Research in the Mathematical Sciences

Petkov, Vesselin

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00301-3

We study the wave equation in the exterior of a bounded domain K with dissipative boundary condition ∂νu-γ(x)∂tu=0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\partial _{\nu } u - \gamma (x) \partial _t u = 0$$\end{document} on the boundary Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document} and γ(x)>0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma (x) > 0.$$\end{document} The solutions are described by a contraction semi-group V(t)=etG,t≥0.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$V(t) = e^{tG}, \, t \ge 0.$$\end{document} The eigenvalues λk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda _k$$\end{document} of G with Reλk<0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{ Re }\,\lambda _k < 0$$\end{document} yield asymptotically disappearing solutions u(t,x)=eλktf(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$u(t, x) = e^{\lambda _k t} f(x)$$\end{document} having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case minx∈Γγ(x)>1.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\min _{x\in \Gamma } \gamma (x) > 1.$$\end{document} For strictly convex obstacles K, this formula concerns all eigenvalues of G.

Deng, Youjun; Duan, Chaohua; Liu, Hongyu

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00299-8

This paper is concerned with the geometric structure of the transmission eigenvalue problem associated with a general conductive transmission condition. We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary. The Herglotz approximation is the Fourier extension of the transmission eigenfunction, and the growth rate of the density function can be used to characterize the regularity of the underlying wave function. The geometric structures derived in this paper include the related results in Diao et al. (Commun Partial Differ Equ 46(4):630–679, 2021) and Blåsten and Liu (J Funct Anal 273:3616–3632, 2017) as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions.

Yang, Hongkang; E, Weinan

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00306-y

The generative adversarial network (GAN) is a well-known model for learning high-dimensional distributions, but the mechanism for its generalization ability is not understood. In particular, GAN is vulnerable to the memorization phenomenon, the eventual convergence to the empirical distribution. We consider a simplified GAN model with the generator replaced by a density and analyze how the discriminator contributes to generalization. We show that with early stopping, the generalization error measured by Wasserstein metric escapes from the curse of dimensionality, despite that in the long term, memorization is inevitable. In addition, we present a hardness of learning result for WGAN.

Monk, Peter; Selgas, Virginia

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00300-4

Target signatures are discrete quantities computed from measured scattering data that could potentially be used to classify scatterers or give information about possible defects in the scatterer compared to an ideal object. Here, we study a class of modified interior transmission eigenvalues that are intended to provide target signatures for an inverse fluid–solid interaction problem. The modification is based on an auxiliary problem parametrized by an artificial diffusivity constant. This constant may be chosen strictly positive, or strictly negative. For both choices, we characterize the modified interior transmission eigenvalues by means of a suitable operator so that we can determine their location in the complex plane. Moreover, for the negative sign choice, we also show the existence and discreteness of these eigenvalues. Finally, no matter the choice of the sign, we analyze the approximation of the eigenvalues from far field measurements of the scattered fluid pressure and provide numerical results which show that, even with noisy data, some of the eigenvalues can be determined from far field data.

Alcántara, Claudia R.

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-022-00311-9

In this work, we construct, for any d≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d \ge 2$$\end{document}, a new foliation on CP2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {CP}^2$$\end{document} of degree d with a unique singular point of multiplicity d-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$d-1$$\end{document} without invariant algebraic curves that contain all its separatrices. We also prove that if X is a foliation on CP2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {CP}^2$$\end{document} with a unique nilpotent singular point, then X has no algebraic leaves. Finally, we characterize logarithmic foliations on CP2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {CP}^2$$\end{document} with a unique singular point. And we give some new examples of this kind of foliations.

