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We investigate Monge–Ampère type fully nonlinear equations on compact almost Hermitian manifolds with boundary and show a priori gradient estimates for a smooth solution of these equations.

Analytical formulas are obtained for all components of the high-frequency field, Poynting vector $$\overrightarrow P $$ , and group velocity vector $$\overrightarrow U $$ of electromagnetic waves propagating in an arbitrary direction in an unbounded bi-gyrotropic medium described by the Hermitian permittivity and permeability tensors. It is proven that the corresponding components of vectors $$\overrightarrow P $$ and $$\overrightarrow U $$ are proportional to each other (therefore, these vectors are parallel) and the ratio between these components is the volume density of the wave energy. The change in the absolute value and orientation of vector $$\overrightarrow U $$ and orientation of vector $$\overrightarrow P $$ depending on the orientation of the wave vector for different types of electromagnetic waves propagating in a ferromagnetic medium (a particular case of a bi-gyrotropic medium) is calculated. It is found that vectors $$\overrightarrow U $$ and $$\overrightarrow P $$ are always identically oriented for waves of all types in a ferromagnetic medium.

A notion of parabolic C-subsolutions is introduced for parabolic equations, extending the theory of C-subsolutions recently developed by B. Guan and more specifically G. Sz\'ekelyhidi for elliptic equations. The resulting parabolic theory provides a convenient unified approach for the study of many geometric flows.

In this paper, we design an interpolation-characteristic scheme for the numerical solution of the inhomogeneous transfer equation. The scheme is based on the Hermitian interpolation to reconstruct the value of an unknown function at the point of intersection of the backward characteristic with the cell faces. The Hermitian interpolation to reconstruct the function values ​​uses both the nodal values ​​of the desired function and its derivative. Unlike previous studies also based on the Hermitian interpolation, not only the differential continuation of the transfer equation but also the relationship between the integral averaged values, nodal values, ​​and derivatives according to the Euler-Maclaurin formula is used to transfer information about the derivatives to the next layer. The third-order difference scheme is shown to converge for smooth solutions. The dissipative and dispersive properties of the scheme are considered using numerical examples of solutions with decreasing smoothness.

We study one-dimensional general non-Hermitian models with asymmetric long-range hopping and explore to analytically solve the systems under some specific boundary conditions. Although the introduction of long-range hopping terms prevents us from finding analytical solutions for arbitrary boundary parameters, we identify the existence of exact solutions when the boundary parameters fulfill some constraint relations, which give the specific boundary conditions. Our analytical results show that the wave functions take simple forms and are independent of hopping range, while the eigenvalue spectra display rich model-dependent structures. Particularly, we find the existence of a special point coined as pseudo-periodic boundary condition, for which the eigenvalues are the same as the periodical system when the hopping parameters fulfill certain conditions, whereas eigenstates display non-Hermitian skin effect., Comment: 7 pages, 4 figures

We study the Hermitian curvature flow of locally homogeneous non-K\"ahler metrics on compact complex surfaces. In particular, we characterize the long-time behavior of the solutions to the flow. We also provide the first example of a compact complex non-K\"ahler manifold admitting a finite time singularity for the Hermitian curvature flow. Finally, we compute the Gromov-Hausdorff limit of immortal solutions after a suitable normalization. Our results follow by a case-by-case analysis of the flow on each complex model geometry., Comment: Some minor changes. A new appendix containing explicit tensor components. To appear on Ann. Mat. Pura Appl

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

This article introduces a new model order reduction method for a linear time-invariant system with symmetric positive-definite matrices. The proposed method allows the construction of a reduced model, represented by a Cauer-equivalent circuit, from the original system. The method is developed by extending the Cauer ladder network method for the quasi-static Maxwell's equations, which is shown to be regarded as the Lanczos algorithm with respect to a self-adjoint matrix. As a numerical example, a Cauer-equivalent circuit is generated from a simple mathematical model as well as the finite-element (FE) model of a magnetic reactor that is driven by a pulsewidth modulation voltage wave. The instantaneous power obtained from the circuit analysis is shown to be in good agreement with that obtained from the original FE model.

