On a problem of Erd\H{o}s and Rothschild on edges in triangles

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for C, as it is known that every N-vertex graph with more than (N^2)/4 edges contains an edge that is in at least N/6 triangles.Comment: 8 page

Local mapping of dissipative vortex motion

We explore, with unprecedented single vortex resolution, the dissipation and motion of vortices in a superconducting ribbon under the influence of an external alternating magnetic field. This is achieved by combing the phase sensitive character of ac-susceptibility, allowing to distinguish between the inductive-and dissipative response, with the local power of scanning Hall probe microscopy. Whereas the induced reversible screening currents contribute only inductively, the vortices do leave a fingerprint in the out-of-phase component. The observed large phase-lag demonstrates the dissipation of vortices at timescales comparable to the period of the driving force (i.e. 13 ms). These results indicate the presence of slow microscopic loss mechanisms mediated by thermally activated hopping transport of vortices between metastable states.Comment: 5 pages, 2 figure

Stock Market Manipulation and Its Regulation

More than eighty years after federal law first addressed stock market manipulation, federal courts remain fractured by disagreement and confusion about manipulation law\u27s most foundational questions. Only last year, plaintiffs petitioned the Supreme Court to resolve a sharp split among the federal circuits concerning manipulation law\u27s central question: whether trading activity alone can ever be considered illegal manipulation under federal law. Academics have been similarly confused economists and legal scholars cannot agree on whether manipulation is possible in principle; let alone on how, if it is, to address it properly in practice

High‐Frequency Trading and the New Stock Market: Sense And Nonsense

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/142455/1/jacf12260.pd

A methodology for distinguishing divergent cell fates within a common progenitor population: adenoma- and neuroendocrine-like cells are confounders of rat ileal epithelial cell (IEC-18) culture

BACKGROUND: IEC-18 cells are a non-transformed, immortal cell line derived from juvenile rat ileal crypt cells. They may have experimental advantages over tumor-derived gastrointestinal lineages, including preservation of phenotype, normal endocrine responses and retention of differentiation potential. However, their proclivity for spontaneous differentiation / transformation may be stereotypical and could represent a more profound experimental confounder than previously realized. We hypothesized that IEC-18 cells spontaneously diverge towards a uniform mixture of epigenetic fates, with corresponding phenotypes, rather than persist as a single progenitor lineage. RESULTS: IEC-18 cells were cultured for 72 hours in serum free media (SFM), with and without various insulin-like growth factor agonists to differentially boost the basal rate of proliferation. A strategy was employed to identify constitutive genes as markers of divergent fates through gene array analysis by cross-referencing fold-change trends for individual genes against crypt cell abundance in each treatment. We then confirmed the cell-specific phenotype by immunolocalization of proteins corresponding to those genes. The majority of IEC-18 cells in SFM alone had a loss in expression of the adenomatous polyposis coli (APC) gene at the mRNA and protein levels, consistent with adenoma-like transformation. In addition, a small subset of cells expressed the serotonin receptor 2A gene and had neuroendocrine-like morphology. CONCLUSIONS: IEC-18 cells commonly undergo a change in cell fate prior to reaching confluence. The most common fate switch that we were able to detect correlates with a down regulation of the APC gene and transformation into an adenoma-like phenotype

High‐Frequency Trading and the New Stock Market: Sense And Nonsense

The stock market has been transformed during the last 25 years. Human suppliers of liquidity like the NASDAQ dealers and NYSE specialists have been replaced by algorithmic market making; stocks that once traded on a single venue now trade across twelve exchanges and a multitude of alternative trading systems. New venues like dark pools, and new participants like high‐frequency traders, have emerged to take on prominent roles. This new market has had more than its share of controversy and regulatory scrutiny, particularly in the wake of Michael Lewis’s bestseller Flash Boys. In this article, the authors analyze five of the most controversial new market practices, including various high‐frequency trading strategies and dark pool activities. They set out a simple conceptual framework based on adverse selection and agency problems, and apply that framework to assess the welfare effects of each of the five practices. While much that is criticized is indeed objectionable, other controversial practices are much more complex than popularly imagined and may in fact be socially desirable. They conclude by evaluating a range of potential reforms to equity market structure

On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi and Trotter states that there exists a constant c(\Delta) such that r(H) \leq c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The important open question is to determine the constant c(\Delta). The best results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta \log^2 \Delta}. We improve this upper bound, showing that there is a constant c for which c(\Delta) \leq 2^{c \Delta \log \Delta}. The induced Ramsey number r_{ind}(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd\H{o}s conjectured the existence of a constant c such that, for any graph H on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in the exponent.Comment: 18 page

Low temperature series expansions for the square lattice Ising model with spin S > 1

We derive low-temperature series (in the variable $u = \exp[-\beta J/S^2]$) for the spontaneous magnetisation, susceptibility and specific heat of the spin-$S$ Ising model on the square lattice for $S=\frac32$, 2, $\frac52$, and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with $S$. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of $S$. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of $S$, though consistent with universality.Comment: 14 pages, LaTeX with IOP style files (ioplppt.sty), epic.sty and eepic.sty. To appear in J. Phys.

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page