Combinatorial Applications of the Subspace Theorem

The Subspace Theorem is a powerful tool in number theory. It has appeared in various forms and been adapted and improved over time. It's applications include diophantine approximation, results about integral points on algebraic curves and the construction of transcendental numbers. But its usefulness extends beyond the realms of number theory. Other applications of the Subspace Theorem include linear recurrence sequences and finite automata. In fact, these structures are closely related to each other and the construction of transcendental numbers. The Subspace Theorem also has a number of remarkable combinatorial applications. The purpose of this paper is to give a survey of some of these applications including sum-product estimates and bounds on unit distances. The presentation will be from the point of view of a discrete mathematician. We will state a number of variants of the Subspace Theorem below but we will not prove any of them as the proofs are beyond the scope of this work. However we will give a proof of a simplified special case of the Subspace Theorem which is still very useful for many problems in discrete mathematics

Extensions of a result of Elekes and R\'onyai

Many problems in combinatorial geometry can be formulated in terms of curves or surfaces containing many points of a cartesian product. In 2000, Elekes and R\'onyai proved that if the graph of a polynomial contains $cn^2$ points of an $n\times n\times n$ cartesian product in $\mathbb{R}^3$, then the polynomial has the form $f(x,y)=g(k(x)+l(y))$ or $f(x,y)=g(k(x)l(y))$. They used this to prove a conjecture of Purdy which states that given two lines in $\mathbb{R}^2$ and $n$ points on each line, if the number of distinct distances between pairs of points, one on each line, is at most $cn$, then the lines are parallel or orthogonal. We extend the Elekes-R\'onyai Theorem to a less symmetric cartesian product. We also extend the Elekes-R\'onyai Theorem to one dimension higher on an $n\times n\times n\times n$ cartesian product and an asymmetric cartesian product. We give a proof of a variation of Purdy's conjecture with fewer points on one of the lines. We finish with a lower bound for our main result in one dimension higher with asymmetric cartesian product, showing that it is near-optimal.Comment: 23 page

Local Peer-to-Peer Communication to Improve Demand Response in Residential Neighborhoods

Increased electricity production from intermittent renewables presents a challenge to utilities. Since the grid has little ability to store energy, fluctuations in solar and wind generation require either an increase in generation from expensive sources or a reduction of demand. Demand Response (DR) programs focus on the latter and are designed to increase grid flexibility by allowing grid operators to modify when or how customers use electricity. For residential customers, this typically means shedding load during periods of high demand through a central controller temporarily shutting off air conditioning (AC) compressors. This type of DR can cause spikes in demand after the units come back online. As the communication and computational capabilities of smart meters and smart thermostats grow, so does the potential to create more decentralized approaches to DR programs. This thesis presents novel thermostat on/off criteria that rely on limited peer to peer communication between a network of residential thermostats. Agent based modeling (ABM) software was used to simulate the emergent behavior that results from thermostat interactions. To demonstrate the benefit of communicating thermostats, the criteria were tested as a means to improve the response following an AC shut off DR event and as an alternative to such events. The introduced criteria, by sharing only the state of neighboring compressors, improved the overall demand profile following a DR event by reducing peak demand up to 21%. However, it was also found to increase the number of cycles an AC unit experiences by 36%, which can reduce its lifetime. Additionally, the stability implications of this approach are explored

Guidelines for Implementing MODEM: An Open-Source, MATLAB-Based Digital Image Correlation Software

MODEM is an open-source, MATLAB-based digital image correlation (DIC) program that was developed at the University of Auckland for small-scale testing of flexible materials. Structural engineering researchers at both the University of Auckland and Cal Poly – San Luis Obispo wanted to expand the uses of the program to study the seismic response of large-scale test specimens. This guide document describes how to implement DIC using MODEM, including the hardware and software needed to run an experiment as well as data collection and post-processing procedures for the program. Additionally, this document includes a case study focusing on a DIC test program consisting of several aluminum coupons subjected to pure tension. A summary of MODEM’s output from one of these tests informs future users of the benefits and pitfalls that can occur while running DIC experiments and prepares them to use this program in their experiments. Furthermore, this work demonstrates that researchers can accurately quantify the full-field deformation of structures at a localized scale and utilize this data to corroborate traditional instrumentation like strain gages and linear potentiometers as well as to calibrate computational finite element models

Simultaneous Arithmetic Progressions on Algebraic Curves

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.Comment: 11 pages, 6 figures, order of email addresses corrected 12 pages, closing remarks, a reference and an acknowledgment adde

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Imaging dielectric relaxation in nanostructured polymers by frequency modulation electrostatic force microscopy

We have developed a method for imaging the temperature-frequency dependence of the dynamics of nanostructured polymer films with spatial resolution. This method provides images with dielectric compositional contrast well decoupled from topography. Using frequency-modulation electrostatic-force-microscopy, we probe the local frequency-dependent (0.1–100 Hz) dielectric response through measurement of the amplitude and phase of the force gradient in response to an oscillating applied electric field. When the phase is imaged at fixed frequency, it reveals the spatial variation in dielectric losses, i.e., the spatial variation in molecular/dipolar dynamics, with 40 nm lateral resolution. This is demonstrated by using as a model system; a phase separated polystyrene/polyvinyl-acetate (PVAc) blend. We show that nanoscale dynamic domains of PVAc are clearly identifiable in phase images as those which light-up in a band of temperature, reflecting the variations in the molecular/dipolar dynamics approaching the glass transition temperature of PVAc

A Preliminary Investigation of Whether High Resolution Cervical Auscultation Signals Present Variations Between Thin Liquid Barium and Water Swallows

Dysphagia, commonly referred to as abnormal swallowing, affects millions of people annually. If not diagnosed expeditiously, dysphagia can lead to more severe complications, such as pneumonia, nutritional deficiency, and dehydration. Bedside screening is the first step of dysphagia characterization and is usually based on pass/fail tests in which a nurse observes the patient performing water swallows to look for overt signs of dysphagia such as coughing. Though quick and convenient, bedside screening provides low-level judgment of impairment, lacks standardization, and suffers from subjectivity. Recently, high resolution cervical auscultation (HRCA) has been investigated as a less expensive and non-invasive method to diagnose dysphagia. It has shown strong preliminary evidence of its effectiveness in penetration-aspiration detection as well as multiple swallow kinematics. HRCA signals have been investigated in conjunction with videofluoroscopy exams performed using barium boluses. An HRCA-based bedside screening is highly desirable to expedite initial dysphagia diagnosis and overcome all drawbacks of current pass/fail screening tests. However, all research conducted using HRCA in dysphagia is based on thin liquid barium boluses and thus not guaranteed to provide valid results for water boluses. If HRCA signals show no significant differences between water and thin liquid barium boluses, then the same algorithms developed from thin liquid barium can be directly applied with water. This study investigates the similarities and differences between HRCA signals from thin liquid barium swallows and water swallows. Multiple features from the time, frequency, time-frequency, and information-theoretic domain were extracted from each type of swallow, and a group of linear mixed models was tested to determine the significance of differences. Machine learning classifiers were fit to the data as well to determine if the swallowed material (thin liquid barium or water) can be correctly predicted from an unlabeled set of HRCA signals. The results demonstrated no systematic difference between the HRCA signals of thin liquid barium swallows and water swallows. While no systematic difference exists, the evidence of complete conformity between HRCA signals of both materials was inconclusive. These results must be validated further to demonstrate similarity between the HRCA signals of thin liquid barium swallows and water swallows