Related papers: Lonely Points in Simplices

A condition for the presence of a "gap" between symmetric spaces sufficient for the inclusion of one of these spaces into the other to be disjointly strictly singular is found. This condition is stated in terms of fundamental functions of…

We investigate indeterminate points in discrete integrable system. They appear in singularity confinement phenomenon naturally. We develop a method to analyse indeterminate points of dynamical maps and using this method we clarify behaviour…

We study separated nets that correspond to substitution tilings of the Euclidean space. We give a simple condition, in terms of the eigenvalues and eigenspaces of the substitution matrix, to know whether the separated net is a bounded…

Let L --> X be a complex line bundle over a compact connected Riemann surface. We consider the abelian vortex equations on L when the metric on the surface has finitely many point degeneracies or conical singularities and the line bundle…

For $A\in\mathbb{Z}^{m\times n}$ we investigate the behaviour of the number of lattice points in $P_A(b)=\{x\in\mathbb{R}^n:Ax\leq b\}$, depending on the varying vector $b$. It is known that this number, restricted to a cone of constant…

We consider singular solutions to quasilinear elliptic equations under zero Dirichlet boundary condition. Under suitable assumptions on the nonlinearity we deduce symmetry and monotonicity properties of positive solutions via an improved…

We introduce and explore a natural rank for totally disconnected locally compact groups called the bounded conjugacy rank. This rank is shown to be a lattice invariant for lattices in sigma compact totally disconnected locally compact…

In this paper, we establish some results about the singular points of certain non-monotone potential operators. Here is a sample: If $X$ is an infinite-dimensional reflexive real Banach space and if $T:X\to X^*$ is a non-monotone, closed,…

We say a lattice point $X=(x_1,\ldots,x_m)$ is visible from the origin, if $\gcd(x_1,...,x_m)=1$. In other word, there are no other lattice point on the line segment from the origin $O$ to $X$. From J.E. Nymann's result, we know that the…

We study isolated points on the modular curves $X_{H}$, for $H$ a subgroup of $\operatorname{GL}_{2}(\mathbb{Z}/n \mathbb{Z})$ for some $n \geq 1$. In particular, we prove a single-sink theorem for such isolated points, which traces the…

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted…

The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds…

In this paper we introduce and study the lattice of normal subgroups of a group $G$ that determine solitary quotients. It is closely connected to the well-known lattice of solitary subgroups of $G$ (see \cite{5}). A precise description of…

We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a…

Determining if an (1+1)-differential-difference equation is integrable or not (in the sense of possessing an infinite number of symmetries) can be reduced to the study of the dependence of the equation on the lattice points, according to…

We consider lattice equations on ${\mathds{Z}}^2$ which are autonomous, affine linear and possess the symmetries of the square. Some basic properties of equations of this type are derived, as well as a sufficient linearization condition and…

We define the modulo-$m$ Toeplitz fixed point generated by Toeplitz substitution and study the lattice subsequence of such fixed point. Moreover, we provide a method to check whether one modulo-$m$ Toeplitz fixed point is a lattice…

We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient…

It was proved by Nill that for any lattice simplex of dimension $d$ with degree $s$ which is not a lattice pyramid, the inequality $d+1 \leq 4s-1$ holds. In this paper, we give a complete characterization of lattice simplices satisfying the…

We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be…