4,995 results on '"Hermitian matrix"'
Search Results
2. The Hermitian Solution to a New System of Commutative Quaternion Matrix Equations
- Author
-
Yue Zhang, Qing-Wen Wang, and Lv-Ming Xie
- Subjects
commutative quaternion algebra ,matrix equations ,Hermitian matrix ,least squares solution ,Mathematics ,QA1-939 - Abstract
This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.
- Published
- 2024
- Full Text
- View/download PDF
3. Some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices
- Author
-
Yan Hong, Dongkyu Lim, and Feng Qi
- Subjects
Generalized eigenvalue ,Hermitian matrix ,Inequality ,Perturbation problem ,Mathematics ,QA1-939 - Abstract
Abstract In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature.
- Published
- 2018
- Full Text
- View/download PDF
4. Hill representations for ∗-linear matrix maps
- Author
-
A. van der Merwe and S. ter Horst
- Subjects
Combinatorics ,Linear map ,Matrix (mathematics) ,General Mathematics ,Nonnegative matrix ,Linear matrix ,Hermitian matrix ,Mathematics - Abstract
In the paper (Hill, 1973) from 1973 R.D. Hill studied linear matrix maps L : ℂ q × q → ℂ n × n which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., L ( V ∗ ) = L ( V ) ∗ , via representations of the form L ( V ) = ∑ k , l = 1 m H k l A l V A k ∗ , V ∈ ℂ q × q , for matrices A 1 , … , A m ∈ ℂ n × q and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and associated matrix reorderings. In this paper we expand the study of such representations, referred to as Hill representations here, in various directions. In particular, we describe which matrices A 1 , … , A m can appear in Hill representations (provided the number m is minimal) and determine the associated Hill matrix H = H k l explicitly. Also, we describe how different Hill representations of L (again with m minimal) are related and investigate further the implication of ∗ -linearity on the linear map L .
- Published
- 2022
5. Dirac series for E6(−14)
- Author
-
Lin-Gen Ding, Chao-Ping Dong, and Haian He
- Subjects
Pure mathematics ,Algebra and Number Theory ,Series (mathematics) ,Infinitesimal ,Simple Lie group ,Dirac (software) ,Type (model theory) ,Hermitian matrix ,Unitary state ,Cohomology ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Mathematics - Abstract
Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered Dirac series of $E_{6(-14)}$ is realized as a composition factor of certain $A_{\mathfrak{q}}(\lambda)$ module. Along the way, we have also obtained all the fully supported irreducible unitary representations of $E_{6(-14)}$ with integral infinitesimal characters., Comment: 32 pages, some strings are folded
- Published
- 2022
6. Central Limit Theorem for Linear Eigenvalue Statistics of <scp>Non‐Hermitian</scp> Random Matrices
- Author
-
Dominik Schröder, László Erdős, and Giorgio Cipolloni
- Subjects
Independent and identically distributed random variables ,Applied Mathematics ,General Mathematics ,Gaussian ,Hermitian matrix ,symbols.namesake ,Matrix (mathematics) ,Distribution (mathematics) ,Statistics ,symbols ,Random matrix ,Eigenvalues and eigenvectors ,Central limit theorem ,Mathematics - Abstract
We consider large non-Hermitian random matrices $X$ with complex, independent, identically distributed centred entries and show that the linear statistics of their eigenvalues are asymptotically Gaussian for test functions having $2+\epsilon$ derivatives. Previously this result was known only for a few special cases; either the test functions were required to be analytic [Rider, Silverstein 2006], or the distribution of the matrix elements needed to be Gaussian [Rider, Virag 2007], or at least match the Gaussian up to the first four moments [Tao, Vu 2016; Kopel 2015]. We find the exact dependence of the limiting variance on the fourth cumulant that was not known before. The proof relies on two novel ingredients: (i) a local law for a product of two resolvents of the Hermitisation of $X$ with different spectral parameters and (ii) a coupling of several weakly dependent Dyson Brownian Motions. These methods are also the key inputs for our analogous results on the linear eigenvalue statistics of real matrices $X$ that are presented in the companion paper [Cipolloni, Erdős, Schroder 2019].
