Dimension and cut vertices: an application of Ramsey theory

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$ is at most $d$ whenever the cover graph of $B$ is a block of the cover graph of $P$, then the dimension of $P$ is at most $d+2$. We also construct examples which show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.Comment: Final published version with updated reference

Dimension of posets with planar cover graphs excluding two long incomparable chains

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.Comment: New section on connections with graph minors, small correction

Tree-width and dimension

Over the last 30 years, researchers have investigated connections between dimension for posets and planarity for graphs. Here we extend this line of research to the structural graph theory parameter tree-width by proving that the dimension of a finite poset is bounded in terms of its height and the tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph

Triangle-free geometric intersection graphs with large chromatic number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.Comment: Small corrections, bibliography updat

Telling time with an intrinsically noisy clock

Intracellular transmission of information via chemical and transcriptional networks is thwarted by a physical limitation: the finite copy number of the constituent chemical species introduces unavoidable intrinsic noise. Here we provide a method for solving for the complete probabilistic description of intrinsically noisy oscillatory driving. We derive and numerically verify a number of simple scaling laws. Unlike in the case of measuring a static quantity, response to an oscillatory driving can exhibit a resonant frequency which maximizes information transmission. Further, we show that the optimal regulatory design is dependent on the biophysical constraints (i.e., the allowed copy number and response time). The resulting phase diagram illustrates under what conditions threshold regulation outperforms linear regulation.Comment: 10 pages, 5 figure

Particle Physics in the LHC era

This book gives a modern introduction to particle physics. The main mathematical tools required for the rest of the book are developed in Chapter 2. A quantitative introduction to accelerator physics is presented in Chapter 3. Chapter 4 covers detector physics, with an emphasis on fundamental physical principles. Chapter 5 covers the static quark model, with applications to light mesons and baryons as well as heavier states containing charm and beauty quarks. Chapter 6 introduces relativistic quantum mechanics and uses spinors to relate Lorentz invariance to the Dirac equation. Chapter 7 covers the basics of the electroweak theory based on broken SU(2) × U(1) symmetry. Chapter 8 reviews some of the key experiments that led to the development of the electroweak theory. Chapter 9 explains the importance of deep inelastic scattering data for providing direct evidence for the existence of quarks. It also gives a brief introduction to quantum chromodynamics (QCD). Chapter 10 considers flavour oscillations in the quark sector and then discusses the evidence for CP violation. Chapter 11 examines the theory of neutrino oscillations as well as the evidence for these oscillations. Chapter 12 gives an elementary introduction to the Higgs mechanism as well as a careful explanation of the experimental evidence for the existence of a Higgs boson. Chapter 13 looks at LHC physics and explains how searches for Beyond the Standard Model Physics are performed. It concludes with a discussion of the evidence for dark matter and dark energy

Maximum entropy models for antibody diversity

Recognition of pathogens relies on families of proteins showing great diversity. Here we construct maximum entropy models of the sequence repertoire, building on recent experiments that provide a nearly exhaustive sampling of the IgM sequences in zebrafish. These models are based solely on pairwise correlations between residue positions, but correctly capture the higher order statistical properties of the repertoire. Exploiting the interpretation of these models as statistical physics problems, we make several predictions for the collective properties of the sequence ensemble: the distribution of sequences obeys Zipf's law, the repertoire decomposes into several clusters, and there is a massive restriction of diversity due to the correlations. These predictions are completely inconsistent with models in which amino acid substitutions are made independently at each site, and are in good agreement with the data. Our results suggest that antibody diversity is not limited by the sequences encoded in the genome, and may reflect rapid adaptation to antigenic challenges. This approach should be applicable to the study of the global properties of other protein families

A mesoscopic ring as a XNOR gate: An exact result

We describe XNOR gate response in a mesoscopic ring threaded by a magnetic flux $\phi$. The ring is attached symmetrically to two semi-infinite one-dimensional metallic electrodes and two gate voltages, viz, $V_a$ and $V_b$, are applied in one arm of the ring which are treated as the inputs of the XNOR gate. The calculations are based on the tight-binding model and the Green's function method, which numerically compute the conductance-energy and current-voltage characteristics as functions of the ring-to-electrode coupling strength, magnetic flux and gate voltages. Our theoretical study shows that, for a particular value of $\phi$ ($=\phi_0/2$) ($\phi_0=ch/e$, the elementary flux-quantum), a high output current (1) (in the logical sense) appears if both the two inputs to the gate are the same, while if one but not both inputs are high (1), a low output current (0) results. It clearly exhibits the XNOR gate behavior and this aspect may be utilized in designing an electronic logic gate.Comment: 8 pages, 5 figure

Laserwire at the Accelerator Test Facility 2 with Sub-Micrometre Resolution

A laserwire transverse electron beam size measurement system has been developed and operated at the Accelerator Test Facility 2 (ATF2) at KEK. Special electron beam optics were developed to create an approximately 1 x 100 {\mu}m (vertical x horizontal) electron beam at the laserwire location, which was profiled using a 150 mJ, 71 ps laser pulse with a wavelength of 532 nm. The precise characterisation of the laser propagation allows the non-Gaussian transverse profiles of the electron beam caused by the laser divergence to be deconvolved. A minimum vertical electron beam size of 1.07 ${\pm}$ 0.06 (stat.) ${\pm}$ 0.05 (sys.) {\mu}m was measured. A vertically focussing quadrupole just before the laserwire was varied whilst making laserwire measurements and the projected vertical emittance was measured to be 82.56 ${\pm}$ 3.04 pm rad.Comment: 17 pages, 26 figures, submitted to Phys. Rev. ST Accel. Beam

First instar larvae of endemic Australian Miltogramminae (Diptera: Sarcophagidae)

The first instar larva of a species of the Australian endemic genus Aenigmetopia Malloch is described for the first time, along with the first instar larvae of three other Australian species representing the genera Amobia Robineau-Desvoidy and Protomiltogramma Townsend. Larval morphology was analysed using a combination of light microscopy, confocal laser scanning microscopy and scanning electron microscopy. The following morphological structures are documented: pseudocephalon, antennal complex, maxillary palpus, facial mask, modifications of thoracic and abdominal segments, anal region, spiracular field, posterior spiracles and details of the cephaloskeleton. Substantial morphological differences are observed between the three genera, most notably in the labrum and mouthhooks of the cephaloskeleton, sensory organs of the pseudocephalon, spinulation, sculpture of the integument and form of the spiracular field. The first instar larval morphology of Aenigmetopia amissa Johnston, Wallman, Szpila & Pape corroborates the close phylogenetic affinity of Aenigmetopia Malloch with Metopia Meigen, inferred from recent molecular analysis. The larval morphology of Amobia auriceps (Baranov), Protomiltogramma cincta Townsend and Protomiltogramma plebeia Malloch is mostly congruent with the morphology of Palaearctic representatives of both genera