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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.

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.

Resonance via Pade (RVP) is an efficient method for calculating autoionization resonance states. It is based on the stabilization technique in which the basis set is scaled. The scaling can be uniform (i.e., all basis functions are scaled) or partial. Herein, we compare the two RVP scaling schemes for calculating an autoionization eigenvalue; moreover, the effect of freezing the core electrons is intertwined within this comparison. In order to study the different behavior of the RVP schemes, we associate each RVP scaling scheme with a complex contour of integration. Similarities between RVP and other non-Hermitian methods emerge from the generated contours, which suggest that RVP introduces similar outgoing boundary conditions as the complex scaling (CS), complex basis function (CBF), and reflection-free complex absorbing potential (RF-CAP) methods. A uniform-RVP contour, unlike a partial one, immediately penetrates the complex plane and influences the interaction region. Hence, uniform scaling within RVP destroys the description of the core electrons, as well as the description of the reference state, and yields less reliable results than partial scaling. The 1s22p3s1P autoionization state of Be, at the equation-of-motion coupled-cluster level, is used as our case study model.

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.

Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a noisy environment may destroy these phases. We investigate the behavior of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian is $\mathcal{PT}$-symmetric. Although the topological nature persists in two-dimensional quantum walks as well, the $\mathcal{PT}$-symmetry has no role to play there. Furthermore, we observe the noise-induced topological phase transition in two-dimensional quantum walks., Comment: 15 pages, 8 figures. Comments or suggestions are most welcome

A finite computational process using arithmetic operations only is called a rational algorithm. Presently, no rational algorithm for checking the congruence of arbitrary complex matrices A and B is known. The situation can be different if both A and B belong to a special matrix class. For instance, there exist rational algorithms for the cases where both matrices are Hermitian, unitary, or accretive. This paper proposes a rational algorithm for checking whether two involutive matrices A and B are congruent.

The class of close-to-convex functions are univalent and so its subclasses. For normalised analytic functions defined on the unit disk, four subclasses of close-to-convex functions are considered and Hermitian-Toeplitz determinants for these classes are investigated. All the results presented in this article are sharp.

For n ≥ 3, we construct a class $$\left\{{{W_{n,{\pi _1},{\pi _2}}}} \right\}$$ of n2 × n2 hermitian matrices by the permutation pairs and show that, for a pair {π1, π2} of permutations on (1, 2, …, n), $${{W_{n,{\pi _1},{\pi _2}}}}$$ is an entanglement witness of the n ⊗ n system if {π1, π2} has the property (C). Recall that a pair {π1, π2} of permutations of (1, 2, …, n) has the property (C) if, for each i, one can obtain a permutation of (1, …, i − 1, i + 1, …, n) from (π1 (1), …, π1 (i − 1), π1(i + 1), …, π1(n)) and (π2(1), …, π2(i − 1), π2(i + 1), …, π2(n)). We further prove that $${{W_{n,{\pi _1},{\pi _2}}}}$$ is not comparable with Wn,π, which is the entanglement witness constructed from a single permutation π; $${{W_{n,{\pi _1},{\pi _2}}}}$$ is decomposable if π1π2 = id or π 1 2 = π 2 2 = id. For the low dimensional cases n ∈ {3, 4}, we give a sufficient and necessary condition on π1, π2 for $${{W_{n,{\pi _1},{\pi _2}}}}$$ to be an entanglement witness. We also show that, for n ∈ {3, 4}, $${{W_{n,{\pi _1},{\pi _2}}}}$$ is decomposable if and only if π1π2 = id or π 1 2 = π 2 2 ; = id; $${{W_{3,{\pi _1},{\pi _2}}}}$$ is optimal if and only if (π1, π2) = (π, π2), where π = (2, 3,1). As applications, some entanglement criteria for states and some decomposability criteria for positive maps are established.

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.

In this paper we describe two methods, both inspired from Godsil-McKay switching on simple graphs, to build cospectral gain graphs whose gain group consists of the complex numbers of modulus 1 (the circle group). The results obtained here can be also applied to the Hermitian matrix of mixed graphs.

Self-dual MDS codes over finite fields are linear codes with important cryptographic and combinatorial applications. In this correspondence, a different approach is proposed to obtain Hermitian self-dual MDS codes from (extended) generalized Reed-Solomon (abbreviation to GRS and EGRS) codes. The necessary and sufficient conditions of a Hermitian self-dual (extended) GRS code are presented. With the help of this new method, we revise and improve the results in Niu and Yue et al. (IEEE Communications Letters, 23(5): 781-784, 2019). Furthermore, we suppose that there exist Hermitian self-dual (extend) GRS codes over $\mathbb {F}_{q^{2}}$ if and only if their code length $n\leq q+1$ .