- Published
- 2021
7. Orbit spaces for torus actions on Hessenberg varieties
- Author
-
Vladislav Vladimirovich Cherepanov
- Subjects
Pure mathematics ,Algebra and Number Theory ,Isospectral ,Torus ,Orbit (control theory) ,Fixed point ,Space (mathematics) ,Hermitian matrix ,Manifold ,Hessenberg variety ,Mathematics - Abstract
In this paper we study effective actions of the compact torus on smooth compact manifolds of even dimension with isolated fixed points. It is proved that under certain conditions on the weight vectors of the tangent representation, the orbit space of such an action is a manifold with corners. In the case of Hamiltonian actions, the orbit space is homotopy equivalent to , the complement to the union of disjoint open subsets of the -sphere. The results obtained are applied to regular Hessenberg varieties and isospectral manifolds of Hermitian matrices of step type. Bibliography: 23 titles.
- Published
- 2021
8. Rank and Kernel of Additive Generalized Hadamard Codes
- Author
-
Steven T. Dougherty, Josep Rifà, and Mercè Villanueva
- Subjects
Trace (linear algebra) ,Kernel (set theory) ,Rank (linear algebra) ,Linear space ,Dimension (graph theory) ,Generalised Hadamard matrix ,Library and Information Sciences ,Rank ,Upper and lower bounds ,Hermitian matrix ,Computer Science Applications ,Combinatorics ,Kernel ,Generalised Hadamard code ,Product (mathematics) ,Additive code ,Nonlinear code ,Information Systems ,Mathematics - Abstract
L'article pertany al grup de recerca Combinatorics, Coding and Security Group (CCSG) A subset of a vector space Fn q is additive if it is a linear space over the field Fp, where q = pe, p prime, and e > 1. Bounds on the rank and dimension of the kernel of additive generalised Hadamard (additive GH) codes are established. For specific ranks and dimensions of the kernel within these bounds, additive GH codes are constructed. Moreover, for the case e = 2, it is shown that the given bounds are tight and it is possible to construct an additive GH code for all allowable ranks and dimensions of the kernel between these bounds. Finally, we also prove that these codes are self-orthogonal with respect to the trace Hermitian inner product, and generate pure quantum codes.
- Published
- 2021
9. Galois self-orthogonal constacyclic codes over finite fields
- Author
-
Hongwei Liu and Yuqing Fu
- Subjects
Combinatorics ,Finite field ,Integer ,Applied Mathematics ,Product (mathematics) ,Euclidean geometry ,Hermitian matrix ,Prime (order theory) ,Computer Science Applications ,Vector space ,Mathematics - Abstract
Let $${\mathbb {F}}_{q}$$ be a finite field with $$q=p^{e}$$ elements, where p is a prime and e is a positive integer. In 2017, Fan and Zhang introduced $$\ell $$ -Galois inner products on the n-dimensional vector space $${\mathbb {F}}_{q}^{n}$$ for $$0\le \ell
- Published
- 2021
10. Integral binary Hamiltonian forms and their waterworlds
- Author
-
Jouni Parkkonen, Frédéric Paulin, University of Jyväskylä (JYU), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and PICS 6950 CNRS
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Binary number ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,waterworld ,differentiaaligeometria ,maximal order ,hyperbolic 5-space ,0103 physical sciences ,0101 mathematics ,Algebraic number ,reduction theory ,Mathematics ,lukuteoria ,Mathematics - Number Theory ,Quaternion algebra ,010102 general mathematics ,Hamilton-Bianchi group ,ryhmäteoria ,Order (ring theory) ,Mathematics::Geometric Topology ,Hermitian matrix ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,Binary quadratic form ,010307 mathematical physics ,Geometry and Topology ,rational quaternion algebra ,Mathematics - Group Theory ,binary Hamiltonian form ,Hamiltonian (control theory) - Abstract
We give a graphical theory of integral indefinite binary Hamiltonian forms $f$ analogous to the one by Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order $\mathcal O$ in a definite quaternion algebra over $\mathbb Q$, we define the waterworld of $f$, analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of $f$ on $\mathcal O\times\mathcal O$. We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), and the $\operatorname{SL}_2(\mathcal O)$-equivariant Ford-Voronoi cellulation of the real hyperbolic $5$-space., Comment: Revised version, 40 pages
- Published
- 2021
11. On Hulls of Some Primitive BCH Codes and Self-Orthogonal Codes
- Author
-
Chunyu Gan, Sihem Mesnager, Chengju Li, and Haifeng Qian
- Subjects