Ambrose, David M.; Cakoni, Fioralba; Moskow, Shari

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00308-w

In this paper, we consider a perturbation problem for real transmission eigenvalues. Real transmission eigenvalues are of particular interest in inverse scattering theory. They can be determined from scattering data and are related to injectivity of the related scattering operators. The goal of this paper is to provide examples of existence of real transmission eigenvalues for inhomogeneities whose refractive index does not satisfy the assumptions for which the (non-self-adjoint) transmission eigenvalue problem is understood. Such “irregular media” are obtained as perturbations of an inhomogeneity for which the existence of real transmission eigenvalues is known. Our perturbation approach uses an application of a version of the implicit function theorem to an appropriate function in the vicinity of an unperturbed real transmission eigenvalue. Several examples of interesting spherical perturbations of spherically symmetric media are included. Partial results are obtained for general media based on our perturbation approach.

Pietromonaco, Stephen

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00298-9

For an Abelian surface A with a symplectic action by a finite group G, one can define the partition function for G-invariant Hilbert schemes ZA,G(q)=∑d=0∞e(Hilbd(A)G)qd.\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$\end{document}We prove the reciprocal ZA,G-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{A,G}^{-1}$$\end{document} is a modular form of weight 12e(A/G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\frac{1}{2}e(A/G)$$\end{document} for the congruence subgroup Γ0(|G|)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma _{0}(|G|)$$\end{document} and give explicit expressions in terms of eta products. Refined formulas for the χy\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi _{y}$$\end{document}-genera of Hilb(A)G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text {Hilb}(A)^{G}$$\end{document} are also given. For the group generated by the standard involution τ:A→A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau : A \rightarrow A$$\end{document}, our formulas arise from the enumerative geometry of the orbifold Kummer surface [A/τ]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$[A/\tau ]$$\end{document}. We prove that a virtual count of curves in the stack is governed by χy(Hilb(A)τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi _{y}(\text {Hilb}(A)^{\tau })$$\end{document}. Moreover, the coefficients of ZA,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Z_{A, \tau }$$\end{document} are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.

Bergvall, Olof

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00305-z

We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document} is acting on the arrangement, the algorithm determines the cohomology groups as representations of Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}. As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

Liu, Shuzhi; Zhang, Qinglong

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-022-00312-8

We construct a family of self-similar and bounded weak solutions for all positive time to the two-dimensional shallow water equations with the bottom topography a quadratic function with respect to the reciprocal of radius under axisymmetry assumptions. The source term generated from the bottom topography leads to a different structure in solutions with that obtained in hyperbolic conservative system. The key point is that the steady-state solution is used to deal with the singularity near the origin in the reduced system under axisymmetric transformation. The obtained results may apply to other systems with similar structures.

Grisalde, Gonzalo; Seceleanu, Alexandra; Villarreal, Rafael H.

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00310-2

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure.

Cogar, Samuel

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00304-0

Eigenvalues arising in scattering theory have been envisioned as a potential source of target signatures in nondestructive testing of materials, whereby perturbations of the eigenvalues computed for a penetrable medium would be used to infer changes in its constitutive parameters relative to some reference values. We consider a recently introduced modification of the class of Stekloff eigenvalues, in which the inclusion of a smoothing operator guarantees that infinitely many eigenvalues exist under minimal assumptions on the medium, and we derive precise formulas that quantify the perturbation of a simple eigenvalue in terms of the coefficients of a perturbed inhomogeneous medium. These formulas rely on the theory of nonlinear eigenvalue approximation and regularity results for elliptic boundary-value problems with heterogeneous coefficients, the latter of which is shown to have a strong influence on the sensitivity of the eigenvalues corresponding to an anisotropic medium. A simple numerical example in two dimensions is used to verify the estimates and suggest future directions of study.

