相关论文: Quite Complete Real Closed fields
We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of…
The decomposition of a quasi-abelian code into shorter linear codes over larger alphabets was given in (Jitman, Ling, (2015)), extending the analogous Chinese remainder decomposition of quasi-cyclic codes (Ling, Sol\'e, (2001)). We give a…
In this note, we study non-standard models of the rational numbers with countably many elements. These are ordered fields, and so it makes sense to complete them, using non-standard Cauchy sequences. The main result of this note shows that…
We show that the class of trapezoid orders in which no trapezoid strictly contains any other trapezoid strictly contains the class of trapezoid orders in which every trapezoid can be drawn with unit area. This is different from the case of…
A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition.…
Proper classes of extensions of real field was defined and topological properties of these extensions were studied. These extensions can be connected, in this case such set is not closed under binary operations (addition and…
In this paper, nil extensions of some special type of ordered semigroups, such as, simple regular ordered semigroups, left simple and right regular ordered semigroup. Moreover, we have characterized complete semilattice decomposition of all…
We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero…
In set theory without the axiom of Choice ZF, we prove that for every commutative field IK, the following statement D_{\IK}: "On every non null IK-vector space, there exists a non null linear form" implies the existence of a IK-linear…
We present a streamlined and (hopefully) accessible proof of the model-completeness of the weak monadic second order version of a dense linear order with left-endpoint but no right-endpoint in a particular finite signature. We also show how…
Given two graphs, a mapping between their edge-sets is cycle-continuous, if the preimage of every cycle is a cycle. The motivation for this notion is Jaeger's conjecture that for every bridgeless graph there is a cycle-continuous mapping to…
We prove that, provided $d > k$, every sufficiently large subset of $\mathbf{F}_q^d$ contains an isometric copy of every $k$-simplex that avoids spanning a nontrivial self-orthogonal subspace. We obtain comparable results for simplices…
A relation extends another relation consistently if its symmetric, respectively its asymmetric, part contains the corresponding part of the smaller relation. It is shown that there exists no finite circular chain made from two transitive…
For any $k \in \Nat$, we show that the cone of $(k+1)$-secant lines of a closed subscheme $Z \subset \mathbb{P}^n_K$ over an algebraically closed field $K$ running through a closed point $p \in \mathbb{P}^n_K$ is defined by the $k$-th…
We consider cubic number fields ordered by their discriminants, and show that there exist arbitrarily long sequences that contain only fields with class numbers greater than a given bound.
Let $p\in\mathbb Z$ be a prime, $\overline{\mathbb Q_p}$ a fixed algebraic closure of the field of $p$-adic numbers and $\overline{\mathbb Z_p}$ the absolute integral closure of the ring of $p$-adic integers. Given a residually algebraic…
While any infimum in a poset can also be computed as a supremum, and vice versa, categorical limits and colimits do not always approximate each other. If I approach a point from below, and you approach it from above, then we will surely…
The "No Gap Conjecture" of Br\"ustle-Dupont-P\'erotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two…
A variety of possible extensions of mappings between posets to their Dedekind order completion is presented. One of such extensions has recently been used for solving large classes of nonlinear systems of partial differential equations with…
Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval…