Let $X$ be a bundle over $S^1$ with fiber 3--manifold $M$ and monodromy $\varphi$. Gay Kirby showed that if $\varphi$ fixes genus $g$ Heegaard splitting of then has $6g+1$ trisection. Genus $3g+1$ trisections have been found in certain special cases, such as the case where is trivial, it known lower than cannot exist general. We generalize these results to prove there exists trisection whenever...

the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

We introduce quantum Markov states (QMS) in a general tree graph $$G= (V, E)$$ , extending the Cayley tree’s case. investigate property w.r.t. finer structure of considered tree. The main result present paper concerns diagonalizability locally faithful QMS $$\varphi $$ on UHF-algebra $${\mathcal {A}}_V$$ over by means suitable conditional expectation into maximal abelian subalgebra. Namely, ...

The study on the shock formation and regularities of resulting surfaces for hyperbolic conservation laws is a basic problem in nonlinear partial differential equations. In this paper, we are concerned with optimal curves 1-D law $\partial_tu+\partial_xf(u)=0$ smooth initial data $u(0,x)=u_0(x)$. If $u_0(x)\in C^{1}(\Bbb R)$ $f(u)\in C^2(\Bbb R)$, it well-known that solution $u$ will blow up tim...

Glucose oxidation to CO(2) in man at the fasted, steady state has been investigated in normal, hypothyroid, patients by monitoring the specific activity of plasma glucose and expired CO(2) after intravenous injection of glucose-1-(14)C, glucose-6-(14)C, and sodium bicarbonate-(24)C in tracer amounts. Making certain stoichiometric assumptions about the oxidation of the C-1 and C-6 carbons of glu...

For a family of graphs G, the G-Contraction problem takes as an input graph G and integer k, goal is to decide if there exists subset F⊆E(G) size at most k such that G/F in G. Here, obtained from by contracting all edges F. In this article, we initiate study Grid Contraction parameterized complexity point view. We present fixed parameter tractable algorithm, running time 2O(k)⋅|V(G)|O(1), for p...

Abstract For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2</m...

A Hankel operator $\mathbf{H}_\varphi$ on the Hardy space $H^2$ of unit circle with analytic symbol $\varphi$ has minimal norm if $\|\mathbf{H}_\varphi\|=\|\varphi \|_2$ and maximal $\|\mathbf{H}_\varphi\| = \|\varphi\|_\infty$. The both only $|\varphi|$ is constant almost everywhere or, equivalently, a multiple an inner function. We show that norm-attaining norm, then norm. If continuous but n...

The paper deals with the problem of convergence branched continued fractions two branches branching which are used to approximate ratios Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. case real parameters $c\geq a\geq 0,$ b\geq $c\neq and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it proved fraction for z}\in G_{\bf h}$, where $G_{\bf h}$ two-dimensional disk. Using this r...