Electromagnetic metasurfaces enable the advanced control of surface-wave propagation by spatially tailoring the local surface reactance. Interestingly, tailoring the surface resistance distribution in space provides new, largely unexplored degrees of freedom. Here, we show that suitable spatial modulations of the surface resistance between positive (i.e., loss) and negative (i.e., gain) values can induce peculiar dispersion effects, far beyond a mere compensation. Taking inspiration from the parity-time symmetry concept in quantum physics, we put forward and explore a class of non-Hermitian metasurfaces that may exhibit extreme anisotropy mainly induced by the gain-loss interplay. Via analytical modeling and full-wave numerical simulations, we illustrate the associated phenomenon of surface-wave canalization, explore nonlocal effects and possible departures from the ideal conditions, and address the feasibility of the required constitutive parameters. Our results suggest intriguing possibilities to dynamically reconfigure the surface-wave propagation, and are of potential interest for applications to imaging, sensing and communications., 11 pages; 10 figures

The change in geometric multiplicity, associated with perturbing a particular pair of off-diagonal entries, in a combinatorially symmetric matrix is investigated. First, we focus upon a Hermitian matrix, with a general graph, when the perturbation is Hermitian. This generalizes prior work in case the graph is a tree; the possibilities are much richer. Then, we turn to general matrices over a field. In both cases, by classifying the incident vertices, all possibilities for the change in geometric multiplicity are identified. Then, examples are given to show that each possibility may actually occur. In some instance, the change in geometric multiplicity depends upon the numerical value of the perturbation, and it matters that both off-diagonal entries change. However, in most instances, the change is qualitatively determined.

The topological properties of photonic microstructures are of great interest because of their experimental feasibility for fundamental study and potential applications. Here, we show that robust guided-mode-resonance states exist in photonic domain-wall structures whenever the complex photonic band structures involve certain topological correlations in general. Using the non-Hermitian photonic analogy of the one-dimensional Dirac equation, we derive essential conditions for photonic Jackiw-Rebbi-state resonances taking advantage of unique spatial confinement and spot-like spectral features which are remarkably robust against random parametric errors. Therefore, the proposed resonance configuration potentially provides a powerful method to create compact and stable photonic resonators for various applications in practice.

We propose an resistors, inductors and capacitors (RLC) electrical circuit to theoretically analyze and fully simulate a new type of non-Hermitian Su-Schrieffer-Heeger (SSH) model with complex hoppings. We formulate its construction and investigate its properties by taking advantage of the circuit's versatility. Rich physical properties can be identified in this system from the normal modes of oscillation of the RLC circuit, including the highly tunable bulk-edge correspondence between topological winding numbers and edge states and the non-Hermitian skin phenomenon originating from a novel complex energy plane topology. The present study is able to show the wide and appealing topological physics inherent to electric circuits, which is readily generalizable to a plenty of both Hermitian and non-Hermitian nontrivial systems.

In this paper, we introduce a new class of additive codes over finite fields, viz. multi-twisted (MT) additive codes, which are generalizations of constacyclic additive codes. We study their algebraic structures by writing a canonical form decomposition and provide an enumeration formula for these codes. By placing ordinary, Hermitian and $$*$$ trace bilinear forms, we further study their dual codes and derive necessary and sufficient conditions under which a MT additive code is self-dual and self-orthogonal. We also derive a necessary and sufficient condition for the existence of a self-dual MT additive code over a finite field, and provide enumeration formulae for all self-dual and self-orthogonal MT additive codes over finite fields with respect to the aforementioned trace bilinear forms. We also obtain several good codes within the family of MT additive codes over finite fields.

We investigate the excited-state population of an open two-level atomic system under the quantum feedback control with non-Hermitian feedback Hamiltonian. We firstly derive the master equation under non-Hermitian feedback controls for the two-level atomic system by using the general measurement theory and then respectively discuss the effect of different feedback parameters on the excited-state population. The results show that the excited-state population can be effectively protected from dissipative environments by adjusting feedback parameters. Furthermore, two schemes to realize long-time excited-state population trapping are proposed. The one is under the quantum feedback control with parity-time (PT)-symmetric feedback Hamiltonian, and the other recovers to the Hermitian quantum-jump-based feedback control. These originally come from the fact that the decay of the open two-level atomic system can be completely balanced by feedback controls with proper feedback parameters.

We present randomized algorithms based on block Krylov subspace methods for estimating the trace and log-determinant of Hermitian positive semi-definite matrices. Using the properties of Chebyshev polynomials and Gaussian random matrix, we provide the error analysis of the proposed estimators and obtain the expectation and concentration error bounds. These bounds improve the corresponding ones given in the literature. Numerical experiments are presented to illustrate the performance of the algorithms and to test the error bounds.

