Search results for "Function composition"
showing 4 items of 4 documents
A matrix of combinatorial numbers related to the symmetric groups
1979
For permutation groups G of finite degree we define numbers t"B(G)=|G|^-^[email protected]?"R"@?"[email protected]?"1(1a"1(g))^b^"^i, where B=(b"1,...,b"1) is a tuple of non-negative integers and a"1(g) denotes the number of i cycles in the element g. We show that t"B(G) is the number of orbits of G, acting on a set @D"B(G) of tuples of matrices. In the case G=S"n we get a natural interpretation for combinatorial numbers connected with the Stiring numbers of the second kind.
Polyomino coloring and complex numbers
2008
AbstractUsually polyominoes are represented as subsets of the lattice Z2. In this paper we study a representation of polyominoes by Gaussian integers. Polyomino {(x1,y1),(x2,y2),…,(xs,ys)}⊂Z2 is represented by the set {(x1+iy1),(x2+iy2),…,(xs+iys)}⊂Z[i]. Then we consider functions of type f:P→G from the set P of all polyominoes to an abelian group G, given by f(x,y)≡(x+iy)m(modv), where v is prime in Z[i],1≤m<N(v) (N(v) is the norm of v). Using the arithmetic of the ring Z[i] we find necessary and sufficient conditions for such a function to be a coloring map.
AnySeq: A High Performance Sequence Alignment Library based on Partial Evaluation
2020
Sequence alignments are fundamental to bioinformatics which has resulted in a variety of optimized implementations. Unfortunately, the vast majority of them are hand-tuned and specific to certain architectures and execution models. This not only makes them challenging to understand and extend, but also difficult to port to other platforms. We present AnySeq - a novel library for computing different types of pairwise alignments of DNA sequences. Our approach combines high performance with an intuitively understandable implementation, which is achieved through the concept of partial evaluation. Using the AnyDSL compiler framework, AnySeq enables the compilation of algorithmic variants that ar…
On the optimal approximation rate of certain stochastic integrals
2010
AbstractGiven an increasing function H:[0,1)→[0,∞) and An(H)≔infτ∈Tn(∑i=1n∫ti−1ti(ti−t)H(t)2dt)12, where Tn≔{τ=(ti)i=0n:0=t0<t1<⋯<tn=1}, we characterize the property An(H)≤cn, and give conditions for An(H)≤cnβ and An(H)≥1cnβ for β∈(0,1), both in terms of integrability properties of H. These results are applied to the approximation of stochastic integrals.