Dimension (graph theory) ,Order (ring theory) ,Library and Information Sciences ,Type (model theory) ,Hermitian matrix ,Upper and lower bounds ,Linear code ,Computer Science Applications ,Combinatorics ,Finite field ,Mathematics::Metric Geometry ,BCH code ,Information Systems ,Mathematics - Abstract
Self-orthogonal codes are an important type of linear codes due to their wide applications in communication and cryptography. The Euclidean (or Hermitian) hull of a linear code is defined to be the intersection of the code and its Euclidean (or Hermitian) dual. It is clear that the hull is self-orthogonal. The main goal of this paper is to obtain self-orthogonal codes by investigating the hulls. Let $\mathcal {C}_{(r,r^{m}-1,\delta,b)}$ be the primitive BCH code over $\mathbb {F}_{r}$ of length $r^{m}-1$ with designed distance $\delta $ , where $\mathbb {F}_{r}$ is the finite field of order $r$ . In this paper, we will present Euclidean (or Hermitian) self-orthogonal codes and determine their parameters by investigating the Euclidean (or Hermitian) hulls of some primitive BCH codes. Several sufficient and necessary conditions for primitive BCH codes with large Hermitian hulls are developed by presenting lower and upper bounds on their designed distances. Furthermore, some Hermitian self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are also investigated. In addition, we determine the dimensions of the code $\mathcal {C}_{(r,r^{2}-1,\delta,1)}$ and its hull in both Hermitian and Euclidean cases for $2 \le \delta \le r^{2}-1$ . We also present two sufficient and necessary conditions on designed distances such that the hull has the largest dimension.
- Published
- 2021
12. CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS
- Author
-
Daniel Oeh
- Subjects
Algebra and Number Theory ,Endomorphism ,Direct sum ,010102 general mathematics ,Mathematics - Operator Algebras ,010103 numerical & computational mathematics ,Type (model theory) ,Automorphism ,01 natural sciences ,Hermitian matrix ,Linear subspace ,Combinatorics ,Lie algebra ,FOS: Mathematics ,Primary 22E45, Secondary 81R05, 81T05 ,Geometry and Topology ,0101 mathematics ,Invariant (mathematics) ,Representation Theory (math.RT) ,ddc:510 ,Mathematics::Representation Theory ,Operator Algebras (math.OA) ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $(\mathfrak{g},\tau)$ be a real simple symmetric Lie algebra and let $W \subset \mathfrak{g}$ be an invariant closed convex cone which is pointed and generating with $\tau(W) = -W$. For elements $h \in \mathfrak{g}$ with $\tau(h) = h$, we classify the Lie algebras $\mathfrak{g}(W,\tau,h)$ which are generated by the closed convex cones \[C_{\pm}(W,\tau,h) := (\pm W) \cap \mathfrak{g}_{\pm 1}^{-\tau}(h),\] where $\mathfrak{g}^{-\tau}_{\pm 1}(h) := \{x \in \mathfrak{g} : \tau(x) = -x, [h,x] = \pm x\}$. These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $\mathfrak{g}(W,\tau,h)$ is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms $\tau$ of $\mathfrak{g}$ with $\tau(W) = -W$ a list of possible subalgebras $\mathfrak{g}(W,\tau,h)$ up to isomorphy., Comment: The title of the paper has been changed; the introduction has been rewritten; some of the proofs have been shortened
- Published
- 2022
13. On stability of the fibres of Hopf surfaces as harmonic maps and minimal surfaces
- Author
-
Liding Huang and Jingyi Chen
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Minimal surface ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Hopf surface ,Harmonic map ,Harmonic (mathematics) ,Torus ,Hermitian matrix ,Cohomology ,Differential Geometry (math.DG) ,FOS: Mathematics ,Mathematics - Abstract
We construct a family of Hermitian metrics on the Hopf surface $ \mathbb{S}^3\times \mathbb{S}^1$, whose fundamental classes represent distinct cohomology classes in the Aeppli cohomology group. These metrics are locally conformally K\"ahler. Among the toric fibres of $\pi:\mathbb{S}^{3} \times \mathbb{S}^1\to\mathbb{C} P^1$ two of them are stable minimal surfaces and each of the two has a neighbourhood so that fibres therein are given by stable harmonic maps from 2-torus and outside, far away from the two tori, there are unstable harmonic ones that are also unstable minimal surfaces., Comment: We generalize Theorem 1.2 to $\mathbb{S}^{2n-1}\times\mathbb{S}^1$ in section 4
- Published
- 2021
14. Kudla–Rapoport cycles and derivatives of local densities
- Author
-
Wei Zhang and Chao Li
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,Dimension (graph theory) ,Hermitian matrix ,Unitary state ,Identity (mathematics) ,symbols.namesake ,Mathematics::Algebraic Geometry ,Intersection ,Eisenstein series ,symbols ,Mathematics::Representation Theory ,Fourier series ,Mathematics - Abstract
We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.
