We define the $S^1$-equivariant Rabinowitz--Floer homology of a bounding contact hypersurface $\Sigma$ in an exact symplectic manifold, and show by a geometric argument that it vanishes if $\Sigma$ is displaceable.

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for operator convex functions on bounded intervals. More precisely, we prove that if $f$ is a nonlinear operator convex function on a bounded interval $(a,b)$ and $A, B$ are bounded linear operators acting on a Hilbert space with spectra in $(a,b)$ and $A-B$ is invertible, then $sf(A)+(1-s)f(B)>f(sA+(1-s)B)$. A short proof for a similar known result concerning a nonconstant operator monotone function on $[0,\infty)$ is presented. Another purpose is to find a lower bound for $f(A)-f(B)$, where $f$ is a nonconstant operator monotone function, by using a key lemma. We also give an estimation of the Furuta inequality, which is an excellent extension of the L\"owner--Heinz inequality.

The solvable graph of a finite group $G$, denoted by ${\Gamma}_{\rm s}(G)$, is a simple graph whose vertices are the prime divisors of $|G|$ and two distinct primes $p$ and $q$ are joined by an edge if and only if there exists a solvable subgroup of $G$ such that its order is divisible by $pq$. Let $p_1<p_2<\cdots<p_k$ be all prime divisors of $|G|$ and let ${\rm D}_{\rm s}(G)=(d_{\rm s}(p_1), d_{\rm s}(p_2), \ldots, d_{\rm s}(p_k))$, where $d_{\rm s}(p)$ signifies the degree of the vertex $p$ in ${\Gamma}_{\rm s}(G)$. We will simply call ${\rm D}_{\rm s}(G)$ the degree pattern of solvable graph of $G$. In this paper, we determine the structure of any finite group $G$ (up to isomorphism) for which ${\Gamma}_{\rm s}(G)$ is star or bipartite. It is also shown that the sporadic simple groups and some of projective special linear groups $L_2(q)$ are characterized via order and degree pattern of solvable graph.

In this paper, we introduce the notion of screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions D₁, D₂ and RadTM on screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds have been obtained. Further we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic. We also study mixed geodesic screen semi-slant lightlike submanifolds of indefinite Sasakian manifolds and obtain a necessary and sufficient condition for induced connection to be metric connection.

We develop a theory of curve diagrams for Artin groups of type $B$. We define the winding number labeling and the wall crossing labeling of curve diagrams, and show that these labelings detect the classical and the dual Garside length, respectively. A remarkable point is that our argument does not require Garside theory machinery like normal forms, and is more geometric in nature.

Networks play important roles in the theory of discrete potentials. Especially, the theory of Dirichlet spaces on networks has become one of the most important tools for the study of potentials on networks. In this paper, first we study some relations between the Dirichlet sums of a function and of its Laplacian. We introduce some conditions to investigate properties of several functional spaces related to Dirichlet potentials and to biharmonic functions. Our goal is to study the growth of the Laplacian related to biharmonic functions on an infinite network. As an application, we prove a Riesz Decomposition theorem for Dirichlet functions satisfying various conditions.

In this paper we construct finite vertex-transitive K\"ahler graphs, which may be considered as discrete models of Hermitian symmetric spaces admitting K\"ahler magnetic fields. We give a condition on cardinality of the set of vertices and the principal and the auxiliary degrees for a vertex-transitive K\"ahler graphs. Also we give some examples of K\"ahler graphs corresponding typical vertex-transitive ordinary graphs.

We prove that a locally symmetric almost Kenmotsu three-manifold is locally isometric to either the hyperbolic space $\mathrm{\Bbb{H}^3(-1)}$ or a product space $\Bbb{H}^2(-4)\times \Bbb{R}$.