We investigate the eigenvalue curves of 1-parameter hermitean and general complex or real matrix flows $A(t)$ in light of their geometry and the uniform decomposability of $A(t)$ for all parameters $t$. The often misquoted and misapplied results by Hund and von Neumann and by Wigner for eigencurve crossings from the late 1920s are clarified for hermitean matrix flows $A(t) = (A(t))^*$. A conjecture on extending these results to general non-normal or non-hermitean 1-parameter matrix flows is formulated and investigated. An algorithm to compute the block dimensions of uniformly decomposable hermitean matrix flows is described and tested. The algorithm uses the ZNN method to compute the time-varying matrix eigenvalue curves of $A(t)$ for $t_o \leq t\leq t_f$. Similar efforts for general complex matrix flows are described. This extension leads to many new and open problems. Specifically, we point to the difficult relationship between the geometry of eigencurves for general complex matrix flows $A(t)$ and a general flow's decomposability into blockdiagonal form via one fixed unitary or general matrix similarity for all parameters $t$., Comment: 15 pages, 11 graphs

We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so well known outside a community of specialists. We give two different proofs of the main eigenvector perturbation theorem. The first, a block-diagonalization technique inspired by the numerical linear algebra research community and based on the implicit function theorem, has apparently not appeared in the literature in this form. The second, based on complex function theory and on eigenprojectors, as is standard in analytic perturbation theory, is a simplified version of well-known results in the literature. The second derivation uses a convenient normalization of the right and left eigenvectors defined in terms of the associated eigenprojector, but although this dates back to the 1950s, it is rarely discussed in the literature. We then show how the eigenvector perturbation theory is easily extended to handle other normalizations that are often used in practice. We also explain how to verify the perturbation results computationally. We conclude with some remarks about difficulties introduced by multiple eigenvalues and give references to work on perturbation of invariant subspaces corresponding to multiple or clustered eigenvalues. Throughout the paper we give extensive bibliographic commentary and references for further reading.

This work develops a simple finite element for the geometrically exact analysis of Bernoulli–Euler rods. Transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. A straight reference configuration is assumed for the rod. The cross-section undergoes a rigid body motion. A rotation tensor with the Rodrigues formula is used to describe the rotation, which makes the updating of the rotational variables very simple. A formula for the Rodrigues parameters in function of the displacements derivative and the torsion angle is for the first time settled down. The consistent connection between elements is thoroughly discussed, and an appropriate approach is developed. Cubic Hermitian interpolation for the displacements together with linear Lagrange interpolation for the torsion incremental angle were employed within the usual Finite Element Method, leading to adequate C1 continuity. A set of numerical benchmark examples illustrates the usefulness of the formulation and numerical implementation.

This paper is devoted to the asymptotic behavior of all eigenvalues of the increasing finite principal sections of an infinite symmetric (in general non-Hermitian) Toeplitz matrix. The symbol of the infinite matrix is supposed to be moderately smooth and to trace out a simple loop in the complex plane. The main result describes the asymptotic structure of all eigenvalues. The asymptotic expansions constructed are uniform with respect to the location of the eigenvalues in the bulk of the spectrum.

We consider real versions of the classical numerical range and higher rank numerical ranges of a matrix. Our motivations come from descriptions of geometric structures in Euclidean space such as ellipsoids. We establish a number of basic results on these numerical ranges, comparing and delineating from their complex counterparts. We prove a two-dimensional elliptical range theorem for the joint real numerical range. We show that the joint numerical range and the real joint numerical range respectively of a set of m Hermitian matrices of order n is contained in an affine subspace of dimension n 2 − 1 and 1 2 ( n 2 + n − 2 ) respectively.