- Published
- 2021
15. Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions
- Author
-
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
16. Hermitian splines with links in the form of the sum of the polynomial and exponents with an odd number of parameters
- Author
-
Yaropolk Pizyur and Bohdan Hnativ
- Subjects
Combinatorics ,Polynomial ,Hermitian matrix ,Mathematics - Abstract
Conditions for the existence of a unique approximation of functions by Hermitian splines with a link in the form of a sum of a polynomial and an exponent with five parameters are established. Formulas for the parameters of the links of these Hermitian splines are derived. A formula for calculating the error and an expression for the kernel of the error of the balance approximation of functions by Hermitian splines with a link in the form of a sum of a polynomial and an exponent are given. Results of approximations are given.
- Published
- 2021
17. Classification of topological phases in one dimensional interacting non-Hermitian systems and emergent unitarity
- Author
-
Wenjie Xi, Zheng-Cheng Gu, Zhi-Hao Zhang, and Wei-Qiang Chen
- Subjects
Partition function (quantum field theory) ,Multidisciplinary ,Strongly Correlated Electrons (cond-mat.str-el) ,Unitarity ,Generalization ,FOS: Physical sciences ,Fixed point ,010502 geochemistry & geophysics ,Topology ,01 natural sciences ,Hermitian matrix ,Condensed Matter - Strongly Correlated Electrons ,Geometric phase ,Energy spectrum ,Mathematics::Differential Geometry ,Well-defined ,0105 earth and related environmental sciences ,Mathematics - Abstract
Topological phases in non-Hermitian systems have become fascinating subjects recently. In this paper, we attempt to classify topological phases in 1D interacting non-Hermitian systems. We begin with the non-Hermitian generalization of the Su-Schrieffer-Heeger (SSH) model and discuss its many-body topological Berry phase, which is well defined for all interacting quasi-Hermitian systems (non-Hermitian systems that have real energy spectrum). We then demonstrate that the classification of topological phases for quasi-Hermitian systems is exactly the same as their Hermitian counterparts. Finally, we construct the fixed point partition function for generic 1D interacting non-Hermitian local systems and find that the fixed point partition function still has a one-to-one correspondence to their Hermitian counterparts. Thus, we conclude that the classification of topological phases for generic 1D interacting non-Hermitian systems is still exactly the same as Hermitian systems., 15 pages, 5 figures
- Published
- 2021
18. Estimates for a function on almost Hermitian manifolds
- Author
-
Masaya Kawamura
- Subjects
chern connection ,Pure mathematics ,Almost complex manifold ,almost hermitian metric ,Function (mathematics) ,almost complex manifold ,Hermitian matrix ,53c15 ,53c55 (secondary) ,QA1-939 ,Geometry and Topology ,Mathematics::Differential Geometry ,32q60 (primary) ,Mathematics - Abstract
We study some estimates for a real-valued smooth function φ on almost Hermitian manifolds. In the present paper, we show that ∂∂∂̄ φ and ∂̄∂∂̄ φ can be estimated by the gradient of the function φ.
- Published
- 2021
19. The metrics of Hermitian holomorphic vector bundles and the similarity of Cowen-Douglas operators
- Author
-
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.
- Published
- 2021
20. Elegant Iterative Methods for Solving a Nonlinear Matrix Equation X-A* eX A=I
- Author
-
Chacha Stephen Chacha
- Subjects
Iterative method ,Fixed-point iteration ,Convergence (routing) ,Applied mathematics ,Nonlinear matrix equation ,Standard algorithms ,Positive-definite matrix ,Approximate solution ,Hermitian matrix ,Mathematics - Abstract
The nonlinear matrix equation was solved by Gao (2016) via standard fixed point method. In this paper, three more elegant iterative methods are proposed to find the approximate solution of the nonlinear matrix equation namely: Newton’s method; modified fixed point method and a combination of Newton’s method and fixed point method. The convergence of Newton’s method and modified fixed point method are derived. Comparative numerical experimental results indicate that the new developed algorithms have both less computational time and good convergence properties when compared to their respective standard algorithms. Keywords: Hermitian positive definite solution; nonlinear matrix equation; modified fixed point method; iterative method
- Published
- 2021
21. Construction of binary LCD codes, ternary LCD codes and quaternary Hermitian LCD codes
- Author
-
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
22. Hermitian Toeplitz determinants for the class S of univalent functions
- Author
-
Nikola Tuneski and Milutin Obradović
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Hermitian matrix ,Toeplitz matrix ,Third order ,0101 mathematics ,Convex function ,Unit (ring theory) ,Mathematics - Abstract
Introducing a new method, we give sharp estimates of the Hermitian Toeplitz determinants of third order for the class $\mathcal{S}$ of functions univalent in the unit disc. The new approach is also illustrated on some subclasses of the class $\mathcal{S}$.
