2,465 research outputs found
Incomplete Transition Complexity of Basic Operations on Finite Languages
The state complexity of basic operations on finite languages (considering
complete DFAs) has been in studied the literature. In this paper we study the
incomplete (deterministic) state and transition complexity on finite languages
of boolean operations, concatenation, star, and reversal. For all operations we
give tight upper bounds for both description measures. We correct the published
state complexity of concatenation for complete DFAs and provide a tight upper
bound for the case when the right automaton is larger than the left one. For
all binary operations the tightness is proved using family languages with a
variable alphabet size. In general the operational complexities depend not only
on the complexities of the operands but also on other refined measures.Comment: 13 page
Collapse and revival of excitations in Bose-Einstein condensates
We study the energies and decay of elementary excitations in weakly
interacting Bose-Einstein condensates within a finite-temperature gapless
second-order theory. The energy shifts for the high-lying collective modes turn
out to be systematically negative compared with the
Hartree-Fock-Bogoliubov-Popov approximation and the decay of the low-lying
modes is found to exhibit collapse and revival effects. In addition,
perturbation theory is used to qualitatively explain the experimentally
observed Beliaev decay process of the scissors mode.Comment: 9 pages, 5 figure
Spectrum of bound fermion states on vortices in He-B
We study subgap spectra of fermions localized within vortex cores in
He-B. We develop an analytical treatment of the low-energy states and
consider the characteristic properties of fermion spectra for different types
of vortices. Due to the removed spin degeneracy the spectra of all singly
quantized vortices consist of two different anomalous branches crossing the
Fermi level. For singular and vortices the anomalous branches are
similar to the standard Caroli-de Gennes -Matricon ones and intersect the Fermi
level at zero angular momentum yet with different slopes corresponding to
different spin states. On the contrary the spectral branches of nonsingular
vortices intersect the Fermi level at finite angular momenta which leads to the
appearance of a large number of zero modes, i.e. energy states at the Fermi
level. Considering the , and vortices with superfluid cores we
show that the number of zero modes is proportional to the size of the vortex
core.Comment: 6 pages, 1 figur
Symmetric Groups and Quotient Complexity of Boolean Operations
The quotient complexity of a regular language L is the number of left
quotients of L, which is the same as the state complexity of L. Suppose that L
and L' are binary regular languages with quotient complexities m and n, and
that the transition semigroups of the minimal deterministic automata accepting
L and L' are the symmetric groups S_m and S_n of degrees m and n, respectively.
Denote by o any binary boolean operation that is not a constant and not a
function of one argument only. For m,n >= 2 with (m,n) not in
{(2,2),(3,4),(4,3),(4,4)} we prove that the quotient complexity of LoL' is mn
if and only either (a) m is not equal to n or (b) m=n and the bases (ordered
pairs of generators) of S_m and S_n are not conjugate. For (m,n)\in
{(2,2),(3,4),(4,3),(4,4)} we give examples to show that this need not hold. In
proving these results we generalize the notion of uniform minimality to direct
products of automata. We also establish a non-trivial connection between
complexity of boolean operations and group theory
Additive decomposability of functions over abelian groups
Abelian groups are classified by the existence of certain additive
decompositions of group-valued functions of several variables with arity gap 2.Comment: 17 page
On the effect of variable identification on the essential arity of functions
We show that every function of several variables on a finite set of k
elements with n>k essential variables has a variable identification minor with
at least n-k essential variables. This is a generalization of a theorem of
Salomaa on the essential variables of Boolean functions. We also strengthen
Salomaa's theorem by characterizing all the Boolean functions f having a
variable identification minor that has just one essential variable less than f.Comment: 10 page
Quantum circuits with uniformly controlled one-qubit gates
Uniformly controlled one-qubit gates are quantum gates which can be
represented as direct sums of two-dimensional unitary operators acting on a
single qubit. We present a quantum gate array which implements any n-qubit gate
of this type using at most 2^{n-1} - 1 controlled-NOT gates, 2^{n-1} one-qubit
gates and a single diagonal n-qubit gate. The circuit is based on the so-called
quantum multiplexor, for which we provide a modified construction. We
illustrate the versatility of these gates by applying them to the decomposition
of a general n-qubit gate and a local state preparation procedure. Moreover, we
study their implementation using only nearest-neighbor gates. We give upper
bounds for the one-qubit and controlled-NOT gate counts for all the
aforementioned applications. In all four cases, the proposed circuit topologies
either improve on or achieve the previously reported upper bounds for the gate
counts. Thus, they provide the most efficient method for general gate
decompositions currently known.Comment: 8 pages, 10 figures. v2 has simpler notation and sharpens some
result
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