arXiv subjects

Eric Peterson

Publications and source records attributed to Eric Peterson.

12 recordsLinked to original sources

arXiv2025

p-adic congruences in iterated derivatives of the Weierstrass elliptic function

We use homotopy theoretic methods to prove congruence relations of number theoretic interest. Specifically, we use the theory of $\mathbb E_\infty$ complex orientations to establish $p$-adic K\"ummer congruences among iterated derivatives of the Weierstrass elliptic function. The machinery of Ando, Hopkins, and Rezk was developed with the intended application of taking congruence relations as input and producing $\mathbb E_\infty$-orientations as output. We run their machine in reverse, using as input the recent results of Carmeli and the first author on the existence of $\mathbb E_\infty$-orientations of Tate fixed-point objects.

math.NT↗

arXiv2022

Techniques for combining fast local decoders with global decoders under circuit-level noise

Implementing algorithms on a fault-tolerant quantum computer will require fast decoding throughput and latency times to prevent an exponential increase in buffer times between the applications of gates. In this work we begin by quantifying these requirements. We then introduce the construction of local neural network (NN) decoders using three-dimensional convolutions. These local decoders are adapted to circuit-level noise and can be applied to surface code volumes of arbitrary size. Their application removes errors arising from a certain number of faults, which serves to substantially reduce the syndrome density. Remaining errors can then be corrected by a global decoder, such as Blossom or Union Find, with their implementation significantly accelerated due to the reduced syndrome density. However, in the circuit-level setting, the corrections applied by the local decoder introduce many vertical pairs of highlighted vertices. To obtain a low syndrome density in the presence of vertical pairs, we consider a strategy of performing a syndrome collapse which removes many vertical pairs and reduces the size of the decoding graph used by the global decoder. We also consider a strategy of performing a vertical cleanup, which consists of removing all local vertical pairs prior to implementing the global decoder. Lastly, we estimate the cost of implementing our local decoders on Field Programmable Gate Arrays (FPGAs).

quant-ph↗

arXiv2022

There aren't that many Morava E-theories

Let $k$ be a perfect field of characteristic $p$. Associated to any (1-dimensional, commutative) formal group law of finite height $n$ over $k$ there is a complex oriented cohomology theory represented by a spectrum denoted $E(n)$ and commonly referred to as Morava $E$-theory. These spectra are known to admit $E_\infty$-structures, and the dependence of the $E_\infty$-structure on the choice of formal group law has been well studied (cf.\ [GH], [R], [L], Section 5, [PV]). In this note we show that the underlying homotopy type of $E(n)$ is independent of the choice of formal group law.

math.AT↗

arXiv2020

Fitting IVIM with Variable Projection and Simplicial Optimization

Fitting multi-exponential models to Diffusion MRI (dMRI) data has always been challenging due to various underlying complexities. In this work, we introduce a novel and robust fitting framework for the standard two-compartment IVIM microstructural model. This framework provides a significant improvement over the existing methods and helps estimate the associated diffusion and perfusion parameters of IVIM in an automatic manner. As a part of this work we provide capabilities to switch between more advanced global optimization methods such as simplicial homology (SH) and differential evolution (DE). Our experiments show that the results obtained from this simultaneous fitting procedure disentangle the model parameters in a reduced subspace. The proposed framework extends the seminal work originated in the MIX framework, with improved procedures for multi-stage fitting. This framework has been made available as an open-source Python implementation and disseminated to the community through the DIPY project.

eess.IV↗

arXiv2018

Cops, robbers, and burning bridges

We consider a variant of Cops and Robbers wherein each edge traversed by the robber is deleted from the graph. The focus is on determining the minimum number of cops needed to capture a robber on a graph $G$, called the {\em bridge-burning cop number} of $G$ and denoted $c_b(G)$. We determine $c_b(G)$ exactly for several elementary classes of graphs and give a polynomial-time algorithm to compute $c_b(T)$ when $T$ is a tree. We also study two-dimensional square grids and tori, as well as hypercubes, and we give bounds on the capture time of a graph (the minimum number of rounds needed for a single cop to capture a robber on $G$, provided that $c_b(G) = 1$).

math.CO↗

arXiv2018

Techniques for Efficiently Handling Power Surges in Fuel Cell Powered Data Centers: Modeling, Analysis, Results