- Published
- 2021
23. The homotopy limit problem and the cellular Picard group of Hermitian K-theory
- Author
-
Drew Heard
- Subjects
Pure mathematics ,Homotopy ,Picard group ,K-Theory and Homology (math.KT) ,Assessment and Diagnosis ,K-theory ,Hermitian matrix ,Base (group theory) ,Tensor (intrinsic definition) ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Geometry and Topology ,Unit (ring theory) ,Analysis ,Descent (mathematics) ,Mathematics - Abstract
We use descent theoretic methods to solve the homotopy limit problem for Hermitian $K$-theory over quasi-compact and quasi-separated base schemes. As another application of these descent theoretic methods, we compute the cellular Picard group of 2-complete Hermitian $K$-theory over $\mathop{Spec}(\mathbb{C})$, showing that the only invertible cellular spectra are suspensions of the tensor unit., Comment: v4: results made unconditional, assumptions on base scheme weakened. Version to appear in Annals of K-theory
- Published
- 2021
24. On Stopping Sets of AG Codes Over Certain Curves With Separated Variables
- Author
-
Guilherme Tizziotti and Wanderson Tenório
- Subjects
Reed–Muller code ,020206 networking & telecommunications ,02 engineering and technology ,Coding theory ,Library and Information Sciences ,Information theory ,Hermitian matrix ,Upper and lower bounds ,Computer Science Applications ,Combinatorics ,Elliptic curve ,Family of curves ,0202 electrical engineering, electronic engineering, information engineering ,Information Systems ,Mathematics - Abstract
In this paper we study stopping sets of AG codes over a family of curves, denoted by $\mathcal {X}_{f,g}$ , that includes several important curves with applications in coding theory. We present results concerning stopping sets of one-point and $m$ -point codes over $\mathcal {X}_{f,g}$ , generalizing some results for Hermitian codes presented by Anderson and Matthews.
- Published
- 2021
25. Structured strong linearizations of structured rational matrices
- Author
-
Ranjan Das and Rafikul Alam
- Subjects
Pure mathematics ,Algebra and Number Theory ,010103 numerical & computational mathematics ,01 natural sciences ,Hermitian matrix ,Rational matrices ,Matrix polynomial ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,0101 mathematics ,System matrix ,Mathematics::Symplectic Geometry ,Eigenvalues and eigenvectors ,Hamiltonian (control theory) ,Mathematics - Abstract
Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational...
- Published
- 2021
26. Quasi-Hemi-Slant Conformal Submersions from Almost Hermitian Manifolds
- Author
-
Şener Yanan
- Subjects
Conformal submersion,quasi-hemi-slant conformal submersion ,Matematik ,Pure mathematics ,General Earth and Planetary Sciences ,Conformal map ,Mathematics::Differential Geometry ,Mathematics::Geometric Topology ,Mathematics::Symplectic Geometry ,Hermitian matrix ,Mathematics ,General Environmental Science - Abstract
In this study, we introduce some geometric properties of quasi-hemi-slant conformal submersions from an almost Hermitian manifold to a Riemannian manifold. We give an explicit example for this type submersions and obtain integrability conditions for certain distributions. Lastly, we search totally geodesicity on base manifold of the map.
- Published
- 2021
27. The $$\partial \overline \partial $$-Bochner Formulas for Holomorphic Mappings between Hermitian Manifolds and Their Applications
- Author
-
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.
