization problems. In this paper, we study CPS tensors with a focus on ranks, computing rank-one decompositions and approximations, as well as their applications. We prove constructively that any CPS tensor can be decomposed into a sum of rank-one CPS tensors, which provides an explicit method to compute such rank-one decompositions. Three types of ranks for CPS tensors are defined and shown to be different in general. This leads to the invalidity of the conjugate version of Comon’s conjecture. We then study rank-one approximations and matricizations of CPS tensors. By carefully unfolding CPS tensors to Hermitian matrices, rank-one equivalence can be preserved. This enables us to develop new convex optimization models and algorithms to compute best rank-one approximations of CPS tensors. Numerical experiments from data sets in radar wave form design, elasticity tensor, and quantum entanglement are performed to justify the capability of our methods.
- Published
- 2021
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