arXiv subjects

Michael Shapiro

Publications and source records attributed to Michael Shapiro.

At least 19 recordsLinked to original sources

arXiv2026

A generalized cluster structure on $GL_n$ via birational Poisson maps

In a recent work, we constructed a rational map from a simple Lie group $\mathcal G$ to itself that intertwines the standard Poisson--Lie structure on $\mathcal G$ with a Poisson homogeneous one defined by a pair of quasi-triangular solutions to the classical Yang--Baxter equation (CYBE) known as R-matrices. We also showed, in the case of $SL_n$, that if the combinatorial Belavin--Drinfeld data associated with these R-matrices satisfies certain aperiodicity conditions, the map is, in fact, birational and can be used to obtain an initial cluster for an exotic cluster structure on $SL_n$ via the pullback of Berenstein--Fomin--Zelevinsky cluster variables. The same strategy was later used by the first author and D.~Voloshyn to describe generalized cluster structures compatible with the Poisson dual of the Poisson--Lie bracket defined by a quasi-triangular R-matrix. In this paper we further promote the use of birational Poisson maps in constructing generalized cluster structures by applying it in the situation when the aperiodicity condition is not satisfied. To this end, we describe a generalized cluster structure on $GL_n$ compatible with the Poisson homogeneous bracket defined by two Cremmer--Gervais solutions to the CYBE related via conjugation by the longest element of the Weyl group. The key ingredient to our construction is a birational map that connects the bracket under consideration with two other Poisson brackets: the Poisson dual to Cremmer--Gervais Poisson--Lie bracket on $GL_{n-1}$ and the bracket on a certain space of complex rational functions of one variable closely related to cluster algebraic interpretation of Coxeter--Toda flows. New notions of a regular pullback of a seed and of an almost-cluster structure whose detailed description are given in a separate note also play an important role in our construction.

math.QA↗

arXiv2026

Regular pullback of generalized cluster structures

We consider the problem of lifting a regular cluster structure on a quasi-affine variety to the ambient affine space and a similar problem of defining a regular pullback of a regular cluster structure under a dominant rational map between affine spaces. We provide sufficient conditions for the existence of the corresponding object, called an almost-cluster structure, study its combinatorics, compatible Poisson bracket and the corresponding upper cluster algebra.

math.AC↗

arXiv2025

Analytical and Numerical Studies of Dark Current in Radiofrequency Structures for Short-Pulse High-Gradient Acceleration

High-gradient acceleration is a key research area that could enable compact linear accelerators for future colliders, light sources, and other applications. In the pursuit of high-gradient operation, RF breakdown limits the attainable accelerating gradient in normal-conducting RF structures. Recent experiments at the Argonne Wakefield Accelerator suggest a promising approach: using short RF pulses with durations of a few nanoseconds. Experimental studies show that these O(1 ns) RF pulses can mitigate breakdown limitations, resulting in higher gradients. For example, an electric field of nearly 400 MV/m was achieved in an X-band photoemission gun driven by 6-ns-long RF pulses, with rapid RF conditioning and low dark current observed. Despite these promising results, the short-pulse regime remains an under-explored parameter space, and RF breakdown physics under nanosecond-long pulses requires further investigation. In this paper, we present analytical and numerical simulations of dark current dynamics in accelerating cavities operating in the short-pulse regime. We study breakdown-associated processes spanning different time scales, including field emission, multipacting, and plasma formation, using simulations of the X-band photogun cavities. The results reveal the advantages of using short RF pulses to reduce dark current and mitigate RF breakdown, offering a path toward a new class of compact accelerators with enhanced performance and reduced susceptibility to breakdown.

physics.acc-ph↗

arXiv2024

Punctured surfaces, quiver mutations, and quotients of Coxeter groups

In 2011, Barot and Marsh provided an explicit construction of presentation of a finite Weyl group $W$ by any quiver mutation-equivalent to an orientation of a Dynkin diagram with Weyl group $W$. The construction was extended by the authors of the present paper to obtain presentations for all affine Coxeter groups, as well as to construct groups from triangulations of unpunctured surfaces and orbifolds, where the groups are invariant under change of triangulation and thus are presented as quotients of numerous distinct Coxeter groups. We extend the construction to include most punctured surfaces and orbifolds, providing a new invariant for almost all marked surfaces.

math.CO↗

arXiv2024

Institutional-Level Monitoring of Immune Checkpoint Inhibitor IrAEs Using a Novel Natural Language Processing Algorithmic Pipeline