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).

In this paper, we introduce $$\omega$$ -constacyclic Type II duadic codes over $${\mathbb {F}}_4$$ of length $$n=3p^t$$ , where p is a prime number, $$p \equiv 1$$ (mod 6), and give their main properties. We also obtain a splitting of n such that extended $$\omega$$ -constacyclic duadic codes are Hermitian self-dual. Moreover, we provide some examples by extended $$\omega$$ -constacyclic duadic codes.

Let $\mathbb {F}_{p^{m}}$ be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ . To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes.

The present paper deals Bochner Ricci pseudo-symmetric mixed generalised quasi-Einstein–Hermitian manifolds, holomorphically projective Ricci pseudo-symmetric mixed generalised quasi-Einstein–Hermitian manifolds and investigated these through different approaches.

We consider the Kac–Bernstein functional equation $$\begin{aligned} f(x+y)g(x-y)=f(x)f(y)g(x)g(-y), \quad x, y\in X, \end{aligned}$$ on an arbitrary Abelian group X. We give a complete description of the solutions of this equation in the class of positive functions. This result is a generalization of a theorem by Pl. Kannappan. We also study the solutions of this equation in the class of complex-valued Hermitian functions.

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

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.

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 .

We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called $C_{ab}$ curves, as well as algorithms for inverting the encoding map, which we call “unencoding”. Some $C_{ab}$ curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ resp. $ \tilde { \mathcal {O}}({\it\text { qn}})$ for AG codes over any $C_{ab}$ curve satisfying very mild assumptions, where n is the code length and q the base field size, and $ \tilde { \mathcal {O}}$ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity $ \tilde { \mathcal {O}}(\text {n})$ for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities $ \tilde { \mathcal {O}}(\text {n}^{5/4})$ and $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ respectively.

In this paper, we study T-eigenvalues of third-order tensors. Definitions of the T-eigenvalues and Hermitian tensors are proposed. We present a commutative tensor family. We prove some T-eigenvalue inequalities for Hermitian tensors, including extensions of Weyl's theorem and Cauchy's interlacing theorem from the matrix case to the tensor case. Finally, we introduce the stability of the T-eigenvalues and study the Lyapunov equation for tensors.

A modulated pseudo-Hermitian optical dimer is investigated. The parameter ranges are determined where the eigenmodes of the dimer conserve their power. It is shown that such a dimer has several exceptional points, and it is found that the dimer modes can conserve their average power over the period even if each of its segments is in the PT-broken phase. On the other hand, even if each segment is in the PT symmetric phase, the average power over the period is not conserved for the dimer. Thisspecial feature is determined by the relationship between the dimer length and the beat period of the signal in the individual segments of the dimer.

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

Blair et al. [7] introduced the notion of bicontact manifold in the context of Hermitian geometry. Bande and Hadjar [1] studied on this notion under the name of contact pairs. These type of structures have important properties and their geometry is some different from classical contact structures. In this paper, we study on some semi-symmetry properties of the normal contact pair manifolds. We prove that a Ricci semi-symmetric (or concircularly Ricci semi-symmetric) normal metric contact pair manifold is a generalized quasi-Einstein manifold. Also, we classify normal metric contact pair manifolds as a generalized quasi-Einstein manifold with certain semi-symmetry conditions and for the concircular curvature tensor , the Riemannian curvature tensor , and an arbitrary vector field .

Matrix---not only the arrays of numbers but also has been used as a tool for many calculations in various subjects. Its inverse, eigen values, eigen vectors are of great importance to know its characters. In this paper I have discussed about some new properties of Hermitian, Skew Hermitian matrices, diagonalisation, eigen values, eigen vectors, spectrum, which will open up a new horizon to the students of Mathematics. Also, in this paper I have authored two totally new theorems for students and researchers . SB’s Theorem 1 is on Normality of a block diagonal matrix and SB’s Theorem 2 is on Spectrum of eigen values. These ideas came to me in course of teaching. Hope, these two theorems will be of great help for the students of Physics and Chemistry as well.

This paper is devoted to the affine-invariant ternary codes defined by Hermitian functions. We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes. Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs. Finally, we determine the dimension and develop a lower bound on the minimum distance of the ternary linear codes.

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.

We show that generically, the degeneracies of a family of Hermitian matrices depending on three parameters have a conical structure. Our result applies to the study of topological phases of matter. It implies that adiabatic deformations of two-dimensional topological insulators come generically with Dirac-like propagating currents, whose total conductivity equals the chiral number of conical points.

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