In this paper, first we consider the numerical range of C ψ , φ , when φ ( z ) = r z with | r | ≤ 1 . Then we study the numerical range of C ψ , φ , where φ is an elliptic automorphism. Next, the exact value of the norm of some weighted composition operators are obtained in order to investigate their numerical radiuses. Finally, we compute the numerical ranges of all hermitian weighted composition operators.

The standard Yee's scheme for the Maxwell eigenvalue problem places the discrete electric field variable at the midpoints of the edges of the grid cells. It performs well when the permittivity is a scalar field. However, when the permittivity is a Hermitian full tensor field it would generate un-physical complex eigenvalues or frequencies. In this paper, we propose a finite element method which can be interpreted as a modified Yee's scheme to overcome this difficulty. This interpretation enables us to create a fast FFT eigensolver that can compute very effectively the band structure of the anisotropic photonic crystal with SC and FCC lattices. Furthermore, we overcome the usual large null space associated with the Maxwell eigenvalue problem by deriving a null-space free discrete eigenvalue problem which involves a crucial Hermitian positive definite linear system to be solved in each of the iteration steps. It is demonstrated that the CG method without preconditioning converges in 37 iterations even when the dimension of a matrix is as large as 5 , 184 , 000 .

The biharmonic operator plays a central role in a wide array of physical models, such as elasticity theory and the streamfunction formulation of the Navier–Stokes equations. Its spectral theory has been extensively studied. In particular the one-dimensional case (over an interval) constitutes the basic model of a high order Sturm–Liouville problem. The need for corresponding numerical simulations has led to numerous works. The present paper relies on a discrete biharmonic calculus. The primary object of this calculus is a high-order compact discrete biharmonic operator (DBO). The DBO is constructed in terms of the discrete Hermitian derivative. However, the underlying reason for its accuracy remained unclear. This paper is a contribution in this direction, expounding the strong connection between cubic spline functions (on an interval) and the DBO. The first observation is that the (scaled) fourth-order distributional derivative of the cubic spline is identical to the action of the DBO on grid functions. It is shown that the kernel of the inverse of the discrete operator is (up to scaling) equal to the grid evaluation of the kernel of $$\left[ \left( \frac{d}{dx}\right) ^4\right] ^{-1}$$ , and explicit expressions are presented for both kernels. As an important application, the relation between the (infinite) set of eigenvalues of the fourth-order Sturm–Liouville problem and the finite set of eigenvalues of the discrete biharmonic operator is studied. The discrete eigenvalues are proved to converge (at an “optimal” $$O(h^4)$$ rate) to the continuous ones. Another consequence is the validity of a comparison principle. It is well known that there is no maximum principle for the fourth-order equation. However, a positivity result is derived, both for the continuous and the discrete biharmonic equation, showing that in both cases the kernels are order preserving.

A generalization of the method developed by Adam, Psarrakos and Tsatsomeros to find inequalities for the eigenvalues of a complex matrix A using knowledge of the largest eigenvalues of its Hermitian part H(A) is presented. The numerical range or field of values of A can be constructed as the intersection of half-planes determined by the largest eigenvalue of H($e^{i\theta}$A). Adam, Psarrakos and Tsatsomeros showed that using the two largest eigenvalues of H(A), the eigenvalues of A satisfy a cubic inequality and the envelope of such cubic curves defines a region in the complex plane smaller than the numerical range but still containing the spectrum of A. Here it is shown how using the three largest eigenvalues of H(A) or more, one obtains new inequalities for the eigenvalues of A and new envelope-type regions containing the spectrum of A., Comment: 19 pages, 11 figures. Version published in Lin. Alg. Appl. 557, 1-21, 2018

We analyze a correlated system in equilibrium with special emphasis on non-Hermitian topology inducing a skin effect. The pseudospectrum, computed by the real-space dynamical mean field theory, elucidates that additional pseudoeigenstates emerge for the open boundary condition in contrast to the dependence of the density of states on the boundary condition. We further discuss how the line-gap topology, another type of non-Hermitian topology, affects the pseudospectrum. Our numerical simulation clarifies that while the damping of the quasiparticles induces the nontrivial point-gap topology, it destroys the nontrivial line-gap topology. The above two effects are also reflected in the temperature dependence of the local pseudospectral weight.

