Odd Power Research Articles

Polynomial phase masks for extending the depth of field of a microscope

A polynomial phase mask is designed and fabricated for enhancing the depth of field of a microscope by more than tenfold. A generic polynomial of degree 31 that consists of 16 odd power terms is optimized by simulated annealing with a realistic average modulation transfer function (MTF) iteratively set as the target MTF. Optical experimental results are shown.

On sharply transitive sets in PG(2,q)

In PG(2, q) a point set K is sharply transitive if the collineation group preserving K has a subgroup acting on K as a sharply transitive permuta- tion group. By a result of Korchmsharply transitive hyperovals only exist for a few values of q, namely q = 2, 4 and 16. In general, sharply transitive complete arcs of even size in PG(2, q) with q even seem to be spo- radic. In this paper, we construct sharply transitive complete 6( p q 1)-arcs for q = 4 2h+1 , h � 4. As far as we are concerned, these are the smallest known complete arcs in PG(2, 4 7 ) and in PG(2, 4 9 ); also, 42 seems to be a new value of the spectrum of the sizes of complete arcs in PG(2, 4 3 ). Our construction applies to any q which is an odd power of 4, but the problem of the completeness of the resulting sharply transitive arc remains open for q � 4 11 . In the second part of this paper, sharply transitive subsets arising as orbits under a Singer subgroup are considered and their characters, that is the possible intersection numbers with lines, are investigated. Subsets of PG(2, q) and certain linear codes are strongly related and the above results from the point of view of coding theory will also be discussed.

Method of self-similar factor approximants

The method of self-similar factor approximants is completed by defining the approximants of odd orders, constructed from the power series with the largest term of an odd power. It is shown that the method provides good approximations for transcendental functions. In some cases, just a few terms in a power series make it possible to reconstruct a transcendental function exactly. Numerical convergence of the factor approximants is checked for several examples. A special attention is paid to the possibility of extrapolating the behavior of functions, with arguments tending to infinity, from the related asymptotic series at small arguments. Applications of the method are thoroughly illustrated by the examples of several functions, nonlinear differential equations, and anharmonic models.

The Role of Binary and Many-Centre Molecular Interactions in Spin Crossover in the Solid State. Part IV. Ordering in Systems with Many-Body Interactions

The theoretical formalism of solid-state spin crossover taking into account many-body interactions and ordering has been extended to the case of 5-centre interactions in a diamond lattice leading to the splitting of the free energy levels into quintets. Consecutive simplification of the obtained equations has been achieved for multiplets with regularly varying splittings: the variation according to the polynomial of the third order corresponds to 5-centre interactions; quadratic and linear variations yield the formalisms of quaternary and ternary interactions respectively; equidistant multiplets correspond to the model of binary interactions. These types of variations have been deduced from a simple microscopic model of the influence of the external molecule on the considered interaction. Parameters of the developed formalism have been expressed in terms of binary potentials and relative efficiencies of external molecules. The expressions obtained provided an adequate description of experimental two-step spin crossover curves. The formalism developed in this way was found to be similar, but not identical, to the phenomenological description based on the Landau theory. Odd power terms in composition (not entering the expansion of free energy derived from Landau theory) were found to be generally non-zero and vital for obtaining adequate descriptions of experimental data.

Recursive Observer Design, Homogeneous Approximation, and Nonsmooth Output Feedback Stabilization of Nonlinear Systems

We present a nonsmooth output feedback framework for local and/or global stabilization of a class of nonlinear systems that are not smoothly stabilizable nor uniformly observable. A systematic design method is presented for the construction of stabilizing, dynamic output compensators that are nonsmooth but Houmllder continuous. A new ingredient of the proposed output feedback control scheme is the introduction of a recursive observer design algorithm, making it possible to construct a reduced-order observer step-by-step, in a naturally augmented manner. Such a nonsmooth design leads to a number of new results on output feedback stabilization of nonlinear systems. One of them is the global stabilizability of a chain of odd power integrators by Holder continuous output feedback. The other one is the local stabilization using nonsmooth output feedback for a wide class of nonlinear systems in the Hessenberg form studied in a previous paper, where global stabilizability by nonsmooth state feedback was already proved to be possible

Almost simple groups of Suzuki type acting on polytopes

Let S = S z ( q ) S = Sz(q) , with q ≠ 2 q\neq 2 an odd power of two. For each almost simple group G G such that S > G ≤ A u t ( S ) S > G \leq Aut(S) , we prove that G G is not a C-group and therefore is not the automorphism group of an abstract regular polytope. For G = S z ( q ) G = Sz(q) , we show that there is always at least one abstract regular polytope P \mathcal {P} such that G = A u t ( P ) G = Aut(\mathcal {P}) . Moreover, if P \mathcal {P} is an abstract regular polytope such that G = A u t ( P ) G = Aut(\mathcal {P}) , then P \mathcal {P} is a polyhedron.

On terraces for abelian groups

It is known that elementary abelian 2-groups of order at least 4 do not have terraces. Bailey's Conjecture is that these are the only groups which do not. We show that abelian groups, except possibly those of order coprime to 3 whose Sylow 2-subgroup is elementary abelian of order an odd power of two, satisfy Bailey's conjecture. A consequence of this is that all 2-nilpotent groups whose Sylow 2-subgroups are abelian, but not elementary abelian, have terraces.

Exact BER computation for cross QAM constellations

When the number of bits per symbol is odd, the peak and average power of transmission can be reduced by using cross quadrature amplitude modulations (QAMs) instead of rectangular QAMs. However, since perfect Gray coding is not possible for cross QAMs, using Smith-style Gray coding, this paper derives the exact bit error rate (BER) for cross QAM constellations over additive white Gaussian noise (AWGN) and Rayleigh fading channels.

