• Volume 127, Issue 5

      November 2017

    • Preface

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    • On the partition dimension of two-component graphs

      D O HARYENI E T BASKORO S W SAPUTRO M BACA A SEMANICOVÁ-FENOVCÍKOVÁ

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      In this paper, we continue investigating the partition dimension for disconnected graphs. We determine the partition dimension for some classes of disconnected graphs $G$ consisting of two components. If $G = G_{1} \cup G_{2}$, then we give the bounds of the partition dimension of $G$ for $G_{1} = P_{n}$ or $G_{1} = C_{n}$ and also for $pd(G_{1}) = pd(G_{2})$.

    • Role of associativity in Ramsey algebras

      ANDREW RAJAH WEN CHEAN TEH ZU YAO TEOH

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      It is known that semigroups are Ramsey algebras. This paper is an attempt to understand the role associativity plays in a binary system being a Ramsey algebra. Specifically,we show that the nonassociative Moufang loop of octonions is not a Ramsey algebra.

    • Some infinite families of Ramsey $(P_{3}, P_{n})$-minimal trees

      D RAHMADANI E T BASKORO M BACA H ASSIYATUN A SEMANICOVÁ-FENOVCÍKOVÁ

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      For any given two graphs G and H, the notation $F \rightarrow (G, H)$ means that for any red–blue coloring of all the edges of $F$ will create either a red subgraph isomorphic to $G$ or a blue subgraph isomorphic to $H$. A graph $F$ is a Ramsey $(G, H)$-minimal graph if $F \rightarrow (G, H)$ but $F − e \nrightarrow (G, H)$, for every $e \in E(F)$. The class of all Ramsey $(G, H)$-minimal graphs is denoted by $\mathcal{R}(G, H)$. In this paper, we construct some infinite families of trees belonging to $\mathcal{R}(P_{3}, P_{n})$, for $n = 8$ and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to $\mathcal{R}(P_{3}, P_{n})$, for $n \geq 10$.

    • Connected size Ramsey number for matchings vs. small stars or cycles

      BUDI RAHADJENG EDY TRI BASKORO HILDA ASSIYATUN

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      The notation $F \rightarrow (G, H)$ means that if the edges of $F$ are colored red and blue, then the red subgraph contains a copy of $G$ or the blue subgraph contains a copy of $H$. The connected size Ramsey number $\hat{r}_{c}(G, H)$ of graphs $G$ and $H$ is the minimum size of a connected graph $F$ satisfying $F \rightarrow (G, H)$. For $m \geq 2$, the graph consisting of $m$ independent edges is called a matching and is denoted by $mK_{2}$. In 1981, Erdös and Faudree determined the size Ramsey numbers for the pair $(mK_{2}, K_{1,t})$. They showed that the disconnected graph $mK_{1,t} \rightarrow (mK_{2}, K_{1,t})$ for $t$, $m \geq 1$. In this paper, we will determine the connected size Ramsey number $\hat{r}_{c}(nK_{2}, K_{1,3})$ for $n \geq 2$ and $\hat{r}_{c}(3K_{2},C_{4})$. We also derive an upper bound of the connected size Ramsey number $\hat{r}_{c}(nK_{2},C_{4})$, for $n \geq 4$.

    • Allowable graphs of the nonlinear Schrödinger equation and their applications

      BICH VAN NGUYEN

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      We construct the definition of allowable graphs of the nonlinear Schrödinger equation of arbitrary degree and use it to verify the separation and irreducibility (over the ring of integers) of the characteristic polynomials of all the possible graphs giving3-dimensional blocks of the normal form of the nonlinear Schrödinger equation. The method is purely algebraic and the obtained results will be useful in further studies of the nonlinear Schrödinger equation.

    • Qualitative behaviour of incompressible two-phase flows with phase transitions: The isothermal case

      JAN PRÜSS SENJO SHIMIZU

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      A thermodynamically consistent model for incompressible two-phase flows with phase transitions is considered mathematically. The model is based on first principles, i.e., balance of mass, momentum and energy. In the isothermal case, this problem is analysed to obtain local well-posedness, stability of non-degenerate equilibria, and global existence and convergence to equilibria of solutions which do not develop singularities.

    • Some functional inequalities on non-reversible Finsler manifolds

      SHIN-ICHI OHTA

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      We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by Ohta and Sturm. Following the approach of the $\Gamma$-calculus of Bakry $\it{et al}$ (2014), we show the dimensional versions of the Poincaré–Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti and Mondino (2015) in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

    • Maximal regularity for non-autonomous stochastic evolution equations in UMD Banach spaces

      TÔN VIÊT TA ATSUSHI YAGI YOSHITAKA YAMAMOTO

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      A non-autonomous stochastic linear evolution equation in UMD Banach spaces of type 2 is considered. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to a stochastic partial differential equation.

    • Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups

      MORITZ WEBER

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      This is a transcript of a series of eight lectures, 90min each, held at IMSc Chennai, India from 5–24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz’s quantum version of the Tannaka–Krein theorem. Building on this, we define Banica–Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions.We sketch the classification of Banica–Speicher quantum groups and we list some applications. We review the state-of-the-artregarding Banica–Speicher quantum groups and we list some open problems.

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