We deal with the modelling a radiator/scatterer using an equivalent radiator. It requires determining the shape and the dimensions of a radiating surface generating, in a certain region of space ${\mathcal{D}}$, an electromagnetic field close to that produced by the radiator/scatterer of interest.We propose, for a fixed equivalent radiator’s shape, an approach for the solution of the dimensioning issue. The approach uses the Singular Value Decomposition (SVDs) of the operators linking the radiator/scatterer to the field on ${\mathcal{D}}$ and the equivalent radiating panel to the field on ${\mathcal{D}}$, again. The approach determines the dimensions of the equivalent radiator minimizing the error arising by approximating the primary radiated/scattered field with that radiated using ${{\mathcal{D}}^\prime }$. The error is expressed as a Hermitian, positive semi-definite quadratic form so that the problem amounts at the maximization of its minimum eigenvalue.Numerical results are presented for a planar radiator of rectangular shape.

Systems with non-Hermitian skin effects are very sensitive to the imposed boundary conditions and lattice size, and thus an important question is whether non-Hermitian skin effects can survive when deviating from the open boundary condition. To unveil the origin of boundary sensitivity, we present exact solutions for one-dimensional non-Hermitian models with generalized boundary conditions and study rigorously the interplay effect of lattice size and boundary terms. Besides the open boundary condition, we identify the existence of non-Hermitian skin effect when one of the boundary hopping terms vanishes. Apart from this critical line on the boundary parameter space, we find that the skin effect is fragile under any tiny boundary perturbation in the thermodynamic limit, although it can survive in a finite size system. Moreover, we demonstrate that the non-Hermitian Su-Schreieffer-Heeger model exhibits a new phase diagram in the boundary critical line, which is different from either open or periodical boundary case., 21 pages, 12 figures

This paper presents finite elements for a higher order steel&ndash, concrete composite beam model developed for the analysis of bridge decks. The model accounts for the slab&ndash, girder partial interaction, the overall shear deformability, and the shear-lag phenomenon in steel and concrete components. The theoretical derivation of the solving balance conditions, in both weak and strong form, is firstly addressed. Then, three different finite elements are proposed, which are characterised by (i) linear interpolating functions, (ii) Hermitian polynomial interpolating functions, and (iii) interpolating functions, respectively, derived from the analytical solution expressed by means of exponential matrices. The performance of the finite elements is analysed in terms of the solution convergence rate for realistic steel&ndash, concrete composite beams with different restraints and loading conditions. Finally, the efficiency of the beam model is shown by comparing the results obtained with the proposed finite elements and those achieved with a refined 3D shell finite element model.

In this paper, we investigate the (2 + 1)-dimensional Hirota&ndash, Satsuma&ndash, Ito (HSI) shallow water wave model. By introducing a small perturbation parameter ϵ, an extended (2 + 1)-dimensional HSI equation is derived. Further, based on the Hirota bilinear form and the Hermitian quadratic form, we construct the rational localized wave solution and discuss its dynamical properties. It is shown that the oblique and skew characteristics of rational localized wave motion depend closely on the translation parameter ϵ. Finally, we discuss two different interactions between a rational localized wave and a line soliton through theoretic analysis and numerical simulation: one is an absorb-emit interaction, and the other one is an emit-absorb interaction. The results show that the delay effect between the encountering and parting time of two localized waves leads to two different kinds of interactions.

We define a parabolic flow of almost balanced metrics. We show that the flow has a unique solution on compact almost Hermitian manifolds.

Using the properties of random Möbius transformations, we investigate the statistical properties of the reflection coefficient in a random chain of lossy scatterers. We explicitly determine the support of the distribution and the condition for coherent perfect absorption to be possible. We show that at its boundaries the distribution has Lifshits-like tails, which we evaluate. We also obtain the extent of penetration of incoming waves into the medium via the Lyapunov exponent. Our results agree well when compared to numerical simulations in a specific random system.

