Local well-posedness for the nonlinear Schr\"odinger equation in the intersection of modulation spaces Mp,qs(Rd)∩M∞,1(Rd)M_{p, q}^s(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$ for $d \in \mathbb{N}$, $p, q \in [1, \infty]$ and $s \geq 0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the above intersection. This improves Theorem 1.1 by B\'enyi and Okoudjou (2009), where only the case $q = 1$ is considered, and closes a gap in the literature. If $q > 1$ and $s > d \left(1 - \frac{1}{q}\right)$ or if $q = 1$ and $s \geq 0$ then $M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty, 1}(\mathbb{R}^d)$ and the above intersection is superfluous. For this case we also reobtain a H\"older-type inequality for modulation spaces.Comment: 14 page

Expansion of anti-AFP Th1 and Tc1 responses in hepatocellular carcinoma occur in different stages of disease

Copyright @ 2010 Cancer Research UK. This work is licensed under the Creative Commons Attribution-NonCommercial-Share Alike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/.Background: α-Fetoprotein (AFP) is a tumour-associated antigen in hepatocellular carcinoma (HCC) and is a target for immunotherapy. However, there is little information on the pattern of CD4 (Th1) and CD8 (Tc1) T-cell response to AFP in patients with HCC and their association with the clinical characteristics of patients. Methods: We therefore analysed CD4 and CD8 T-cell responses to a panel of AFP-derived peptides in a total of 31 HCC patients and 14 controls, using an intracellular cytokine assay for IFN-γ. Results: Anti-AFP Tc1 responses were detected in 28.5% of controls, as well as in 25% of HCC patients with Okuda I (early tumour stage) and in 31.6% of HCC patients with stage II or III (late tumour stages). An anti-AFP Th1 response was detected only in HCC patients (58.3% with Okuda stage I tumours and 15.8% with Okuda stage II or III tumours). Anti-AFP Th1 response was mainly detected in HCC patients who had normal or mildly elevated serum AFP concentrations (P=0.00188), whereas there was no significant difference between serum AFP concentrations in these patients and the presence of an anti-AFP Tc1 response. A Th1 response was detected in 44% of HCC patients with a Child–Pugh A score (early stage of cirrhosis), whereas this was detected in only 15% with a B or C score (late-stage cirrhosis). In contrast, a Tc1 response was detected in 17% of HCC patients with a Child–Pugh A score and in 46% with a B or C score. Conclusion: These results suggest that anti-AFP Th1 responses are more likely to be present in patients who are in an early stage of disease (for both tumour stage and liver cirrhosis), whereas anti-AFP Tc1 responses are more likely to be present in patients with late-stage liver cirrhosis. Therefore, these data provide valuable information for the design of vaccination strategies against HCC.Association for International Cancer Research and Polkemmet Fund, London Clinic

Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology

This paper is an extended account of my "Introductory Plenary talk at Knots in Hellas 2016" conference We start from the short introduction to Knot Theory from the historical perspective, starting from Heraclas text (the first century AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram colorings (1956). In the second section we describe how Fox work was generalized to distributive colorings (racks and quandles) and eventually in the work of Jones and Turaev to link invariants via Yang-Baxter operators, here the importance of statistical mechanics to topology will be mentioned. Finally we describe recent developments which started with Mikhail Khovanov work on categorification of the Jones polynomial. By analogy to Khovanov homology we build homology of distributive structures (including homology of Fox colorings) and generalize it to homology of Yang-Baxter operators. We speculate, with supporting evidence, on co-cycle invariants of knots coming from Yang-Baxter homology. Here the work of Fenn-Rourke-Sanderson (geometric realization of pre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariants of links) will be discussed and expanded. Dedicated to Lou Kauffman for his 70th birthday.Comment: 35 pages, 31 figures, for Knots in Hellas II Proceedings, Springer, part of the series Proceedings in Mathematics & Statistics (PROMS

Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity

In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant $\Lambda>0$ in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection $A$ and the canonically conjugate triad field $e$. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of $A+=A + \sqrt\Lambda e$. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection $A_{\lambda}=A+\lambda e$ on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity. The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.Comment: 19 pages, references added. Published versio

Asynchronous optical sampling of on-chip terahertz devices for real-time sensing and imaging applications

We demonstrate that asynchronous optical sampling (ASOPS) can be used to measure the propagation of terahertz (THz) bandwidth pulses in a coplanar waveguide device with integrated photoconductive switches used for signal excitation and detection. We assess the performance of the ASOPS technique as a function of measurement duration, showing the ability to acquire full THz time-domain traces at rates up to 100 Hz. We observe a peak dynamic range of 40 dB for the shortest measurement duration of 10 ms, increasing to 88 dB with a measurement time of 500 s. Our work opens a route to real-time video-rate imaging via modalities using scanned THz waveguides, as well as real-time THz sensing of small volume analytes; we benchmark our on-chip ASOPS measurements against previously published simulations of scanning THz sensor devices, demonstrating sufficient dynamic range to underpin future video-rate THz spectroscopy measurements with these devices

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v

Persistence of Diophantine flows for quadratic nearly-integrable Hamiltonians under slowly decaying aperiodic time dependence

The aim of this paper is to prove a Kolmogorov-type result for a nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists of the possibility to choose an arbitrarily small decaying coefficient, consistently with the perturbation size.Comment: Several corrections in the proof with respect to the previous version. Main statement unchange

Spatially Resolved On-Chip Picosecond Pulse Detection Using Graphene

We present an on-chip time domain terahertz (TD-THz) system in which picosecond pulses are generated in low-temperature-grown gallium arsenide (LT-GaAs) and detected in graphene. The detected pulses were found to vary in amplitude, full width at half maximum (FWHM), and DC offset when sampled optically at different locations along a 50-μm-long graphene photoconductive (PC) detector. The results demonstrate the importance of detection location and switch design in graphene-based on-chip PC detectors

Knots and Particles

Using methods of high performance computing, we have found indications that knotlike structures appear as stable finite energy solitons in a realistic 3+1 dimensional model. We have explicitly simulated the unknot and trefoil configurations, and our results suggest that all torus knots appear as solitons. Our observations open new theoretical possibilities in scenarios where stringlike structures appear, including physics of fundamental interactions and early universe cosmology. In nematic liquid crystals and 3He superfluids such knotted solitons might actually be observed.Comment: 9 pages, 4 color eps figures and one b/w because of size limit (color version available from authors

The Combinatorics of Alternating Tangles: from theory to computerized enumeration

We study the enumeration of alternating links and tangles, considered up to topological (flype) equivalences. A weight $n$ is given to each connected component, and in particular the limit $n\to 0$ yields information about (alternating) knots. Using a finite renormalization scheme for an associated matrix model, we first reduce the task to that of enumerating planar tetravalent diagrams with two types of vertices (self-intersections and tangencies), where now the subtle issue of topological equivalences has been eliminated. The number of such diagrams with $p$ vertices scales as $12^p$ for $p\to\infty$. We next show how to efficiently enumerate these diagrams (in time $\sim 2.7^p$) by using a transfer matrix method. We give results for various generating functions up to 22 crossings. We then comment on their large-order asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200