Certificates for nonnegativity of polynomials with zeros on compact semialgebraic sets

We prove a criterion for an element of a commutative ring A to be contained in an archimedean semiring T A. It can be used to investigate the question whether nonnegativity of a polynomial on a compact semialgebraic set can be certified in a certain way. In case of (strict) positivity instead of nonnegativity, our criterion simplifies to classical results of Stone, Kadison, Krivine, Handelman, Schmudgen et al. As an application of our result, we give a new proof of the following result of Handelman: If an odd power of a real polynomial in several variables has only nonnegative coecients, then so do all suciently high powers.

Higher-order semiclassical energy expansions for potentials with non-integer powers

In this paper, we present a semiclassical eigenenergy expansion for the potential |x|α when α is a positive rational number of the form2n/m (n is a positive integer and m is an odd positive integer). Remarkably, this expansion is found to be identical to the WKB expansion obtained for the potentialx N(N-even), if2n/m is replaced byN. Taking the limitm → 2 of the above expansion, we obtain an explicit asymptotic energy expansion of symmetric odd power potentials |x|2j+1 (j- positive integer). We then show how to develop approximate semiclassical expansions for potentials |x|α when α is any positive real number.

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Let G a simple group of type 2 B 2(q) or 2 G 2(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = 2 B 2(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = 2 G 2(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = 2 B 2(q), and containing M if G = 2 G 2(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.

Partial Fractions, Binomial Coefficients, and the Integral of an Odd Power of sec θ

(2002). Partial Fractions, Binomial Coefficients, and the Integral of an Odd Power of sec θ. The American Mathematical Monthly: Vol. 109, No. 8, pp. 746-749.

Coverings of Curves with Asymptotically many Rational Points,

Ihara defined the quantity A ( q ), which is the lim sup as g approaches ∞ of the ratio N q ( g )/ g , where N q ( g ) is the maximum number of rational points a curve of genus g defined over a finite field F q may have. A ( q ) is of great relevance for applications to algebraic–geometric codes. It is known that A ( q )⩽√q−1 and equality holds when q is a square. By constructing class field towers with good parameters, in this paper we present improvements of lower bounds of A ( q ) for q an odd power of a prime.

Coverings of Curves with Asymptotically many Rational Points

Ihara defined the quantity A( q), which is the lim sup as g approaches ∞ of the ratio N q ( g)/ g, where N q ( g) is the maximum number of rational points a curve of genus g defined over a finite field F q may have. A( q) is of great relevance for applications to algebraic–geometric codes. It is known that A( q)⩽√q−1 and equality holds when q is a square. By constructing class field towers with good parameters, in this paper we present improvements of lower bounds of A( q) for q an odd power of a prime.

Partial Fractions, Binomial Coefficients, and the Integral of an Odd Power of sec q

Partial Fractions, Binomial Coefficients, and the Integral of an Odd Power of sec q

A Geometric Construction of Partial Geometries with a Hermitian Point Graph

In 1998 Mathon constructed algebraically a class of partial geometriespg (q− 1, (q2− 1)/2, (q− 1)/2), where q is an even power of 3. The point graph of these partial geometries is the Hermitian graph constructed by Taylor. In this paper a geometric construction of Mathon’s partial geometries is given. The construction also works if q is an odd power of 3, yielding new partial geometries with a Hermitian point graph.

Quantitative Structure-Property Relationships Generated with Optimizable Even/Odd Wiener Polynomial Descriptors

Chemical structures of organic compounds are characterized numerically by a variety of structural descriptors, one of the earliest and most widely used being the Wiener index W, derived from the interatomic distances in a molecular graph. Extensive use of distance-based structural descriptors or topological indices has been made in QSPR and QSAR models, drug design, toxicology, virtual screening of combinatorial libraries, similarity and diversity assessment. Novel topological indices are introduced representing a partitioning of the Wiener polynomial based on counts of even and odd molecular graph distances. During the QSAR/QSPR modeling process the variables of the even and odd power functions are optimized in order to offer an improved mapping of the investigated property. These novel topological indices are tested in QSPR models for the boiling temperature, molar heat capacity, standard Gibbs energy of formation, vaporization enthalpy, refractive index, and density of alkanes. In many cases, the even/odd Wiener polynomial indices proposed here give notably improved correlations or suggest simpler QSPR models.

Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization

Without imposing any growth condition, we prove that every chain of odd power integrators perturbed by a C 1 triangular vector field is globally stabilizable via non-Lipschitz continuous state feedback, although it is not stabilizable, even locally, by any smooth state feedback because the Jacobian linearization may have uncontrollable modes whose eigenvalues are on the right half-plane. The proof is constructive and accomplished by developing a machinery – adding a power integrator – that enables one to explicitly design a C 0 globally stabilizing feedback law as well as a C 1 control Lyapunov function which is positive definite and proper.

HIGHER POWER SQUEEZING EFFECTS FOR ODD AND EVEN q-COHERENT STATES

The higher power squeezing effects of photon field for odd and even q coherent states have been studied. We find that the squeezing effects are influenced by the values of parameter q, and the odd and even q-deformed coherent states can exhibit odd power squeezing effect but no even power squeezing effect when parameter q is taken values departure from 1 greatly, and these squeezing characteristics are different from those of the conventional odd and even coherent states.

Describing the symmetry of one-dimensional figures by power sums of point coordinates

The article examines the problem of describing the symmetry of a one-dimensional figure by power sums of the coordinates of its points. In case ofn points of unit mass, the figure symmetry requires [(n+1)/2] values of odd power sums; in case of massive points,n values are required in general. A one-to-one correspondence is established between the coordinates ofn points of the figure and the values of power sums to order (2n−1) inclusive.