Hilbert-Schmidt distance is one of the prominent distance measures in quantum information theory which finds applications in diverse problems, such as construction of entanglement witnesses, quantum algorithms in machine learning, and quantum state tomography. In this work, we calculate exact and compact results for the mean square Hilbert-Schmidt distance between a random density matrix and a fixed density matrix, and also between two random density matrices. In the course of derivation, we also obtain corresponding exact results for the distance between a Wishart matrix and a fixed Hermitian matrix, and two Wishart matrices. We verify all our analytical results using Monte Carlo simulations. Finally, we apply our results to investigate the Hilbert-Schmidt distance between reduced density matrices generated using coupled kicked tops., Published version

In [4], the author introduced two parabolic flows; the almost Hermitian flow and the almost Hermitian curvature flow on a compact almost complex manifold. In this paper, we will develop some regularity results for solutions to these flows. We will derive higher order derivative estimates in the presence of a curvature bound. We also exhibit a long-time existence obstruction for solutions to the almost Hermitian curvature flow by showing smoothing estimates for the curvature and torsion.

In this paper, shape functions with regularity up to $$C^{2}$$ were developed for the four-node quadrilateral finite element considering the partition of unity property. This high-regularity approximation space was applied to approach the natural frequencies of free vibration of some in-plane elastic problems as well as the elastic wave response of forced vibration caused by the application of impulsive loading. The obtained results show that it was possible to numerically approximate a greater number of accurate natural frequencies with the devised procedure when a comparison is established with the $$C^{0}$$ Lagrangian and serendipity elements of 4, 8 (serendipity), 16, and 25 nodes. Also, the numerical predictions for the elastic wave propagation problem using the derived approximation space presented small oscillations improving the representation of the displacement field.

We provide a systematic and self-consistent method to calculate the generalized Brillouin Zone (GBZ) analytically in one dimensional non-Hermitian systems, which helps us to understand the non-Hermitian bulk-boundary correspondence. In general, a n-band non-Hermitian Hamiltonian is constituted by n distinct sub-GBZs, each of which is a piecewise analytic closed loop. Based on the concept of resultant, we can show that all the analytic properties of the GBZ can be characterized by an algebraic equation, the solution of which in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a systematic method to obtain the GBZ from aGBZ. Two physical applications are also discussed. Our method provides an analytic approach to the spectral problem of open boundary non-Hermitian systems in the thermodynamic limit., 6+16 pages, 5+5 figures. To appear in Physical Review Letters

We consider a 3-dimensional (3D) non-Hermitian exceptional line semimetal model and take open boundary conditions in x, y, and z directions separately. In each case, we calculate the parameter regions where the bulk-boundary correspondence is broken. The breakdown of the bulk-boundary correspondence is manifested by the deviation from unit circles of generalized Brillouin zones (GBZ) and the discrepancy between spectra calculated with open boundary conditions (OBC) and periodic boundary conditions (PBC). The consistency between OBC and PBC spectra can be recovered if the PBC spectra are calculated with GBZs. We use both unit-circle Brillouin zones (BZ) and GBZs to plot the topological phase diagrams. The systematic analysis of the differences between the two phase diagrams suggests that it is necessary to use GBZ to characterize the bulk-boundary correspondence of non-Hermitian models., 9 pages, 5 figures

Green's functions are fundamental quantities that determine the linear responses of physical systems. The recent developments of non-Hermitian systems, therefore, call for Green's function formulas of non-Hermitian bands. This task is complicated by the high sensitivity of energy spectrums to boundary conditions, which invalidates the straightforward generalization of Hermitian formulas. Here, based on the non-Bloch band theory, we obtain simple Green's function formulas of general one-dimensional non-Hermitian bands. Furthermore, in the large-size limit, these formulas dramatically reduce to finding the roots of a simple algebraic equation. As an application, our formulation provides the desirable formulas for the defining quantities, the gain and directionality, of directional amplification. Thus, our formulas provide an efficient guide for designing directional amplifiers., Comment: 11 pages, 7 figures, including Supplemental Material

