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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 .

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes DM-CC, a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.

We introduce the concept of Almost-Companion Matrix (ACM) by relaxing the non-derogatory property of the standard Companion Matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and Unitary ACMs starting from appropriate third degree polynomials, with implications for their use in physical-mathematical problems such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degree., Comment: 26 pages

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

Unlike the original Cloude-van Zyl decomposition of reflection symmetric polSAR data, a recently suggested version of the decomposition for full/quad pol data relies on the Cloude-Pottier mean alpha angle (ᾱ) to characterize the scattering mechanism. ᾱ can be calculated from the eigenvectors of the coherency matrix. By means of the eigenvector-eigenvalue identity (EEI) we can avoid the calculation of the eigenvectors. The EEI finds ᾱ by means of eigenvalues of the 3×3 coherency matrix and its 2×2 minor(s) only and is well suited for fast array based computer implementation. In this paper with focus on computational aspects we demonstrate fast EEI based determination of ᾱ on X-band F-SAR image data over Vejers, Denmark, including a detailed example of calculations and computer code.

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

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].

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.

Strictly unstable linear systems under additive and nonvanishing stochastic noise with unbounded supports are known to be impossible to stabilize by using deterministically constrained control inputs. In this paper, similar impossibility results are obtained for the scenarios where the control input is probabilistically constrained and the support of the noise distribution is not necessarily unbounded. In particular, control policies that have bounded time-averaged second moments are considered. It is shown that for such control policies, there are critical average moment bounds, below which second moment stabilization of a linear stochastic system is not possible, and moreover, second moment of the state diverges regardless of the choice of control policy and the initial state distribution. Nonnegative-definite Hermitian matrices are exploited to extract sufficient instability conditions that can be assessed by using the eigenstructure of the system matrix and the distribution of the noise. The results indicate that in certain networked control system settings with noise, designing stabilizing constrained controllers is an impossible task, if the probability of successful transmissions of control commands over the network is known to be too small in average.

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.

Intense light traps and binds small particles, offering unique control to the microscopic world. With incoming illumination and radiative losses, optical forces are inherently nonconservative, thus non-Hermitian. Contrary to conventional systems, the operator governing time evolution is real and asymmetric (i.e., non-Hermitian), which inevitably yield complex eigenvalues when driven beyond the exceptional points, where light pumps in energy that eventually “melts” the light-bound structures. Surprisingly, unstable complex eigenvalues are prevalent for clusters with ~10 or more particles, and in the many-particle limit, their presence is inevitable. As such, optical forces alone fail to bind a large cluster. Our conclusion does not contradict with the observation of large optically-bound cluster in a fluid, where the ambient damping can take away the excess energy and restore the stability. The non-Hermitian theory overturns the understanding of optical trapping and binding, and unveils the critical role played by non-Hermiticity and exceptional points, paving the way for large-scale manipulation., Non-conservativeness plays a mysterious role in optical trapping. Applying the non-Hermitian theory, the authors showed that the existence of exceptional points drives the Lorentz force to lose its ability to bind clusters of ~10 or more microparticles, unless remedied by dissipative forces.

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

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

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.

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.

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

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.

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.

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.

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

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.

We introduce the exceptional topological insulator (ETI), a non-Hermitian topological state of matter that features exotic non-Hermitian surface states which can only exist within the three-dimensional topological bulk embedding. We show how this phase can evolve from a Weyl semimetal or Hermitian three-dimensional topological insulator close to criticality when quasiparticles acquire a finite lifetime. The ETI does not require any symmetry to be stabilized. It is characterized by a bulk energy point gap, and exhibits robust surface states that cover the bulk gap as a single sheet of complex eigenvalues or with a single exceptional point. The ETI can be induced universally in gapless solid-state systems, thereby setting a paradigm for non-Hermitian topological matter., 6+ pages, 3 figures; 21 pages Supplemental Material