Cutkosky, Steven Dale; Sarkar, Parangama

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00307-x

We develop a theory of multiplicities and mixed multiplicities of filtrations, extending the theory for filtrations of m-primary ideals to arbitrary (not necessarily Noetherian) filtrations. The mixed multiplicities of r filtrations on an analytically unramified local ring R come from the coefficients of a suitable homogeneous polynomial in r variables of degree equal to the dimension of the ring, analogously to the classical case of the mixed multiplicities of m-primary ideals in a local ring. We prove that the Minkowski inequalities hold for arbitrary filtrations. The characterization of equality in the Minkowski inequality for m-primary ideals in a local ring by Teissier, Rees and Sharp and Katz does not extend to arbitrary filtrations, but we show that they are true in a large and important subcategory of filtrations. We define divisorial and bounded filtrations. The filtration of powers of a fixed ideal is a bounded filtration, as is a divisorial filtration. We show that in an excellent local domain, the characterization of equality in the Minkowski equality is characterized by the condition that the integral closures of suitable Rees like algebras are the same, strictly generalizing the theorem of Teissier, Rees and Sharp and Katz. We also prove that a theorem of Rees characterizing the inclusion of ideals with the same multiplicity generalizes to bounded filtrations in excellent local domains. We give a number of other applications, extending classical theorems for ideals.

Kawamura, Hanamichi; Maesaka, Takumi; Seki, Shin-ichiro

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00297-w

We introduce the multivariable connected sum which is a generalization of Seki–Yamamoto’s connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating multivariable connected sums and for giving relations among special values of multiple polylogarithms. In particular, our class of relations contains Ohno’s relations for multiple polylogarithms.

Fornerod, Jean; Nguyen, Hoai-Minh

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00303-1

Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreteness of the set of eigenvalues of the transmission eigenvalue problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert–Schmidt operators.

Andruskiewitsch, N.; Angiono, I.; Pevtsova, J.; Witherspoon, S.

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00287-y

We show that the cohomology ring of a finite-dimensional complex pointed Hopf algebra with an abelian group of group-like elements is finitely generated. Our strategy has three major steps. We first reduce the problem to the finite generation of cohomology of finite dimensional Nichols algebras of diagonal type. For the Nichols algebras, we do a detailed analysis of cohomology via the Anick resolution reducing the problem further to specific combinatorial properties. Finally, to check these properties, we turn to the classification of Nichols algebras of diagonal type due to Heckenberger. In this paper, we complete the verification of these combinatorial properties for major parametric families, including Nichols algebras of Cartan and super types and develop all the theoretical foundations necessary for the case-by-case analysis. The remaining discrete families are addressed in a separate publication. As an application of the main theorem, we deduce finite generation of cohomology for other classes of finite-dimensional Hopf algebras, including basic Hopf algebras with abelian groups of characters and finite quotients of quantum groups at roots of one.

Gehrmann, Lennart

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00302-2

Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld’s upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note, we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon–Vonk can be computed in terms of L\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {L}$$\end{document}-invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups.

Sutthimat, Phiraphat; Rujivan, Sanae; Mekchay, Khamron; Rakwongwan, Udomsak

2022 Research in the Mathematical Sciences

doi: 10.1007/s40687-021-00309-9

This paper proposes an analytical formula for the conditional expectations of path-dependent product of polynomial and exponential function in the form of ∑j=0nλj(l)rtlje∑k=1mαk(l)rtk\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \left( \sum _{j=0}^{n}\lambda _j^{(l)}r_{t_l}^j\right) e\,^{\sum \limits _{k=1}^m\alpha _k^{(l)}r_{t_k}} \end{aligned}$$\end{document}for n,m∈N\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n,m \in \mathbb {N}$$\end{document}, l=1,2,...,m\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$l=1,2,...,m$$\end{document}, 0≤t1<t2<⋯<tm=T<∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0 \le t_{1}<t_{2}< \cdots<t_{m} = T <\infty $$\end{document} and λj(l),αk(l)∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lambda _{j}^{(l)}, \alpha _{k}^{(l)}\in \mathbb {R}$$\end{document}, where {rt}t∈[0,T]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{r_{t}\}_{t\in [0,T]}$$\end{document} corresponds to the extended Cox–Ingersoll–Ross (ECIR) process. The validation of the analytical formula is illustrated for several examples by comparing the results from the formula with those from Monte Carlo (MC) simulations. The efficiency of the formula is also presented via the computational run-times as compared with MC simulation. Moreover, the application of the analytical formula of this work is demonstrated for pricing arrears interest rate swaps under the ECIR process.