Narrow Results

We show that the log canonical bundle, κ, of is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of ℙ¹ × ⋯ × ℙ^n-3, permuted by the symmetric group. The homogeneous ideal of is the sum of the homogeneous ideals of these Segre embeddings.

Let h₁, h₂,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w₁, w₂, …, where each w_i is a word in h_i and possibly other h_j, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that , where is the subgroup generated by the roots of the elements in H. If H₀ is finitely generated and the sequence of subgroups H₀, H₁, H₂, … satisfies then the sequence stabilizes, i.e. for some m, H_i=H_i+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" the exponents a_i satisfy gcd(a₁,…,a_n)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

This paper addresses the question of simultaneously solving a set of equations in one variable over torsion-free groups.

We show that the existential theory of free partially commutative monoids with involution is decidable. As a consequence the existential theory of graph groups is also decidable. If the underlying alphabet of generators is fixed, we obtain a PSPACE-completeness result, otherwise (in the uniform setting) our decision procedure is in EXPSPACE. Our proof is a reduction to the main result of [6].

It is known that the problem of determining consistency of a finite system of equations in a free group or a free monoid is decidable, but the corresponding problem for systems of equations in a free inverse monoid of rank at least two is undecidable. Any solution to a system of equations in a free inverse monoid induces a solution to the corresponding system of equations in the associated free group in an obvious way, but solutions to systems of equations in free groups do not necessarily lift to solutions in free inverse monoids. In this paper, we show that the problem of determining whether a solution to a finite system of equations in a free group can be extended to a solution of the corresponding system in the associated free inverse monoid is decidable. We are able to use this to solve the consistency problem for certain classes of single-variable equations in free inverse monoids.

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by 2s for an orientable equation and by 12s⁴ for a non-orientable equation, where s is the sum of the lengths of the coefficients.

In this paper, we investigate the reducibility property of semidirect products of the form $V\ast D$ relatively to (pointlike) systems of equations of the form $x_1=\cdots =x_n$, where $D$ d̃enotes the pseudovariety of definite semigroups. We establish a connection between pointlike reducibility of $V\ast D$ and the pointlike reducibility of the pseudovariety $V$. In particular, for the canonical signature $\kappa$ consisting of the multiplication and the $(\omega -1)$-power, we show that $V\ast D$ is pointlike $\kappa$-reducible when $V$ is pointlike $\kappa$-reducible.

In this paper, we investigate properties of varieties of algebras described by a novel concept of equation that we call commutator equation. A commutator equation is a relaxation of the standard term equality obtained substituting the equality relation with the commutator relation. Namely, an algebra $A$ satisfies the commutator equation $p{\approx }_{C}q$ if for each congruence $𝜃$ in $Con(A)$ and for each substitution $p^A,q^A$ of elements in the same $𝜃$-class, we have $(p^A,q^A)\in [𝜃,𝜃]$. This notion of equation draws inspiration from the definition of a weak difference term and allows for further generalization of it. Furthermore, we present an algorithm that establishes a connection between congruence equations valid in the variety generated by the abelian algebras of the idempotent reduct of a given variety and congruence equations that hold in the entire variety. Additionally, we provide a proof that if the variety generated by the abelian algebras of the idempotent reduct of a variety satisfies a nontrivial idempotent Mal’cev condition, then also the entire variety satisfies a nontrivial idempotent Mal’cev condition, a statement that follows also from [12, Theorem 3.13].

Generalization is an important operation for programs that learn. Anti-unification guarantees the existence of a term which is the most specific generalization of a collection of terms. This provides a formal basis for learning from examples. However, such generalization may be "too" general and therefore counterexamples are required to restrict them. In this case, the learner has to check whether the resulting formula provides a concept to learn, i.e. whether a formula of the form t/{t1,…,tn} is a generalization, where t is viewed as a generalization of a set of examples and t1,…,tn are counterexample. The formula t/{t1,…,tn} is often called an implicit representation. The implicit representation t/{t1,…,tn} is a generalization iff there exist ground (variable-free) instances of t which are not instances of t1,…,tn. This is a non-trivial task even when there is no background knowledge (see [15]).

We present here a method to check whether an implicit representation t/{t1,…,tn} is a generalization with respect to a finite set of equations which describes the background knowledge problem, i.e. whether there exists a ground instance of t which is not equivalent to any ground instance of t1,…,tn with respect to a set E of equations. Whereas this problem is in general undecidable since the equality is so, we show in this paper that in this case where the set E of equations is compiled into a ground convergent term rewriting system, we can discover concepts in Universe of Discourse with background knowledge described by a finite set of equations. We show how the method applies to induce concepts in theories of elementary arithmetic.

It is known that orbit reduction can be performed in one or two stages and it has been proven that the two processes are symplectically equivalent. In the context of orbit reduction by one stage, we shall write an expression for the reduced two-form in the general case and obtain the equations of motion derived from this theory. Then we shall develop the same process in the case in which the symmetry group has a normal subgroup to get the reduced symplectic form by two stages and the consequent orbit reduced equations. In both cases, we shall illustrate the method with three physical examples.

Let G be a group, t an element distinct from G and r(t)= g₁t^l₁ ⋯ g_kt^l_k∈ G ∗ 〈t 〉, where each g_i is an element of G of order greater than 2 and the l_i are non-zero integers such that l₁+l₂+ ⋯ +l_k≠ 0 and |l_i| ≠ |l_j| for i ≠ j. It is known that if k≤ 2, then the natural map from G to the one-relator product 〈G,t | r(t)〉 is injective. In this paper, we prove that the same holds for all k ∉ {4, 5}.

We consider velocity structure functions in turbulence through an approach using cumulants, and for a fixed value of the distance ℓ. This allows to consider the cumulant generating function Φ_ℓ(q) = log〈|ΔV_ℓ|^q〉. Using an atmospheric turbulent database, we show that the cumulant generating function is nonanalytic, with a development compatible with a log-stable model of the form Φ_ℓ(q) = A_ℓq + B_ℓq^α and a parameter value of α = 1.5. The parameters A_ℓ, and B_ℓ are experimentally estimated: they are respectively increasing and decreasing functions of ℓ; their scaling ranges correspond to the scaling range of the velocity fluctuations. The dependence between these two functions is studied in relation to Extended Self Similarity and Generalized Extended Self Similarity properties.

Data from time lapse microscopy of live embryonic rat hippocampal neurons growing in cell culture are used to study the dynamics of axonal growth.¹ We analyze axonal trajectory data based on cells growing in a homogeneous medium. Due to the noisy nature of the data we develop filtering algorithms to smoothen out the paths while maintaining the underlying dynamics of the axon growth process. We analyze the new paths and propose a model for growth cone kinematics during axonogenesis without a gradient field. In this work we present a simple renewal process with the aim of reproducing certain path behaviors of the growth cone. Future development will include, renewal process simulation, and gradients effects.