- Published
- 2021
28. The Hermitian Kirchhoff Index and Robustness of Mixed Graph
- Author
-
Qianru Zhou, Min Li, Shuming Zhou, Gaolin Chen, and Wei Lin
- Subjects
Algebraic connectivity ,Social graph ,Article Subject ,Resistance distance ,Computer science ,General Mathematics ,General Engineering ,Mixed graph ,020206 networking & telecommunications ,02 engineering and technology ,Engineering (General). Civil engineering (General) ,Hermitian matrix ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,Graph (abstract data type) ,Applied mathematics ,020201 artificial intelligence & image processing ,TA1-2040 ,Laplacian matrix ,Mathematics ,Connectivity - Abstract
Large-scale social graph data poses significant challenges for social analytic tools to monitor and analyze social networks. The information-theoretic distance measure, namely, resistance distance, is a vital parameter for ranking influential nodes or community detection. The superiority of resistance distance and Kirchhoff index is that it can reflect the global properties of the graph fairly, and they are widely used in assessment of graph connectivity and robustness. There are various measures of network criticality which have been investigated for underlying networks, while little is known about the corresponding metrics for mixed networks. In this paper, we propose the positive walk algorithm to construct the Hermitian matrix for the mixed graph and then introduce the Hermitian resistance matrix and the Hermitian Kirchhoff index which are based on the eigenvalues and eigenvectors of the Hermitian Laplacian matrix. Meanwhile, we also propose a modified algorithm, the directed traversal algorithm, to select the edges whose removal will maximize the Hermitian Kirchhoff index in the general mixed graph. Finally, we compare the results with the algebraic connectivity to verify the superiority of the proposed strategy.
- Published
- 2021
29. On spectrally convex ordinary algebras
- Author
-
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.
- Published
- 2021
30. An Efficient Iterative Approach to Large Sparse Nonlinear Systems with Non-Hermitian Jacobian Matrices
- Author
-
Min-Hong Chen
- Subjects
Nonlinear system ,symbols.namesake ,Applied Mathematics ,Jacobian matrix and determinant ,symbols ,Applied mathematics ,Hermitian matrix ,Mathematics - Published
- 2021
31. A partial ordering approach to characterize properties of a pair of orthogonal projectors
- Author
-
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
32. On Quasi-Hemi-Slant Riemannian Maps
- Author
-
Rajendra Prasad, Sushil Kumar, Aysel Turgut Vanli, and Sumeet Kumar
- Subjects
Riemannian maps,Semi-invariant maps,Quasi bi-slant maps,Quasi hemi-slant ,Pure mathematics ,Multidisciplinary ,Integrable system ,020209 energy ,Mühendislik ,General Engineering ,02 engineering and technology ,Hermitian matrix ,Submersion (mathematics) ,Engineering ,0202 electrical engineering, electronic engineering, information engineering ,Totally geodesic ,020201 artificial intelligence & image processing ,Mathematics::Differential Geometry ,Mathematics - Abstract
In this paper, quasi-hemi-slant Riemannian maps from almost Hermitian manifolds onto Riemannian manifolds are introduced. The geometry of leaves of distributions that are involved in the definition of the submersion and quasi-hemi-slant Riemannian maps are studied. In addition, conditions for such distributions to be integrable and totally geodesic are obtained. Also, a necessary and sufficient condition for proper quasi-hemi-slant Riemannian maps to be totally geodesic is given. Moreover, structured concrete examples for this notion are given.
- Published
- 2021
33. Unified relational-theoretic approach in metric-like spaces with an application
- Author
-
Reena Jain, Hemant Kumar Nashine, Choonkil Park, and Jung Rye Lee
- Subjects
Physics ,positive definite matrix ,nonlinear matrix equation ,General Mathematics ,Trace norm ,Positive-definite matrix ,Type (model theory) ,Fixed point ,Hermitian matrix ,Surface plot ,Combinatorics ,fixed point ,Error analysis ,relational metric space ,Metric (mathematics) ,metric-like space ,QA1-939 ,Mathematics - Abstract
In this paper, we introduce a modified implicit relation and obtain some new fixed point results for $ \sigma $-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation $ \mathcal{U} = \mathcal{D} + \sum_{i = 1}^{m}\mathcal{A}_{i}^{*}\mathcal{G} \mathcal{(U)}\mathcal{A}_{i} $, where $ \mathcal{D} $ is an $ n\times n $ Hermitian positive definite matrix, $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $, $\dots$, $ \mathcal{A}_{m} $ are $ n \times n $ matrices, and $ \mathcal{G} $ is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot.
