Search results for "Integer"
showing 10 items of 250 documents
On a Linear Diophantine Problem of Frobenius: Extending the Basis
1998
LetXk={a1, a2, …, ak},k>1, be a subset of N such that gcd(Xk)=1. We shall say that a natural numbernisdependent(onXk) if there are nonnegative integersxisuch thatnhas a representationn=∑ki=1 xiai, elseindependent. The Frobenius numberg(Xk) ofXkis the greatest integer withnosuch representation. Selmer has raised the problem of extendingXkwithout changing the value ofg. He showed that under certain conditions it is possible to add an elementc=a+kdto the arithmetic sequencea,a+d,a+2d, …, a+(k−1) d, gcd(a, d)=1, without alteringg. In this paper, we give the setCof all independent numberscsatisfyingg(A, c)=g(A), whereAcontains the elements of the arithmetic sequence. Moreover, ifa>kthen we give …
The irregularity strength of circulant graphs
2005
AbstractThe irregularity strength of a simple graph is the smallest integer k for which there exists a weighting of the edges with positive integers at most k such that all the weighted degrees of the vertices are distinct. In this paper we study the irregularity strength of circulant graphs of degree 4. We find the exact value of the strength for a large family of circulant graphs.
Complexity of decision trees for boolean functions
2004
For every positive integer k we present an example of a Boolean function f/sub k/ of n = (/sub k//sup 2k/) + 2k variables, an optimal deterministic tree T/sub k/' for f/sub k/ of complexity 2k + 1 as well as a nondeterministic decision tree T/sub k/ computing f/sub k/. with complexity k + 2; thus of complexity about 1/2 of the optimal deterministic decision tree. Certain leaves of T/sub k/ are called priority leaves. For every input a /spl isin/ {0, 1}/sup n/ if any of the parallel computation reaches a priority leaves then its label is f/sub k/ (a). If the priority leaves are not reached at all then the label on any of the remaining leaves reached by the computation is f/sub k/. (a).
A Loopless Generation of Bitstrings without p Consecutive Ones
2001
Let F n (p) be the set of all n-length bitstrings such that there are no p consecutive ls. F n (p) is counted with the pth order Fibonacci numbers and it may be regarded as the subsets of {1, 2,…, n} without p consecutive elements and bitstrings in F n (p) code a particular class of trees or compositions of an integer. In this paper we give a Gray code for F n (p) which can be implemented in a recursive generating algorithm, and finally in a loopless generating algorithm.
Linear Diophantine Problems
1996
The Frobenius number g(A k ) Let A k \({A_k} = \{ {a_1},...,{a_k}\}\subset\) IN with gcd(A k ) = 1, n\( \in I{N_0}.\) If $$n = \sum\limits_{i = 1}^k {{x_i}{a_i},{x_i}}\in I{N_0}$$ (1) we call this a representation or a g-representation of n by Ak (in order to distinguish between several types of representations that will be considered in the sequel). Then the Frobenius number g(A k ) is the greatest integer with no g-representation.
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.
The case of equality in the dichotomy of Mohammadi–Oh
2019
If $n \geq 3$ and $\Gamma$ is a convex-cocompact Zariski-dense discrete subgroup of $\mathbf{SO}^o(1,n+1)$ such that $\delta_\Gamma=n-m$ where $m$ is an integer, $1 \leq m \leq n-1$, we show that for any $m$-dimensional subgroup $U$ in the horospheric group $N$, the Burger-Roblin measure associated to $\Gamma$ on the quotient of the frame bundle is $U$-recurrent.
Generalised norms in finite soluble groups
2014
Abstract We give a framework for a number of generalisations of Baerʼs norm that have appeared recently. For a class C of finite nilpotent groups we define the C -norm κ C ( G ) of a finite group G to be the intersection of the normalisers of the subgroups of G that are not in C . We show that those groups for which the C -norm is not hypercentral have a very restricted structure. The non-nilpotent groups G for which G = κ C ( G ) have been classified for some classes. We give a classification for nilpotent classes closed under subgroups, quotients and direct products of groups of coprime order and show the known classifications can be deduced from our classification.
A Note on a Conjecture of Duval and Sturmian Words
2002
We prove a long standing conjecture of Duval in the special case of Sturmian words. Mathematics Subject Classication. ??????????????. Let U be a nonempty word on a nite alphabet A: A nonempty word B dierent from U is called a border of U if B is both a prex and sux of U: We say U is bordered if U admits a border, otherwise U is said to be unbordered. For example, U = 011001011 is bordered by the factor 011; while 00010001001 is unbordered. An integer 1 k n is a period of a word U = U1 :::U n if and only if for all 1 i n k we have Ui = Ui+k. It is easy to see that k is a period of U if and only if the prex B of U of length n k is a border of U or is empty. Let (U) denote the smallest period …
Periodic and Nil Polynomials in Rings
1980
Let R be an associative ring and f(x1,…, xd) a polynomial in noncommuting variables. We say that f is periodic or nil in R if for all r1,…, rd ∈ R we have that f(r1,…, rd) is periodic, respectively nilpotent (recall that a ∈ R is periodic if for some integer ).