Search results for "Complex."
showing 10 items of 5824 documents
Asymptotics for Graded Capelli Polynomials
2014
The finite dimensional simple superalgebras play an important role in the theory of PI-algebras in characteristic zero. The main goal of this paper is to characterize the T 2-ideal of graded identities of any such algebra by considering the growth of the corresponding supervariety. We consider the T 2-ideal Γ M+1,L+1 generated by the graded Capelli polynomials C a p M+1[Y,X] and C a p L+1[Z,X] alternanting on M+1 even variables and L+1 odd variables, respectively. We prove that the graded codimensions of a simple finite dimensional superalgebra are asymptotically equal to the graded codimensions of the T 2-ideal Γ M+1,L+1, for some fixed natural numbers M and L. In particular csupn(Γk2+l2+1…
Pattern Matching and Pattern Discovery Algorithms for Protein Topologies
2001
We describe algorithms for pattern-matching and pattern-learning in TOPS diagrams (formal descriptions of protein topologies). These problems can be reduced to checking for subgraph isomorphism and finding maximal common subgraphs in a restricted class of ordered graphs. We have developed a subgraph isomorphism algorithm for ordered graphs, which performs well on the given set of data. The maximal common subgraph problem then is solved by repeated subgraph extension and checking for isomorphisms. Despite its apparent inefficiency, this approach yields an algorithm with time complexity proportional to the number of graphs in the input set and is still practical on the given set of data. As a…
Embeddings of Danielewski surfaces
2003
A Danielewski surface is defined by a polynomial of the form P=x nz −p(y). Define also the polynomial P ′ =x nz −r(x)p(y) where r(x) is a non-constant polynomial of degree ≤n−1 and r(0)=1. We show that, when n≥2 and deg p(y)≥2, the general fibers of P and P ′ are not isomorphic as algebraic surfaces, but that the zero fibers are isomorphic. Consequently, for every non-special Danielewski surface S, there exist non-equivalent algebraic embeddings of S in ℂ3. Using different methods, we also give non-equivalent embeddings of the surfaces xz=(y d n >−1) for an infinite sequence of integers d n . We then consider a certain algebraic action of the orthogonal group $\mathcal O(2)$ on ℂ4 which was…
On the decision problem for the guarded fragment with transitivity
2002
The guarded fragment with transitive guards, [GF+TG], is an extension of GF in which certain relations are required to be transitive, transitive predicate letters appear only in guards of the quantifiers and the equality symbol may appear everywhere. We prove that the decision problem for [GF+TG] is decidable. This answers the question posed in (Ganzinger et al., 1999). Moreover, we show that the problem is 2EXPTIME-complete. This result is optimal since the satisfiability problem for GF is 2EXPTIME-complete (Gradel, 1999). We also show that the satisfiability problem for two-variable [GF+TG] is NEXPTIME-hard in contrast to GF with bounded number of variables for which the satisfiability pr…
On the Finite Satisfiability Problem for the Guarded Fragment with Transitivity
2005
We study the finite satisfiability problem for the guarded fragment with transitivity. We prove that in case of one transitive predicate the problem is decidable and its complexity is the same as the general satisfiability problem, i.e. 2Exptime-complete. We also show that finite models for sentences of GF with more transitive predicate letters used only in guards have essentially different properties than infinite ones.
Words and forbidden factors
2002
AbstractGiven a finite or infinite word v, we consider the set M(v) of minimal forbidden factors of v. We show that the set M(v) is of fundamental importance in determining the structure of the word v. In the case of a finite word w we consider two parameters that are related to the size of M(w): the first counts the minimal forbidden factors of w and the second gives the length of the longest minimal forbidden factor of w. We derive sharp upper and lower bounds for both parameters. We prove also that the second parameter is related to the minimal period of the word w. We are further interested to the algorithmic point of view. Indeed, we design linear time algorithm for the following two p…
Two shortest path metrics on well-formed parentheses strings
1996
We present an analysis of two transformations on well-formed parentheses strings. Using a lattice approach, the corresponding least-move distances are computable, the first in linear time and the second in quadratic time.
Exponential Codimension Growth of PI Algebras: An Exact Estimate
1999
Abstract LetAbe an associative PI-algebra over a fieldFof characteristic zero. By studying the exponential behavior of the sequence of codimensions {cn(A)} ofA, we prove thatInv(A)=limn→∞ c n ( A ) always exists and is an integer. We also give an explicit way for computing such integer: letBbe a finite dimensionalZ2-graded algebra whose Grassmann envelopeG(B) satisfies the same identities ofA; thenInv(A)=Inv(G(B))=dim C(0)+dim C(1)whereC(0)+C(1)is a suitableZ2-graded semisimple subalgebra ofB.
Asymptotics for the standard and the Capelli identities
2003
Let {c n (St k )} and {c n (C k )} be the sequences of codimensions of the T-ideals generated by the standard polynomial of degreek and by thek-th Capelli polynomial, respectively. We study the asymptotic behaviour of these two sequences over a fieldF of characteristic zero. For the standard polynomial, among other results, we show that the following asymptotic equalities hold: $$\begin{gathered} c_n \left( {St_{2k} } \right) \simeq c_n \left( {C_{k^2 + 1} } \right) \simeq c_n \left( {M_k \left( F \right)} \right), \hfill \\ c_n \left( {St_{2k + 1} } \right) \simeq c_n \left( {M_{k \times 2k} \left( F \right) \oplus M_{2k \times k} \left( F \right)} \right), \hfill \\ \end{gathered} $$ wher…
Varieties with at most quadratic growth
2010
Let V be a variety of non necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions cn(V); n = 1; 2, … and here we study varieties of polynomial growth. Recently, for any real number a, 3 < a < 4, a variety V was constructed satisfying C1n^a < cn(V) < C2n^a; for some constants C1;C2. Motivated by this result here we try to classify all possible growth of varieties V such that cn(V) < Cn^a; with 0 < a < 2, for some constant C. We prove that if 0 < a < 1 then, for n large, cn(V) ≤ 1, whereas if V is a commutative variety and 1 < a < 2, then lim logn cn(V) = 1 o…