Local Properties via Color Energy Graphs and Forbidden Configurations

The local properties problem of Erdős and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants, by extending the color energy technique---a variant of the additive energy technique from additive combinatorics (color energy was originally introduced by the last two authors [C. Pohoata and A. Sheffer, Combinatorica, 39 (2019), pp. 705-714]). We generalize the concept of color energy to higher color energies and combine these with bounds on the extremal numbers of even cycles. Let f(n,k,l) denote the minimum number of colors required to color the edges of K_n such that every k vertices span at least l colors. It can be easily shown that f(n,k,(k/2)–[k/2]+2 = Θ(n²). Erdős and Gyárfás asked what happens when l = (k/2)-[k/2]+1, one away from the easy case, and derived the bound f(n,k,l) = Ω(n^(4/3)). Our technique significantly improves this to f(n,k,(k/2))–[k/2]+1) = Ω(n^(2-8/k))

Identifying and Evaluating Equity Provisions in State Health Care Reform

Identifies state policies that promote equitable access to quality health care and analyzes whether reform proposals in five states meet those equity benchmarks. Discusses innovative measures and the need to implement truly universal health insurance

An Ultrastructural Model to Test Microburst Stimulation of Nerves

Some patients that suffer from epilepsy may become refractory to pharmaceutical treatment. An option with these patients is vagus nerve stimulation (VNS) therapy with neuro-cybernetic medical devices. The purpose of this research is two-fold: 1.) to determine if a recovery technique can be used with formalin-fixed samples of nerve tissue for transmission electron microscopy (TEM) and 2.) to determine if there is an ultrastructural difference in tissue exposed to the neuro-cybernetic device. If successful, the study will reduce animal studies and expense by establishing a mechanism to perform retrospective TEM studies on formalin-fixed tissue. Additionally, TEM allows examination of specimens in much greater detail than light microscopy. Therefore, using TEM to compare ultrastructural differences in tissue that was exposed to the medical device and healthy tissue will help determine with precision if any damage is caused by the medical device. To complete these objectives, formalin-fixed vagus nerve tissue from goats that were exposed to the medical device is collected, recovered, evaluated by TEM, and compared to healthy traditionally fixed vagus nerve tissue. Results show that the recovery technique makes it possible to achieve quantitative data from formalin-fixed tissue samples. This method establishes a mechanism to execute retrospective TEM studies on formalin-fixed tissue, thereby reducing future animal studies. Results also show that there are some differences in goat nerve that has been exposed to the medical device. These subtle ultrastructural changes (potentially reversible) do not appear to have clinical impact

A Construction for Difference Sets with Local Properties

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size $k$ satisfies $|A'-A'|\ge k^{\log_2 3}$. This construction leads to the first non-trivial upper bound for the problem of distinct distances with local properties

Local Properties via Color Energy Graphs and Forbidden Configurations

The local properties problem of Erd\H{o}s and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants. We do this by continuing to develop the color energy technique --- a variant of the concept of additive energy from Additive Combinatorics. In particular, we generalize the concept of color energy to higher color energies, and combine these with Extremal Graph Theory results about graphs with no cycles or subdivisions of size $k$

An Ultrastructural Model to Test Microburst Stimulation of Nerves

Some patients that suffer from epilepsy may become refractory to pharmaceutical treatment. An option with these patients is vagus nerve stimulation (VNS) therapy with neuro-cybernetic medical devices. The purpose of this research is two-fold: 1.) to determine if a recovery technique can be used with formalin-fixed samples of nerve tissue for transmission electron microscopy (TEM) and 2.) to determine if there is an ultrastructural difference in tissue exposed to the neuro-cybernetic device. If successful, the study will reduce animal studies and expense by establishing a mechanism to perform retrospective TEM studies on formalin-fixed tissue. Additionally, TEM allows examination of specimens in much greater detail than light microscopy. Therefore, using TEM to compare ultrastructural differences in tissue that was exposed to the medical device and healthy tissue will help determine with precision if any damage is caused by the medical device. To complete these objectives, formalin-fixed vagus nerve tissue from goats that were exposed to the medical device is collected, recovered, evaluated by TEM, and compared to healthy traditionally fixed vagus nerve tissue. Results show that the recovery technique makes it possible to achieve quantitative data from formalin-fixed tissue samples. This method establishes a mechanism to execute retrospective TEM studies on formalin-fixed tissue, thereby reducing future animal studies. Results also show that there are some differences in goat nerve that has been exposed to the medical device. These subtle ultrastructural changes (potentially reversible) do not appear to have clinical impact

Extensions of Autocorrelation Inequalities with Applications to Additive Combinatorics

Barnard and Steinerberger [‘Three convolution inequalities on the real line with connections to additive combinatorics’, Preprint, 2019, arXiv:1903.08731] established the autocorrelation inequality Min_(0≤t≤1)∫_Rf(x)f(x+t) dx ≤ 0.411||f||²L¹, for fϵL¹(R), where the constant 0.4110.411 cannot be replaced by 0.370.37. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics. We show that for f to be extremal for this inequality, we must have max min_(x₁∈R 0≤t≤1)[f(x₁−t)+f(x₁+t)] ≤ min_max(x₂∈ R0≤t≤1)[f(x₂−t)+f(x₂+t)]. Our central technique for deriving this result is local perturbation of f to increase the value of the autocorrelation, while leaving ||f||L¹|| unchanged. These perturbation methods can be extended to examine a more general notion of autocorrelation. Let d, n∈Z⁺, f∈L¹, A be a d×n matrix with real entries and columns a_i for 1≤i≤n and C be a constant. For a broad class of matrices A, we prove necessary conditions for f to extremise autocorrelation inequalities of the form Min_(t∈ [0,1]^d)∫R∏_(i=1)^n f(x+t⋅a_i)dx≤C||f||^nL¹

Algorithmic Collusion by Large Language Models

The rise of algorithmic pricing raises concerns of algorithmic collusion. We conduct experiments with algorithmic pricing agents based on Large Language Models (LLMs), and specifically GPT-4. We find that (1) LLM-based agents are adept at pricing tasks, (2) LLM-based pricing agents autonomously collude in oligopoly settings to the detriment of consumers, and (3) variation in seemingly innocuous phrases in LLM instructions ("prompts") may increase collusion. These results extend to auction settings. Our findings underscore the need for antitrust regulation regarding algorithmic pricing, and uncover regulatory challenges unique to LLM-based pricing agents