K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

(See e.g. [6] for background.) When X is equipped with a distinguished basis D for its topology, closed under finite meets and joins, one can investigate situations where D is also closed under the implication (2), i.e., where D is a Heyting subalgebra of O (X). Recall that X is a spectral space if it is compact and T0, its collection D of compact open subsets forms a basis which is closed unde...

Let G be a linear algebraic group defined over Q, and assume that G(R) is compact and meets every connected component of G(C). Let Q̂ := Ẑ ⊗ Q be the ring of finite adèles. Every arithmetic subgroup Γ of G(Q) is finite, and is obtained by choosing an open, compact subgroup K of G(Q̂) and defining Γ = K ∩G(Q) in G(Q̂). We note that G(Q) is discrete and co-compact in G(Q̂). In this paper, we consider...

(See e.g. [6] for background.) When X is equipped with a distinguished basis D for its topology, closed under finite meets and joins, one can investigate situations where D is also closed under the implication (2), i.e., where D is a Heyting subalgebra of O (X). Recall that X is a spectral space if it is compact and T0, its collection D of compact open subsets forms a basis which is closed unde...

In this paper, we propose a regular layout geometry called 3×3 lattice. The main difference of this geometry compared to the known 2×2 regular layout geometry is that it allows the cofactors on a level to propagate to three rather than two nodes on the lower level. This gives additional freedom to synthesize compact functional representations. We propose a SAT-based algorithm, which exploits th...

In this work we report on the Landau gauge photon propagator computed for pure 4D compact QED in confined and deconfined phases large lattice volumes: ${32}^{4}$, ${48}^{4}$ ${96}^{4}$. phase, develops mass scales that render finite at all momentum no volume dependence is observed simulations performed. Furthermore, phase compatible with a Yukawa massive type functional form. For seems to appro...

Compact U(1) theory in 4 dimensions is used to compare the modified iterative and the Laplacian fixing to lattice Landau gauge in a controlled setting, since in the Coulomb phase the lattice theory must reproduce the perturbative prediction. It turns out that on either side of the phase transition clear differences show up and in the Coulomb phase the ability to remove double Dirac sheets prove...

The phase diagram of the two-dimensional extended q−states Potts model is investigated in the q → 1 limit. This is equivalent to studying the phase diagram of a two-dimensional infinite interacting lattice animal. An exact solution on the Bethe lattice and a Migdal-Kadanoff renormalization group calculation predict a line of θ transitions from an extended to a compact phase in the lattice anima...