Search results for "Finite set"
showing 10 items of 101 documents
Deciding reachability for planar multi-polynomial systems
1996
In this paper we investigate the decidability of the reachability problem for planar non-linear hybrid systems. A planar hybrid system has the property that its state space corresponds to the standard Euclidean plane, which is partitioned into a finite number of (polyhedral) regions. To each of these regions is assigned some vector field which governs the dynamical behaviour of the system within this region. We prove the decidability of point to point and region to region reachability problems for planar hybrid systems for the case when trajectories within the regions can be described by polynomials of arbitrary degree.
Einklassige Geschlechter totalpositiver quadratischer Formen in totalreellen algebraischen Zahlkörpern
1971
Abstract It is proved that totally positive quadratic forms with three or more variables and class number h = 1 exist only in a finite number of algebraic number fields. Each field allows only a finite number of such forms with bounded scale. To prove this, upper estimates for all local factors in Siegel's analytic formula are constructed by calculating explicitly numbers of solutions of quadratic congruences.
Polynomial codimension growth of algebras with involutions and superinvolutions
2017
Abstract Let A be an associative algebra over a field F of characteristic zero endowed with a graded involution or a superinvolution ⁎ and let c n ⁎ ( A ) be its sequence of ⁎-codimensions. In [4] , [12] it was proved that if A is finite dimensional such sequence is polynomially bounded if and only if A generates a variety not containing a finite number of ⁎-algebras: the group algebra of Z 2 and a 4-dimensional subalgebra of the 4 × 4 upper triangular matrices with suitable graded involutions or superinvolutions. In this paper we focus our attention on such algebras since they are the only finite dimensional ⁎-algebras, up to T 2 ⁎ -equivalence, generating varieties of almost polynomial gr…
The parameterized local deduction theorem for quasivarieties of algebras and its application
1996
Let τ be an algebraic type. To each classK of τ-algebras a consequence relation ⊧ K defined on the set of τ-equations is assigned. Some weak forms of the deduction theorem for ⊧ K and their algebraic counterparts are investigated. The (relative) congruence extension property (CEP) and its variants are discussed.CEP is shown to be equivalent to a parameter-free form of the deduction theorem for the consequence ⊧ K .CEP has a strong impact on the structure ofK: for many quasivarietiesK,CEP implies thatK is actually a variety. This phenomenon is thoroughly discussed in Section 5. We also discuss first-order definability of relative principal congruences. This property is equivalent to the fact…
Characterizing varieties of colength ≤4
2009
Let A be an associative algebra over a field F of characteristic zero, and let χ n (A), n = 1,2,…, be the sequence of cocharacters of A. For every n ≥ 1, let l n (A) denote the nth colength of A, counting the number of S n -irreducibles appearing in χ n (A). In this article, we classify the algebras A such that the sequence of colengths l n (A), n = 1,2,…, is bounded by four. Moreover we construct a finite number of algebras A 1,…, A d , such that l n (A) ≤ 4 if and only if A 1,…, A d ∉ var(A).
Partial Finitely Generated Bi-Ideals
2016
Partial words have been studied by Blanchet-Sadri et al., but bi-ideals or reccurrent words have been studied for centuries by many researchers. This paper gives a solution for some problems for partial reccurrent words. This paper gives an algorithm for a given finitely generated bi-ideal, how to construct a new basis of ultimately finitely generated bi-ideal, which generates the same given bi-ideal. The paper states that it is always possible to find a basis for a given finitely generated bi-ideal. The main results of this paper are presented in third section. At first, we show that if two irreduciable bi-ideals are different, they will differ in infinitely many places. This led to the st…
Sharpness of Rickman’s Picard theorem in all dimensions
2015
We show that given \({n \geqslant 3}\), \({q \geqslant 1}\), and a finite set \({\{y_1, \ldots, y_q \}}\) in \({\mathbb{R}^n}\) there exists a quasiregular mapping \({\mathbb{R}^n\to \mathbb{R}^n}\) omitting exactly points \({y_1, \ldots, y_q}\).
Empirical definition of social types in the analysis of inequality of opportunity: a latent classes approach
2014
The empirical analysis of inequality of opportunity centres on disparities between social types, defined by the exposure to circumstances beyond individual control. Despite this, its main theoretical foundation—the Roemer model—does not indicate how to carry out, in practice, the required partition of the population into such types. This paper operationalises this definition of social types using a latent classes approach. Our specification is embedded in a probabilistic extension of the canonical Roemer model, which assumes that the relevant population consists of a finite number of latent types, from which each individual can be treated as a random draw. This makes possible the use of the…
On the Greedy Algorithm for the Shortest Common Superstring Problem with Reversals
2015
We study a variation of the classical Shortest Common Superstring (SCS) problem in which a shortest superstring of a finite set of strings $S$ is sought containing as a factor every string of $S$ or its reversal. We call this problem Shortest Common Superstring with Reversals (SCS-R). This problem has been introduced by Jiang et al., who designed a greedy-like algorithm with length approximation ratio $4$. In this paper, we show that a natural adaptation of the classical greedy algorithm for SCS has (optimal) compression ratio $\frac12$, i.e., the sum of the overlaps in the output string is at least half the sum of the overlaps in an optimal solution. We also provide a linear-time implement…
Finite Alphabet Control of Logistic Networks with Discrete Uncertainty
2014
We consider logistic networks in which the control and disturbance inputs take values in finite sets. We derive a necessary and sufficient condition for the existence of robustly control invariant (hyperbox) sets. We show that a stronger version of this condition is sufficient to guarantee robust global attractivity, and we construct a counterexample demonstrating that it is not necessary. Being constructive, our proofs of sufficiency allow us to extract the corresponding robust control laws and to establish the invariance of certain sets. Finally, we highlight parallels between our results and existing results in the literature, and we conclude our study with two simple illustrative exampl…