Fuel cells are a promising power source for future data centers, offering high energy efficiency, low greenhouse gas emissions, and high reliability. However, due to mechanical limitations related to fuel delivery, fuel cells are slow to adjust to sudden increases in data center power demands, which can result in temporary power shortfalls. To mitigate the impact of power shortfalls, prior work has proposed to either perform power capping by throttling the servers, or to leverage energy storage devices (ESDs) that can temporarily provide enough power to make up for the shortfall while the fuel cells ramp up power generation. Both approaches have disadvantages: power capping conservatively limits server performance and can lead to service level agreement (SLA) violations, while ESD-only solutions must significantly overprovision the energy storage device capacity to tolerate the shortfalls caused by the worst-case (i.e., largest) power surges, which greatly increases the total cost of ownership (TCO). We propose SizeCap, the first ESD sizing framework for fuel cell powered data centers, which coordinates ESD sizing with power capping to enable a cost-effective solution to power shortfalls in data centers. SizeCap sizes the ESD just large enough to cover the majority of power surges, but not the worst-case surges that occur infrequently, to greatly reduce TCO. It then uses the smaller capacity ESD in conjunction with power capping to cover the power shortfalls caused by the worst-case power surges. As part of our new flexible framework, we propose multiple power capping policies with different degrees of awareness of fuel cell and workload behavior, and evaluate their impact on workload performance and ESD size. Using traces from Microsoft's production data center systems, we demonstrate that SizeCap significantly reduces the ESD size without violating any SLAs.

cs.DC↗

arXiv2016

Unit Hypercube Visibility Numbers of Trees

A visibility representation of a graph $G$ is an assignment of the vertices of $G$ to geometric objects such that vertices are adjacent if and only if their corresponding objects are "visible" each other, that is, there is an uninterrupted channel, usually axis-aligned, between them. Depending on the objects and definition of visibility used, not all graphs are visibility graphs. In such situations, one may be able to obtain a visibility representation of a graph $G$ by allowing vertices to be assigned to more than one object. The {\it visibility number} of a graph $G$ is the minimum $t$ such that $G$ has a representation in which each vertex is assigned to at most $t$ objects. In this paper, we explore visibility numbers of trees when the vertices are assigned to unit hypercubes in $\mathbb{R}^n$. We use two different models of visibility: when lines of sight can be parallel to any standard basis vector of $\mathbb{R}^n$, and when lines of sight are only parallel to the $n$th standard basis vector in $\mathbb{R}^n$. We establish relationships between these visibility models and their connection to trees with certain cubicity values.

math.CO↗

arXiv2013

A Relative Lubin-Tate Theorem via Meromorphic Formal Geometry

We formulate a theory of punctured affine formal schemes, suitable for certain problems within algebraic topology. As an application, we show that the Morava K-theoretic localizations of Morava E-theory corepresent a version of the Lubin-Tate moduli problem in this framework.

math.AT↗

arXiv2011

The Morava E-theory of Eilenberg-Mac Lane spaces

We deform the Ravenel-Wilson computation of the Morava K-homology of Eilenberg-Mac Lane spaces to obtain a similar description of their completed Morava E-homology. This yields both a cohomological description and an interpretation on the level of formal schemes: the scheme associated to the E-cohomology of the space K(Z/p^infty, q) is the qth exterior power of the p-divisible group associated to the versal infinitesimally deformed formal group over Lubin-Tate space.

math.AT↗

arXiv2011

Multiplicative 2-cocycles at the prime 2

Using a previous classification result on symmetric additive 2-cocycles, we collect a variety of facts about the Lubin-Tate cohomology of formal groups to compute the 2-primary component of the scheme of symmetric multiplicative 2-cocycles. This scheme classifies certain kinds of highly symmetric multiextensions, as studied in general by Mumford or Breen. A low-order version of this computation has previously found application in homotopy theory through the sigma-orientation of Ando, Hopkins, and Strickland, and the complete computation is reflective of certain structure found in the homotopy type of connective K-theory. This paper has been completely rewritten from its first posted draft, including a correction of the statement of the main result.

math.AT↗

arXiv2008

An Elementary Classification of Symmetric 2-Cocycles

We present a classification of the so-called "additive symmetric 2-cocycles" of arbitrary degree and dimension over Z/p, along with a partial result and some conjectures for m-cocycles over Z/p, m > 2. This expands greatly on a result originally due to Lazard and more recently investigated by Ando, Hopkins, and Strickland, which together with their work culminates in a complete classification of 2-cocycles over an arbitrary commutative ring. The ring classifying these polynomials finds application in algebraic topology, including generalizations of formal group laws and of cubical structures.

math.AC↗

arXiv2007

Cooperative Gating and Spatial Organization of Membrane Proteins through Elastic Interactions

Biological membranes are elastic media in which the presence of a transmembrane protein leads to local bilayer deformation. The energetics of deformation allow two membrane proteins in close proximity to influence each other's equilibrium conformation via their local deformations, and spatially organize the proteins based on their geometry. We use the mechanosensitive channel of large conductance (MscL) as a case study to examine the implications of bilayer-mediated elastic interactions on protein conformational statistics and clustering. The deformations around MscL cost energy on the order of 10 kT and extend ~3nm from the protein edge, as such elastic forces induce cooperative gating and we propose experiments to measure these effects. Additionally, since elastic interactions are coupled to protein conformation, we find that conformational changes can severely alter the average separation between two proteins. This has important implications for how conformational changes organize membrane proteins into functional groups within membranes.

q-bio.SC↗