More Like this

1. Reflectionless excitation of arbitrary photonic structures: a general theory

   https://doi.org/10.1515/nanoph-2020-0403  

   Stone, A. Douglas; Sweeney, William R.; Hsu, Chia Wei; Wisal, Kabish; Wang, Zeyu (October 2020, Nanophotonics)

   null (Ed.)

   Abstract We outline and interpret a recently developed theory of impedance matching or reflectionless excitation of arbitrary finite photonic structures in any dimension. The theory includes both the case of guided wave and free-space excitation. It describes the necessary and sufficient conditions for perfectly reflectionless excitation to be possible and specifies how many physical parameters must be tuned to achieve this. In the absence of geometric symmetries, such as parity and time-reversal, the product of parity and time-reversal, or rotational symmetry, the tuning of at least one structural parameter will be necessary to achieve reflectionless excitation. The theory employs a recently identified set of complex frequency solutions of the Maxwell equations as a starting point, which are defined by having zero reflection into a chosen set of input channels, and which are referred to as R-zeros. Tuning is generically necessary in order to move an R-zero to the real frequency axis, where it becomes a physical steady-state impedance-matched solution, which we refer to as a reflectionless scattering mode (RSM). In addition, except in single-channel systems, the RSM corresponds to a particular input wavefront, and any other wavefront will generally not be reflectionless. It is useful to consider the theory as representing a generalization of the concept of critical coupling of a resonator, but it holds in arbitrary dimension, for arbitrary number of channels, and even when resonances are not spectrally isolated. In a structure with parity and time-reversal symmetry (a real dielectric function) or with parity–time symmetry, generically a subset of the R-zeros has real frequencies, and reflectionless states exist at discrete frequencies without tuning. However, they do not exist within every spectral range, as they do in the special case of the Fabry–Pérot or two-mirror resonator, due to a spontaneous symmetry-breaking phenomenon when two RSMs meet. Such symmetry-breaking transitions correspond to a new kind of exceptional point, only recently identified, at which the shape of the reflection and transmission resonance lineshape is flattened. Numerical examples of RSMs are given for one-dimensional multimirror cavities, a two-dimensional multiwaveguide junction, and a multimode waveguide functioning as a perfect mode converter. Two solution methods to find R-zeros and RSMs are discussed. The first one is a straightforward generalization of the complex scaling or perfectly matched layer method and is applicable in a number of important cases; the second one involves a mode-specific boundary matching method that has only recently been demonstrated and can be applied to all geometries for which the theory is valid, including free space and multimode waveguide problems of the type solved here. 

   more » « less
2. Positivity and Universal Plücker Coordinates for Spaces of Quasi-Exponentials

   https://doi.org/10.1093/imrn/rnaf235  

   Karp, Steven N; Mukhin, Evgeny; Tarasov, Vitaly (August 2025, International Mathematics Research Notices)

   A quasi-exponential is an entire function of the form $$e^{cu}p(u)$$, where $p(u)$ is a polynomial and $$c \in \mathbb{C}$$. Let $$V = \langle e^{h_{1}u}p_{1}(u), \dots , e^{h_{N}u}p_{N}(u) \rangle $$ be a vector space with a basis of quasi-exponentials. We show that if $$h_{1}, \dots , h_{N}$$ are nonnegative and all of the complex zeros of the Wronskian $$\operatorname{Wr}(V)$$ are real, then $$V$$ is totally nonnegative in the sense that all of its Grassmann–Plücker coordinates defined by the Taylor expansion about $u=t$ are nonnegative, for any real $$t$$ greater than all of the zeros of $$\operatorname{Wr}(V)$$. Our proof proceeds by showing that the higher Gaudin Hamiltonians $$T_\lambda ^{G}(t)$$ introduced in [1] are universal Plücker coordinates about $u=t$ for the Wronski map on spaces of quasi-exponentials. The result that $$V$$ is totally nonnegative follows from the fact that $$T_\lambda ^{G}(t)$$ is positive semidefinite, which we establish using partial traces. We also show that if $$h_{1} = \cdots = h_{N} = 0$$ then $$T_\lambda ^{G}(t)$$ equals $$\beta ^\lambda (t)$$, which is the universal Plücker coordinate for the Wronski map on spaces of polynomials introduced in [4]. 

   more » « less
3. Non-commutative $$L^p$$ spaces and Grassmann stochastic analysis

   https://doi.org/10.1007/s00440-025-01379-4  

   De_Vecchi, Francesco; Fresta, Luca; Gordina, Maria; Gubinelli, Massimiliano (August 2025, Probability Theory and Related Fields)

   Abstract We introduce a theory of non-commutative$$L^p$$ $L^p$spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect toItô–Grassmannprocesses, Girsanov’s formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables including a Grassmann analog of the $$\Phi ^4_2$$ ${\Phi }_2^4$Euclidean QFT in a bounded region and weak solution to singular SPDEs in the spirit of the early work of Jona-Lasinio and Mitter on the stochastic quantisation of $$\Phi ^4_2$$ ${\Phi }_2^4$. 

   more » « less
4. Expected Number of Real Zeros for Random Orthogonal Polynomials

   Lubinsky, Doron; Pritsker, Igor; Xie, Xiaoyou (January 2018, Mathematical proceedings of the Cambridge Philosophical Society)

   We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $$(2/\pi + o(1))\log{n}$$ expected real zeros in terms of the degree $$n$$. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight, then random linear combinations have $$n/\sqrt{3} + o(n)$$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution. 

   more » « less
5. On the parameters of line Schubert symplectic Grassmann codes

   https://doi.org/10.1142/S021949882641001X  

   Bookamer, Mackenzie; Jaramillo, Susana; Piñero_González, Fernando; Southern, Lani (April 2026, Journal of Algebra and Its Applications)

   The Grassmannian [Formula: see text], is the collection of all two-dimensional subspaces of a vector space of dimension [Formula: see text]. It is one of the most widely studied objects in Algebraic Geometry and has interesting algebraic, geometric, and combinatorial properties. Since subspaces of dimension [Formula: see text] are known as lines the Grassmannian [Formula: see text], is known as the Grassmannian of lines. A class of linear codes, known as Grassmann codes, are used to understand the geometric and algebraic properties of the Grassmannian. Codes from Schubert subvarieties and polar subvarieties of the Grassmannian are known. For the case [Formula: see text], the parameters of Schubert codes have been established by Chen and the parameters of polar Grassmann codes have been established by Cardinali and Giuzzi. For general [Formula: see text], the parameters for Schubert codes are known, but the parameters of polar Grassmann codes are not known for general [Formula: see text]. The case in which either a polarity condition or a Schubert condition is applied to the Grassmannian on their own is simpler. These conditions have been studied independently but not concurrently. In this work, we study linear codes derived from subvarieties of the Grassmannian defined from the intersection of Schubert and symplectic varieties. We determine the length, dimension and minimum distance for several cases of line Schubert symplectic Grassmann codes when both the Schubert conditions and some special symplectic polarity conditions are combined in different ways. 

   more » « less