Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

The singular continuous diffraction measure of the Thue-Morse chain

The paradigm for singular continuous spectra in symbolic dynamics and in mathematical diffraction is provided by the Thue-Morse chain, in its realisation as a binary sequence with values in $\{\pm 1\}$. We revisit this example and derive a functional equation together with an explicit form of the corresponding singular continuous diffraction measure, which is related to the known representation as a Riesz product.Comment: 6 pages, 1 figure; revised and improved versio

The Administration of Xultophy for Diabetic Patients on Hemodialysis

Background: Recent diabetic treatments include Insulin Degludec/ liraglutide (IDeg/Lira, Xultophy) in clinical practice. Authors have continued clinical research concerning diabetes, chronic renal failure, dialysis, and others. Subjects and Methods: Ten patients with type 2 diabetes mellitus (T2DM) undergoing hemodialysis were investigated. They showed that ages 74.5 ± 5.9 years, M/F=6/4, BMI 21.1± 3.8kg/m2, hemodialysis duration 8.1 ± 5.7 years. At the beginning, fundamental data were Cre 8.2 ± 1.9 mg/dL, HbA1c 6.5 ± 0.8%. Xultophy was started on 5-12 doses and continued for 6 months with the same or 1-4 increased doses for better glycemic variability. Results: Out of 10 subjects, the changes in HbA1c showed a decrease in 7, stable in 2, and an increase in 1. HbA1c value was 6.2 ± 0.8% in average at 6 months. There were no remarkable adverse effects by Xultophy for 6 months. Discussion and Conclusion: Xultophy was started at 5-12 doses, which were remarkably lower doses than usual doses with satisfactory efficacy. One of the reasons may be from the characteristic of the patients, who were diabetic with undergoing hemodialysis. Another factor is possibly from liraglutide, which has hepatic clearance with potential vascular protective effects. These results are expected to become reference data for future research

Evaluation of effective resistances in pseudo-distance-regular resistor networks

In Refs.[1] and [2], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on the stratification of the network and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity. In this paper, evaluation of effective resistances on more general networks called pseudo-distance-regular networks [21] or QD type networks \cite{obata} is investigated, where we use the stratification of these networks and show that the effective resistances between a given node such as $\alpha$ and all of the nodes $\beta$ belonging to the same stratum with respect to $\alpha$ ($R_{\alpha\beta^{(m)}}$, $\beta$ belonging to the $m$-th stratum with respect to the $\alpha$) are the same. Then, based on the spectral techniques, an analytical formula for effective resistances $R_{\alpha\beta^{(m)}}$ such that $L^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta}$ (those nodes $\alpha$, $\beta$ of the network such that the network is symmetric with respect to them) is given in terms of the first and second orthogonal polynomials associated with the network, where $L^{-1}$ is the pseudo-inverse of the Laplacian of the network. From the fact that in distance-regular networks, $L^{-1}_{\alpha\alpha}=L^{-1}_{\beta\beta}$ is satisfied for all nodes $\alpha,\beta$ of the network, the effective resistances $R_{\alpha\beta^{(m)}}$ for $m=1,2,...,d$ ($d$ is diameter of the network which is the same as the number of strata) are calculated directly, by using the given formula.Comment: 30 pages, 7 figure

Changes in Global Gene Expression in Response to Chemical and Genetic Perturbation of Chromatin Structure

DNA methylation is important for controlling gene expression in all eukaryotes. Microarray analysis of mutant and chemically-treated Arabidopsis thaliana seedlings with reduced DNA methylation revealed an altered gene expression profile after treatment with the DNA methylation inhibitor 5-aza-2′ deoxycytidine (5-AC), which included the upregulation of expression of many transposable elements. DNA damage-response genes were also coordinately upregulated by 5-AC treatment. In the ddm1 mutant, more specific changes in gene expression were observed, in particular for genes predicted to encode transposable elements in centromeric and pericentromeric locations. These results confirm that DDM1 has a very specific role in maintaining transcriptional silence of transposable elements, while chemical inhibitors of DNA methylation can affect gene expression at a global level

Conceptualisation, Development, Fabrication and In Vivo Validation of a Novel Disintegration Tester for Orally Disintegrating Tablets

Disintegration time is the key critical quality attribute for a tablet classed as an Orally Disintegrating Tablet (ODT). The currently accepted in vitro testing regimen for ODTs is the standard United States Pharmacopeia (USP) test for disintegration of immediate release tablets, which requires a large volume along with repeated submergence of the dosage form within the disintegration medium. The aim of this study was to develop an in vivo relevant ODT disintegration test that mimicked the environment of the oral cavity, including lower volume of disintegration medium, with relevant temperature and humidity that represent the conditions of the mouth. The results showed that the newly developed Aston test was able to differentiate between different ODTs with small disintegration time windows, as well as between immediate release tablets and ODTs. The Aston test provided higher correlations between ODT properties and disintegration time compared to the USP test method and most significantly, resulted in a linear in vitro/in vivo correlation (IVIVC) (R 2 value of 0.98) compared with a "hockey stick" profile of the USP test. This study therefore concluded that the newly developed Aston test is an accurate, repeatable, relevant and robust test method for assessing ODT disintegration time which will provide the pharmaceutical industry and regulatory authorities across the world with a pragmatic ODT testing regime

A quasi-sure non-degeneracy property for the Brownian rough path

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-selfintersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization

Nash embedding and equilibrium in pure quantum states

With respect to probabilistic mixtures of the strategies in non-cooperative games, quantum game theory provides guarantee of fixed-point stability, the so-called Nash equilibrium. This permits players to choose mixed quantum strategies that prepare mixed quantum states optimally under constraints. In this letter, we show that fixed-point stability of Nash equilibrium can also be guaranteed for pure quantum strategies via an application of the Nash embedding theorem, permitting players to prepare pure quantum states optimally under constraints.Comment: 7 pages, 1 figure. arXiv admin note: text overlap with arXiv:1609.0836

Property (T) and rigidity for actions on Banach spaces

We study property (T) and the fixed point property for actions on $L^p$ and other Banach spaces. We show that property (T) holds when $L^2$ is replaced by $L^p$ (and even a subspace/quotient of $L^p$), and that in fact it is independent of $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when 1. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.Comment: Many minor improvement