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    Hermitian Forms in Function Theory
    Transactions of the American Mathematical Society (1984)
    Christine R. Leverenz
    Let $f$ and $g$ be analytic in the unit disk $|z|\; < 1$. We give a new derivation of the positive semidefinite Hermitian form equivalent to $|g(z)| \leq |f(z)|$, for $| z | < 1$, and use it to derive Hermitian forms for various classes of univalent functions. Sharp coefficient bounds for these classes are obtained from the Hermitian forms. We find the specific functions required to make the Hermitian forms equal to zero.
    Hermitian function
    Zero (linguistics)
    Unit disk
    Citations (16)
    Non-Hermitian Hamiltonians with real and complex eigenvalues: An sl(2,C) approach
    arXiv (Cornell University) (2002)
    Bijan BagchiC. Quesne
    Potential algebras are extended from Hermitian to non-Hermitian Hamiltonians and shown to provide an elegant method for studying the transition from real to complex eigenvalues for a class of non-Hermitian Hamiltonians associated with the complex Lie algebra A$_1$.
    Citations (0)
    Use of Elastodynamic Reciprocity Theorems for Field Calculations
    Birkhäuser Boston eBooks (2002)
    J. D. Achenbach
    Reciprocity is a good thing. Something is given and something else, equally or more valuable, is returned. So it is in reciprocity for states of deformation of elastic bodies. What is received in return is the main benefit from the reciprocal relationship. From a solution to one loading case, some important aspects or the complete solution to another loading case are returned. However, the return is not always a complete solution, but sometimes an equation for computing such a solution.
    Reciprocity
    Reciprocal
    Citations (2)
    Work statistics in non-Hermitian evolutions with Hermitian endpoints
    Physical review. E (2021)
    Zheng-Yang ZhouZe-Liang XiangJ. Q. YouFranco Nori
    Non-Hermitian systems with specific forms of Hamiltonians can exhibit novel phenomena. However, it is difficult to study their quantum thermodynamical properties. In particular, the calculation of work statistics can be challenging in non-Hermitian systems due to the change of state norm. To tackle this problem, we modify the two-point measurement method in Hermitian systems. The modified method can be applied to non-Hermitian systems which are Hermitian before and after the evolution. In Hermitian systems, our method is equivalent to the two-point measurement method. When the system is non-Hermitian, our results represent a projection of the statistics in a larger Hermitian system. As an example, we calculate the work statistics in a non-Hermitian Su-Schrieffer-Heeger model. Our results reveal several differences between the work statistics in non-Hermitian systems and the one in Hermitian systems.
    Hermitian function
    Citations (8)
    Edge modes in subwavelength resonators in one dimension
    arXiv (Cornell University) (2023)
    Habib AmmariS BarandunJinghao CaoFlorian Feppon
    We present the mathematical theory of one-dimensional infinitely periodic chains of subwavelength resonators. We analyse both Hermitian and non-Hermitian systems. Subwavelength resonances and associated modes can be accurately predicted by a finite dimensional eigenvalue problem involving a capacitance matrix. We are able to compute the Hermitian and non-Hermitian Zak phases, showing that the former is quantised and the latter is not. Furthermore, we show the existence of localised edge modes arising from defects in the periodicity in both the Hermitian and non-Hermitian cases. In the non-Hermitian case, we provide a complete characterisation of the edge modes.
    Matrix (chemical analysis)
    Citations (0)
    Hermitian dynamics in a class of pseudo-Hermitian networks
    Physical Review A (2011)
    L. JinZ. Song
    We investigate a pseudo-Hermitian lattice system, which consists of a set of isomorphic pseudo-Hermitian clusters coupled together in a Hermitian manner. We show that such non-Hermitian systems can act as Hermitian systems. This is made possible by considering the dynamics of a state involving an identical eigenmode of each isomorphic cluster. It still holds when multiple eigenmodes are involved and additional restrictions on the state are imposed. This Hermitian dynamics is demonstrated for the case of an exactly solvable $\mathcal{PT}$-symmetric ladder system.
    Hermitian function
    Citations (33)
    Rank-bmRpositive Hermitian approximation algorithms and positive Hermitian decompositions of Hermitian tensors
    Scientia Sinica Mathematica (2023)
    Bo YangYing LiNi GuyanMengshi Zhang
    Rank (graph theory)
    Hermitian function
    Citations (1)
    A Model of Reciprocal Fairness: Application to the Labour Contract
    RePEc: Research Papers in Economics (2006)
    Stéphane Mahuteau
    We investigate to what extent reciprocity, exhibited by employers and employees, lead to stable gift exchange practices in the labour contract, giving rise to non-compensating wage differentials among industries and firms. We use the concept of Sequential Reciprocity Equilibrium (Dufwenberg and Kirchsteiger 1998, 2004) to incorporate players’ preferences for reciprocity in their utility function. We show that successful gift exchange practices may arise if both players actually care for reciprocity. We test the predictions of the model using a matched employer-employee French dataset. Our results show that French employers and employees’ decisions are influenced by reciprocity concerns.
    Reciprocity
    Reciprocal
    Norm of reciprocity
    Social exchange theory
    Incomplete contracts
    Citations (0)
    Edge Modes in Subwavelength Resonators in One Dimension
    Multiscale Modeling and Simulation (2023)
    Habib AmmariS BarandunJinghao CaoFlorian Feppon
    .We present the mathematical theory of one-dimensional infinitely periodic chains of subwavelength resonators. We analyze both Hermitian and non-Hermitian systems. Subwavelength resonances and associated modes can be accurately predicted by a finite dimensional eigenvalue problem involving a capacitance matrix. We are able to compute the Hermitian and non-Hermitian Zak phases, showing that the former is quantized and the latter is not. Furthermore, we show the existence of localized edge modes arising from defects in the periodicity in both the Hermitian and non-Hermitian cases. In the non-Hermitian case, we provide a complete characterization of the edge modes.Keywordssubwavelength resonancesnon-Hermitian topological systemstopologically protected edge modesone-dimensional periodic chains of subwavelength resonatorsMSC codes35B3435P2535J0535C2046T2578A40
    Matrix (chemical analysis)
    Citations (9)
    Hermitian Tensor Decompositions
    arXiv (Cornell University) (2019)
    Jiawang NieZi Yang
    Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.
    Hermitian function
    Rank (graph theory)
    Hermitian symmetric space
    Citations (2)