Waves typically propagate very differently through a homogeneous medium like free space than through an inhomogeneous medium like a complex dielectric structure. Here we present the surprising result that wave solutions in two-dimensional free space can be mapped to a solution inside a suitably designed non-Hermitian potential landscape such that both solutions share the same spatial distribution of their wave intensity. The mapping we introduce here is broadly applicable as a design protocol for a special class of non-Hermitian media across which specific incoming waves form scattering-free propagation channels. This protocol naturally enables the design of structures with a broadband unidirectional invisibility for which outgoing waves are indistinguishable from those of free space. We illustrate this concept through the example of a beam that maintains its Gaussian shape while passing through a randomly assembled distribution of scatterers with gain and loss.

Two novel chiral block lattice topologies are here conceived having interesting auxetic and acoustic behavior. The architectured chiral material is made up of a periodic repetition of square or hexagonal rigid and heavy blocks connected by linear elastic interfaces, whose chirality results from an equal rotation of the blocks with respect to the line connecting their centroids. The governing equation of the Lagrangian model is derived and a hermitian eigenproblem is formulated to obtain the frequency band structure. An equivalent micropolar continuum is analytically derived through a standard continualization approach in agreement with the procedure proposed by Bacigalupo and Gambarotta (2017) from which an approximation of the frequency spectrum is obtained. Moreover, the overall elastic moduli of the equivalent Cauchy continuum are obtained in closed form via a proper condensation procedure. The parametric analysis involving the overall elastic moduli of the Cauchy equivalent continuum model and the frequency band structure is carried out to catch the influence of the chirality angle and of the ratio between the tangential and normal stiffness of the interface. Finally, it is shown how chirality and interface stiffness may affect strong auxeticity and how the equivalent micropolar model provides dispersion curves in excellent agreement with the current ones for a wide range of the wave vector magnitude.

In this paper, we proposed a new method for numerical conformal mapping function. In order to obtain the highly accurate numerical results of this method, we reduce calculations of the charge points to improved Hermitian and skew-Hermitian splitting method based on (k, j)-Pade iteration. Experimental examples show that the proposed method has high-precision.

This paper focuses on the invariance of deficiency indices of second-order symmetric linear difference equations under perturbations. By applying the perturbation theory of Hermitian linear relations, the invariance of deficiency indices of the corresponding minimal subspaces under bounded and relatively bounded perturbations is built. As a consequence, the invariance of limit types of second-order symmetric linear difference equations under bounded and relatively bounded perturbations is obtained.

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian phase flow (TLPF) on the space of almost calibrated $(1,1)$-forms that fits into the GIT framework for dHYM metrics recently discovered by Collins-Yau. The TLPF has some special properties that are not seen for the line bundle mean curvature flow (i.e. the mirror of the Lagrangian mean curvature flow for graphs). We show that the TLPF starting from any initial data exists for all positive time. Moreover, we show that the TLPF converges smoothly to a dHYM metric assuming the existence of a $C$-subsolution, which gives a new proof for the existence of dHYM metrics in the highest branch., Comment: 24 pages, final version, to appear in Internat. J. Math

A plane Hermitian wavelet finite element method is presented in this paper. Wave motion can be used to analyze plane structures with small defects such as cracks and obtain results. By using the tensor product of modified Hermitian wavelet shape functions, the plane Hermitian wavelet shape functions are constructed. Scale functions of Hermitian wavelet shape functions can replace the polynomial shape functions to construct new wavelet plane elements. As the scale of the shape functions increases, the precision of the new wavelet plane element will be improved. The new Hermitian wavelet finite element method which can be used to simulate wave motion analysis can reveal the law of the wave motion in plane. By using the results of transmitted and reflected wave motion, the cracks can be easily identified in plane. The results show that the new Hermitian plane wavelet finite element method can use the fewer elements to simulate the plane structure effectively and accurately and detect the cracks in plane.