Background: Immune checkpoint inhibitors (ICIs) have revolutionized cancer treatment but can result in severe immune-related adverse events (IrAEs). Monitoring IrAEs on a large scale is essential for personalized risk profiling and assisting in treatment decisions. Methods: In this study, we conducted an analysis of clinical notes from patients who received ICIs at the Tel Aviv Sourasky Medical Center. By employing a Natural Language Processing algorithmic pipeline, we systematically identified seven common or severe IrAEs. We examined the utilization of corticosteroids, treatment discontinuation rates following IrAEs, and constructed survival curves to visualize the occurrence of adverse events during treatment. Results: Our analysis encompassed 108,280 clinical notes associated with 1,635 patients who had undergone ICI therapy. The detected incidence of IrAEs was consistent with previous reports, exhibiting substantial variation across different ICIs. Treatment with corticosteroids varied depending on the specific IrAE, ranging from 17.3% for thyroiditis to 57.4% for myocarditis. Our algorithm demonstrated high accuracy in identifying IrAEs, as indicated by an area under the curve (AUC) of 0.89 for each suspected note and F1 scores of 0.87 or higher for five out of the seven IrAEs examined at the patient level. Conclusions: This study presents a novel, large-scale monitoring approach utilizing deep neural networks for IrAEs. Our method provides accurate results, enhancing understanding of detrimental consequences experienced by ICI-treated patients. Moreover, it holds potential for monitoring other medications, enabling comprehensive post-marketing surveillance to identify susceptible populations and establish personalized drug safety profiles.

cs.CL↗

arXiv2024

Cluster structure on genus 2 spherical DAHA: seven-colored flower

We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type $A_1$ as sum over lattice points in a convex polyhedron in 7-dimensional space.

math.RT↗

arXiv2023

Non-deterministic linear thresholding systems reveal their deterministic origins

Linear thresholding systems have been used as a model of neural activation and have more recently been proposed as a model of gene activation. Deterministic linear thresholding systems can be turned into non-deterministic systems by the introduction of noise. Under mild conditions on the noise, we show that the deterministic model can be deduced from the probabilities of the non-deterministic model.

q-bio.NC↗

arXiv2023

Long cycles in linear thresholding systems

Linear thresholding systems have been used as a model of neural activation and more recently proposed as a model of gene regulation. Here we exhibit linear thresholding systems whose dynamics produce surprisingly long cycles.

q-bio.NC↗

arXiv2023

A unified approach to exotic cluster structures on simple Lie groups

We propose a new approach to building log-canonical coordinate charts for any simply-connected simple Lie group $\G$ and arbitrary Poisson-homogeneous bracket on $\G$ associated with Belavin--Drinfeld data. Given a pair of representatives $r, r'$ from two arbitrary Belavin--Drinfeld classes, we build a rational map from $\G$ with the Poisson structure defined by two appropriately selected representatives from the standard class to $\G$ equipped with the Poisson structure defined by the pair $r, r'$. In the $A_n$ case, we prove that this map is invertible whenever the pair $r, r'$ is drawn from aperiodic Belavin--Drinfeld data, as defined in~\cite{GSVple}. We further apply this construction to recover the existence of a regular complete cluster structure compatible with the Poisson structure associated with the pair $r, r'$ in the aperiodic case.

math.QA↗

arXiv2023

Symplectic groupoid and cluster algebras

We consider the symplectic groupoid of pairs $(B,\mathbb{A})$ with $\mathbb A$ unipotent upper-triangular matrices and $B\in GL_n$ being such that $\widetilde {\mathbb A}=B{\mathbb A} B^{\text{T}}$ are also unipotent upper-triangular matrices. We explicitly solve this groupoid condition using Fock--Goncharov--Shen cluster variables and show that for $B$ satisfying the standard semiclassical Lie--Poisson algebra, the matrices $B$, $\mathbb A$, and $\widetilde{\mathbb A}$ satisfy the closed Poisson algebra relations expressible in the $r$-matrix form. Identifying entries of $\mathbb A$ and $\widetilde {\mathbb A}$ with geodesic functions for geodesics on the two halves of a closed Riemann surface of genus $g=n-1$ separated by the Markov element, we are able to construct the geodesic function $G_B$ ``dual'' to the Markov element. We thus obtain the complete cluster algebra description of Teichm\"uller space $\mathcal T_{2,0}$ of genus two. We discuss also the generalization of our construction for higher genera. For genus larger than three we need a Hamiltonian reduction based on the rank condition $\hbox{rank\,}({\mathbb A}+{\mathbb A}^{\text{T}})\le 4$; we present the example of such a reduction for $\mathcal T_{4,0}$.

