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Wayne Lewis

Publications and source records attributed to Wayne Lewis.

7 recordsLinked to original sources
arXiv2024

Finite-Dimensional Protori Are Adelic Tori

We show the category of finite-dimensional compact connected abelian groups (finite-dimensional protori) {\it is} the category of {\it adelic tori}, for which there is a complete Lie theory: for each adelic torus $G$ there is a proper short exact sequence $0\ra\q^n\ra \mathcal{L}(G)\xrightarrow{\exp} G\ra 0$ with $\exp$ the {\it adelic exponential}. Extending the definition of nonarchimedean dimension to apply to number fields, the capstone result is the fact that all number fields have nonarchimedean dimension 1.

math.NT↗
arXiv2022

Digital Low-Level RF control system for Accumulator Ring at Advanced Light Source Upgrade Project

Currently ALS is undergoing an upgrade to ALSU to produce 100 times brighter soft X-ray light. The LLRF system for Accumulator Ring (AR) is composed of two identical LLRF stations, for driving RF amplifiers. The closed loop RF amplitude and phase stability is measured as $< 0.1\%$ and $< 0.1^\circ$ respectively, using the non-IQ digital down conversion together with analog up/down conversion, under a system-on-chip architecture. Realtime interlock system is implemented with $< 2 \mu$s latency, for machine protection against arc flash and unexpected RF power. Control interfaces are developed to enable PLC-FPGA-EPICS communication to support operation, timing, cavity tuning, and interlock systems. The LLRF system handles alignment of buckets to swap beams between AR and Storage Ring by synchronous phase loop ramping between the two cavities. The system also includes an optimization routine to characterize the loop dynamics and determine optimal operating point using a built-in network analyzer feature. A cavity emulator of 31 kHz bandwidth is integrated with the LLRF system to validate the performance of the overall system being developed.

physics.acc-ph↗
arXiv2019

Structure of Finite-Dimensional Protori

A Structure Theorem for Protori is derived for the category of finite-dimensional protori(compact connected abelian groups), which details the interplay between the properties of density, discreteness, torsion, and divisibility within a finite-dimensional protorus. The spectrum of resolutions for a finite-dimensional protorus are parameterized in the structure theorem by the dual category of finite rank torsion-free abelian groups. A consequence is a universal resolution for a finite-dimensional protorus, independent of a choice of a particular subgroup. A resolution is also given strictly in terms of the path component of the identity and the union of all zero-dimensional subgroups. The structure theorem is applied to show that a morphism of finite-dimensional protori lifts to a product morphism between products of periodic locally compact groups and real vector spaces.

math.GR↗
arXiv2019

Protori and Torsion-Free Abelian Groups

The Resolution Theorem for Compact Abelian Groups is applied to show that the profinite subgroups of a finite-dimensional compact connected abelian group (protorus) which induce tori quotients comprise a lattice under intersection (meet) and $+$ (join), facilitating a proof of the existence of a universal resolution. A finite rank torsion-free abelian group $X$ is algebraically isomorphic to a canonical dense subgroup $X_G$ of its Pontryagin dual $G$. A morphism between protori lifts to a product morphism between the universal covers, so morphisms in the category can be studied as pairs of maps: homomorphisms between finitely generated profinite abelian groups and linear maps between finite-dimensional real vector spaces. A concept of non-Archimedean dimension is introduced which acts a useful invariant for classifying protori.

math.GR↗
arXiv2018

The Lattice of Profinite Subgroups of Protori

Compact connected abelian groups, or protori, have intrinsic structural characteristics that present for the entire category. In the case of finite-dimensional torus-free protori, The Resolution Theorem for Compact Abelian Groups sets the stage for demonstrating that the profinite subgroups inducing tori quotients comprise an isogeny class of finitely generated modules over the profinite integers, which is a lattice under intersection (meet) and + (join). The structural results enable the formulation of a universal resolution in the category of protori under morphisms of compact abelian groups. A single profinite subgroup from the lattice in the Resolution Theorem is replaced by the direct limit of the lattice of such subgroups and effects a covering morphism in which the discrete torsion-free Pontryagin dual of the protorus organically emerges as the kernel of the quotient map resolving the protorus. Among other advantages, this enables the study of a finite rank torsion-free abelian group as a canonical subgroup of its dual protorus, with the concomitant topological, analytical, and number-theoretic insights availed by the compact abelian setting.

math.GR↗
arXiv2018

The Main Decomposition of Finite-Dimensional Protori

A protorus is a compact connected abelian group. We use a result on finite rank torsion-free abelian groups and Pontryagin Duality to considerably generalize a well-known factorization of a finite-dimensional protorus into a product of a torus and a torus-free complementary factor. We also classify by types the solenoids of Hewitt and Ross.

math.GR↗