This paper presents the formulation of exact stiffness matrices applied in linear generalized beam theory (GBT) under constant and/or linear loading distribution in the longitudinal direction. Also, the assortment of the correct exact stiffness matrix and the corresponding shape function are presented based on main transversal deformation mode, which can be divided into: (1) dominant distortion mode; (2) dominant torsion mode; (3) and critical distortion–torsion mode. Special attention is given to the hyperbolic–trigonometric shape functions, which are organized in a system of vector in function of longitudinal direction and a coefficient matrix obtained from the completeness requirement. This approach has the benefit of compacting the terms of the stiffness matrix and systematizing the boundary conditions of an element by applying the completeness coefficient matrix as a transformation matrix. As a result, in linear analysis, a single element can represent the stress and displacement fields. Moreover, due to the higher-order continuous derivatives properties of hyperbolic–trigonometric shape functions, the generalized internal shear is obtained without the typical discontinuity of Hermitian shape functions. A full and detailed example, applied in a thin-walled circular hollow cross section, provides not only an illustration of the presented approach, but also a quick introduction point in GBT.

In this paper, we describe a possible generalization of the Wasserstein 2-metric, originally defined on the space of scalar probability densities, to the space of Hermitian matrices with trace one, and to the space of matrix-valued probability densities. Our approach follows a computational fluid dynamical formulation of the Wasserstein-2 metric and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. It allows determining the gradient flow for the quantum entropy relative to this matricial Wasserstein metric. This may have implications to some key issues in quantum information theory., Comment: 19 pages

In this paper we study a class of second order fully nonlinear elliptic equations containing gradient terms on compact Hermitian manifolds and obtain a priori estimates under proper assumptions close to optimal. The analysis developed here should be useful to deal with other Hessian equations containing gradient terms in other contexts.

Motivated by the recent work of Wu and Yau on the ampleness of canonical line bundle for compact K\"ahler manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called $\textbf{real bisectional curvature}$ for Hermitian manifolds. When the metric is K\"ahler, this is just the holomorphic sectional curvature $H$, and when the metric is non-K\"ahler, it is slightly stronger than $H$. We classify compact Hermitian manifolds with constant non-zero real bisectional curvature, and also slightly extend Wu-Yau's theorem to the Hermitian case. The underlying reason for the extension is that the Schwarz lemma of Wu-Yau works the same when the target metric is only Hermitian but has nonpositive real bisectional curvature.

We consider a complex-valued $L \times L$ exponential correlation matrix. Such a matrix has unit diagonal elements; each lower off-diagonal element is the correlation coefficient raised to the power of the modulus of the difference of the row and column indices, while each upper off-diagonal element is the complex conjugate of the correlation coefficient raised to the power of the modulus of the difference of the row and column indices. This makes it a Hermitian Toeplitz matrix. Analytical expressions for the eigenvectors of the exponential correlation matrix are presented, and closed form approximations of the eigenvalues for the low and high correlation cases and for the cases of linear interpolation and large matrix size are derived. Closed form expressions for the eigenvalues of exponential correlation matrices of sizes ranging from 3 to 8 in terms of the correlation coefficient, by a novel method of transformation of the characteristic polynomial and subsequent factorization of the transformed characteristic polynomial, are also derived. Furthermore, applications of the results obtained to the performance evaluation of wireless communication systems employing diversity are presented.

We establish exact relations between the winding of "energy" (eigenvalue of Hamiltonian) on the complex plane as momentum traverses the Brillouin zone with periodic boundary condition, and the presence of "skin modes" with open boundary condition in non-hermitian systems. We show that the nonzero winding with respect to any complex reference energy leads to the presence of skin modes, and vice versa. We also show that both the nonzero winding and the presence of skin modes share the common physical origin that is the non-vanishing current through the system., 22 pages, 10 figures. To appear in Physical Review Letters