math.QA↗

arXiv2022

Introducing isodynamic points for binary forms and their ratios

The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Mobius group on the set of triangles. Generalizing this classical result, we introduce below the isodynamic map associating to a univariate polynomial of degree d at least 3 with at most double roots a polynomial of degree (at most) 2d-4 such that this map commutes with the action of the Mobius group on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called the isodynamic points of the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

math.CV↗

arXiv2022

Advanced RF Structures for Wakefield Acceleration and High-Gradient Research

Structure wakefield acceleration (SWFA) is one of the most promising AAC schemes in several recent strategic reports, including DOE's 2016 AAC Roadmap, report on the Advanced and Novel Accelerators for High Energy Physics Roadmap (ANAR), and report on Accelerator and Beam Physics Research Goals and Opportunities. SWFA aims to raise the gradient beyond the limits of conventional radiofrequency (RF) accelerator technology, and thus the RF to beam energy efficiency, by reducing RF breakdowns from confining the microwave energy in a short (on the order of about 10 ns) and intense pulse excited by a drive beam. We envision that the following research topics, within the scope of AF7, are of great interest in the next decade: advanced wakefield structures, terahertz and sub-terahertz (THz) structures, and RF breakdown physics. Research on SWFA in the above directions would directly contribute to long-term large-scale applications, including AAC-based linear colliders and compact light sources. There is also potentially a strong synergy between SWFA and other AAC concepts, when structures are combined with plasmas into hybrid AAC schemes. Research on novel structures is at the core of advancing SWFA, and is critical to future AAC-based linear colliders; at the same, it has a strong synergy with other directions, such as cavity designs, high-power microwave systems and sources, and compact light sources.

physics.acc-ph↗

arXiv2021

Finiteness of rank for Grassmann convexity

The Grassmann convexity conjecture gives a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. The conjecture can be reformulated in terms of convex curves in the nilpotent lower triangular group. The formula has already been shown to be a correct lower bound and to give a correct upper bound in several small dimensional cases. In this paper we obtain a general explicit upper bound.

math.CA↗

arXiv2021

Characteristic equation for symplectic groupoid and cluster algebras

We use the Darboux coordinate representation found by two of the authors (L.Ch. and M.Sh.) for entries of general symplectic leaves of the $\mathcal A_n$-groupoid of upper-triangular matrices to express roots of the characteristic equation $\det(\mathbb A-\lambda \mathbb A^{\text{T}})=0$, with $\mathbb A\in \mathcal A_n$, in terms of Casimirs of this Darboux coordinate representation, which is based on cluster variables of Fock--Goncharov higher Teichm\"uller spaces for the algebra $sl_n$. We show that roots of the characteristic equation are simple monomials of cluster Casimir elements. This statement remains valid in the quantum case as well. We consider a generalization of $\mathcal A_n$-groupoid to a $\mathcal A_{Sp_{2m}}$-groupoid.

math.RT↗

arXiv2020

Cluster algebras from surfaces and extended affine Weyl groups

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with every triangulation a basis in $V$, such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type $A$, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types $E$.

math.CO↗

arXiv2020

Generalized Cluster Structures Related to the Drinfeld Double of $GL_n$

We prove that the regular generalized cluster structure on the Drinfeld double of $GL_n$ constructed in arXiv:1912.00453 is complete and compatible with the standard Poisson--Lie structure on the double. Moreover, we show that for $n=4$ this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in arXiv:1912.00453 possesses similar compatibility and completeness properties.

math.RT↗

arXiv2019

Periodic staircase matrices and generalized cluster structures

As is well-known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plucker relations, Desnanot--Jacobi identities and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin--Drinfeld Poisson--Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators.

math.AC↗

arXiv2019

Grassmann convexity and multiplicative Sturm theory, revisited

In this paper we settle a special case of the Grassmann convexity conjecture formulated earlier by B.and M.Shapiro. We present a conjectural formula for the maximal total number of real zeros of the consecutive Wronskians of an arbitrary fundamental solution to a disconjugate linear ordinary differential equation with real time. We show that this formula gives the lower bound for the required total number of real zeros for equations of an arbitrary order and, using our results on the Grassmann convexity, we prove that the aforementioned formula is correct for equations of orders $4$ and $5$.

math.CA↗