- Published
- 2021
34. Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces
- Author
-
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
35. Virasoro Versus Superintegrability. Gaussian Hermitian Model
- Author
-
A. Morozov, V. Mishnyakov, A. D. Mironov, and R. Rashkov
- Subjects
High Energy Physics - Theory ,Pure mathematics ,Physics and Astronomy (miscellaneous) ,Gaussian ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Space (mathematics) ,01 natural sciences ,Hermitian matrix ,010305 fluids & plasmas ,Matrix (mathematics) ,symbols.namesake ,High Energy Physics - Theory (hep-th) ,Simple (abstract algebra) ,0103 physical sciences ,symbols ,010306 general physics ,Mathematical Physics ,Mathematics - Abstract
Relation between the Virasoro constraints and KP integrability (determinant formulas) for matrix models is a lasting mystery. We elaborate on the claim that the situation is improved when integrability is enhanced to super-integrability, i.e. to explicit formulas for Gaussian averages of characters. In this case, the Virasoro constraints are equivalent to simple recursive formulas, which have appropriate combinations of characters as their solutions. Moreover, one can easily separate dependence on the size of matrix, and deduce superintegrability from the Virasoro constraints. We describe one of the ways to do so for the Gaussian Hermitian matrix model. The result is a spectacularly elegant reformulation of Virasoro constraints as identities for the Schur functions evaluated at appropriate loci in the space of time-variables., Comment: 6 pages
- Published
- 2021
36. Nadel–Nakano vanishing theorems of vector bundles with singular Hermitian metrics
- Author
-
Masataka Iwai
- Subjects
Pure mathematics ,Rank (linear algebra) ,Mathematics::Complex Variables ,Generalization ,010102 general mathematics ,Holomorphic function ,Vector bundle ,General Medicine ,01 natural sciences ,Hermitian matrix ,Mathematics::Algebraic Geometry ,Square-integrable function ,0103 physical sciences ,Sheaf ,Hermitian manifold ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generalization of Griffiths' vanishing theorem. Finally, we show a generalization of Ohsawa's vanishing theorem.
- Published
- 2021
37. On Ovoids of the Generalized Quadrangle $$H(3,q^2)$$
- Author
-
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.
- Published
- 2021
38. Definite determinantal representations via orthostochastic matrices
- Author
-
Papri Dey
- Subjects
Semidefinite programming ,Difficult problem ,Polynomial ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,010102 general mathematics ,Scalar (mathematics) ,Regular polygon ,010103 numerical & computational mathematics ,01 natural sciences ,Hermitian matrix ,Computational Mathematics ,Bivariate polynomials ,Optimization and Control (math.OC) ,15A75, 15B10, 15B51, 90C22 ,FOS: Mathematics ,Computer Science::Symbolic Computation ,0101 mathematics ,Mathematics - Optimization and Control ,Monic polynomial ,Mathematics - Abstract
Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate polynomials are definite determinantals. Indeed, in general, it is a difficult problem to decide whether a given polynomial is definite determinantal, and if it is, it is of paramount interest to determine a definite determinantal representation of that polynomial. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree d. In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points.
- Published
- 2021
39. Representations of the necklace braid group $${{\mathcal {N}}{\mathcal {B}}}_n$$ of dimension 4 ($$n=2,3,4$$)
- Author
-
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
40. Spectral theory for self-adjoint quadratic eigenvalue problems - a review
- Author
-
Ion Zaballa and Peter Lancaster
- Subjects
Pure mathematics ,Algebra and Number Theory ,Spectral theory ,010103 numerical & computational mathematics ,01 natural sciences ,Hermitian matrix ,Section (fiber bundle) ,Matrix (mathematics) ,Canonical form ,0101 mathematics ,Algebraic number ,Self-adjoint operator ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).
- Published
- 2021
41. The continuity equation of the Gauduchon metrics
- Author
-
Tao Zheng
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Conjecture ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Hermitian matrix ,Differential Geometry (math.DG) ,Continuity equation ,0103 physical sciences ,FOS: Mathematics ,Interval (graph theory) ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,53C55, 35J60, 32W20, 58J05 ,Mathematics - Abstract
We study the continuity equation of the Gauduchon metrics and establish its interval of maximal existence, which extends the continuity equation of the K��hler metrics introduced by La Nave \& Tian for and of the Hermitian metrics introduced by Sherman \& Weinkove. Our method is based on the solution to the Gauduchon conjecture by Sz��kelyhidi, Tosatti \& Weinkove., 15pages
- Published
- 2021
42. Stable Solutions to the Abelian Yang–Mills–Higgs Equations on $$S^2$$ and $$T^2$$
- Author
-
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.
