Search results for "Quadratic growth"
showing 10 items of 16 documents
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…
Error-Free Affine, Unitary, and Probabilistic OBDDs
2018
We introduce the affine OBDD model and show that zero-error affine OBDDs can be exponentially narrower than bounded-error unitary and probabilistic OBDDs on certain problems. Moreover, we show that Las Vegas unitary and probabilistic OBDDs can be quadratically narrower than deterministic OBDDs. We also obtain the same results for the automata versions of these models.
Quadratic speedup for finding marked vertices by quantum walks
2020
A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk.
Stationary states of a two-state defect quadratically coupled to a few bosonic modes
1998
Abstract A fully quantistic microscopic two-phonon interaction model between an active centre and localized modes of an irradiated insulating material is introduced. Its exact diagonalization is accomplished with the help of a suitable unitary operator. Explicit expressions for the eigenvalues and eigenvectors are reported. The possible relevance of such a model in the context of the material science area is briefly pointed out.
Words with the Maximum Number of Abelian Squares
2015
An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain \(\varTheta (n^2)\) distinct factors that are abelian squares. We study infinite words such that the number of abelian square factors of length n grows quadratically with n.
Relaxation of Quasilinear Elliptic SystemsviaA-quasiconvex Envelopes
2002
We consider the weak closure WZof the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems div s0 s=1 s(x)F 0 s(ru(x )+ g(x)) f(x) =0i n; u =( u1;:::;um)2 H 1 0 (; R m ) ; =( 1;:::;s 0 )2 S; where R n is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and S =f measurable j s(x )=0o r 1 ;s =1 ;:::;s0 ;1(x )+ +s0 (x )=1 g .W e show that WZis the zero level set for an integral functional with the integrand QF being the A-quasiconvex envelope for a certain functionF and the operator A = (curl,div) m . If the functions Fs are isotropic, then on the characteristic cone (dened by the operator A) QF coincides with the A-p…
Comments on `A new efficient method for calculating perturbation energies using functions which are not quadratically integrable'
1996
The recently proposed method of calculating perturbation energies using a non-normalizable wavefunction by Skala and Cizek is analysed and rigorously proved.
Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term
2006
We study existence and regularity of distributional solutions for possibly degenerate quasi-linear parabolic problems having a first order term which grows quadratically in the gradient. The model problem we refer to is the following (1){ut−div(α(u)∇u)=β(u)|∇u|2+f(x,t),in Ω×]0,T[;u(x,t)=0,on ∂Ω×]0,T[;u(x,0)=u0(x),in Ω. Here Ω is a bounded open set in RN, T>0. The unknown function u=u(x,t) depends on x∈Ω and t∈]0,T[. The symbol ∇u denotes the gradient of u with respect to x. The real functions α, β are continuous; moreover α is positive, bounded and may vanish at ±∞. As far as the data are concerned, we require the following assumptions: ∫ΩΦ(u0(x))dx<∞ where Φ is a convenient function which …
On the accurate determination of nonisolated solutions of nonlinear equations
1981
A simple but efficient method to obtain accurate solutions of a system of nonlinear equations with a singular Jacobian at the solution is presented. This is achieved by enlarging the system to a higher dimensional one whose solution in question is isolated. Thus it can be computed e. g. by Newton's method, which is locally at least quadratically convergent and selfcorrecting, so that high accuracy is attainable.
Unitary decoupling treatment of a quadratic bimodal cavity quantum electrodynamics model
2013
We consider a two-photon quantum model of radiation–matter interaction between a single two-level atom and a degenerate bimodal high-Q cavity field. Within this tripartite system, the explicit construction of two collective radiation modes, one of which is freely evolving and the other one quadratically coupled to the matter subsystem, is reported. The meaning and advantages of such a decoupling treatment are carefully discussed.