- Published
- 2021
43. Quaternionic contact 4n + 3-manifolds and their 4n-quotients
- Author
-
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
44. Gain-line graphs via G-phases and group representations
- Author
-
Daniele D'Angeli, Matteo Cavaleri, and Alfredo Donno
- Subjects
Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,Gain graph ,Group algebra ,Hermitian matrix ,Group representation ,law.invention ,Matrix (mathematics) ,law ,Line graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,Abelian group ,Equivalence class ,Mathematics - Abstract
Let $G$ be an arbitrary group. We define a gain-line graph for a gain graph $(\Gamma,\psi)$ through the choice of an incidence $G$-phase matrix inducing $\psi$. We prove that the switching equivalence class of the gain function on the line graph $L(\Gamma)$ does not change if one chooses a different $G$-phase inducing $\psi$ or a different representative of the switching equivalence class of $\psi$. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of $G$ on the set $\mathcal H_\Gamma$ of $G$-phases of $\Gamma$ allows us to characterize gain functions on $\Gamma$, gain functions on $L(\Gamma)$, their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph., Comment: 28 pages, 6 figures, 1 table
- Published
- 2021
45. Minimal linear codes from Hermitian varieties and quadrics
- Author
-
Stefano Lia, Marco Timpanella, and Matteo Bonini
- Subjects
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.
- Published
- 2021
46. Deformed Hermitian Yang–Mills connections, extended gauge group and scalar curvature
- Author
-
Jacopo Stoppa and Enrico Schlitzer
- Subjects
Mathematics - Differential Geometry ,Kaehler metrics, scalar curvature, moment maps ,General Mathematics ,Holomorphic function ,Yang–Mills existence and mass gap ,01 natural sciences ,Mathematics - Algebraic Geometry ,Complex geometry ,Gauge group ,0103 physical sciences ,FOS: Mathematics ,scalar curvature ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Moment map ,Mathematical physics ,Mathematics ,010102 general mathematics ,Hermitian matrix ,Kaehler metrics ,moment maps ,Differential Geometry (math.DG) ,Settore MAT/03 - Geometria ,010307 mathematical physics ,Scalar curvature - Abstract
The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background K\"ahler metric, to be constant. In this paper we introduce and study dHYM equations with variable K\"ahler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and mirror to Thomas' moment map for special Lagrangians, to the Donaldson-Fujiki picture of scalar curvature as a moment map. As a consequence one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the K\"ahler-Yang-Mills system of \'Alvarez-C\'onsul, Garcia-Fernandez and Garc\'ia-Prada, and the coupled K\"ahler-Einstein equations of Hultgren-Witt Nystr\"om. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, following ideas of Feng and Sz\'ekelyhidi., Comment: 42 pages, some corrections to Section 4
- Published
- 2021
47. The J-equation and the supercritical deformed Hermitian–Yang–Mills equation
- Author
-
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
48. A proof of Sørensen’s conjecture on Hermitian surfaces
- Author
-
Mrinmoy Datta, Masaaki Homma, and Peter Beelen
- Subjects
Combinatorics ,Conjecture ,Intersection of surfaces ,Applied Mathematics ,General Mathematics ,Rational points ,Hermitian matrix ,Hermitian surfaces ,Mathematics - Abstract
In this article we prove a conjecture formulated by A. B. Sørensen in 1991 on the maximal number of F q 2 \mathbb {F}_{q^2} -rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree d ≤ q . d \le q.
- Published
- 2021
49. On Transitive Ovoids of Finite Hermitian Polar Spaces
- Author
-
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
50. The continuity equation of almost Hermitian metrics
- Author
-
Chang Li and Tao Zheng
- Subjects
Pure mathematics ,Homogeneous manifolds ,Applied Mathematics ,010102 general mathematics ,01 natural sciences ,Hermitian matrix ,010101 applied mathematics ,Continuity equation ,Interval (graph theory) ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
We extend the continuity equation of the Kahler metrics introduced by La Nave & Tian and the Hermitian metrics introduced by Sherman & Weinkove to the almost Hermitian metrics, and establish its interval of maximal existence. As an example, we study the continuity equation on the (locally) homogeneous manifolds in more detail.
- Published
- 2021
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.