Asymptotic Basis Research Articles

Partitions of bases into disjoint unions of bases

Let A be an asymptotic basis of order h in the sense of additive number theory, and let f( n) denote the maximum number of pairwise disjoint representations of n in the form n = a i 1 + a i 2 + … + a i n , where a i j ∈ A and a i 1 ≤ a i 2 ≤ … ≤ a i h . Let t ≥ 2. If f( n) ≥ clog n for c sufficiently large, then A can be written in the form A = A 1 ⌣ … ⌣ A t , where A i ⌢ A j ≠ ⊘ for 1 ≤ i < j ≤ t and A j is an asumptotic basis of order h for j = 1, …, t. If lim n → ∞ f(n) log n = ∞ , then A = ∪ j = 1 ∞ A j , where A i ⌢ A j = ⊘ for i ≠ j and each A j is an asymptotic basis of order h. These results are obtained by means of some purely combinatorial theorems. Related open problems are also discussed.

Coulomb pair-creation I

Electron — positron pair creation in strong Coulomb fields is outlined. It is shown that the singular behaviour of the adiabatic basis can be removed if solutions of the time dependent external field Dirac equation are used as a basis to expand the fermion field operator. This latter “asymptotic basis” makes it possible to introduce Feynman-propagator. Applying the reduction technique, the computation of all of the basic quantities can be reduced to the solution of an integral equation. The positron spectrum for separable potential model with Lorentzian time dependence and for potential jump is analyzed in the pole approximation in the second part of our series.

Exact order of subsets of asymptotic bases in additive number theory

Let A be an asymptotic basis of order h. Define I k(A)={F|F⫅A,|F|=k andA⧹F is a basis} where | F| indicates the number of elements in the finite set F. We denote by g( A) the exact order of A. Define ▪ An open problem in additive number theory is to calculate G k ( h). We prove in this paper that G k(h)⩾( 4 3 )( h k+1 k+1+O(h k) as h tends to infinity, which is an improvement of a result of Melvyn B. Nathanson (The exact order of subsets of additive bases, in “Proceedings, New York Number Theory Seminar, 1982,” Lecture Notes in Mathematics, Vol. 1052, pp. 273–277, Springer-Verlag, 1984) . On the other hand, we estimate G k ( h) as k tends to infinity for any fixed integer h. It is proved that G k ( h) has order of magnitude k h−1 as k tends to infinity for any fixed h ≥ 2.

An extremal problem for least common multiples

Let A be a set of natural numbers, and let [ A] h denote the set of all least common multiples [ a 1,…, a h ] with a i εA. If nε[ A] h for all sufficiently large integers n, then A is an asymptotic LCM basis of order h. If n∉[ A] h for infinitely many n⩾1, then A is an asymptotic LCM nonbasis of order h. The nonbasis A is maximal if A∪{ b} is an asymptotic LCM basis of order h for every natural number b∉ A. In this paper the structure of all maximal asymptotic LCM bases of order h is determined.

Relativistic photon-maxwellian electron cross sections

Abstract Temperature-corrected cross sections, complementing the Klein-Nishina set, are developed for astrophysical, plasma, and transport applications. The set is obtained from a nonlinear least squares fit to the exact photon-Maxwellian electron cross sections, using the static formula as the asymptotic basis. Two parameters are sufficient (two decimal places) to fit the exact cross sections over a range of 0–100 keV in electron temperature, and 0–1 MeV in incident photon energy. The Et is made to the total cross sections, yet the parameters predict both total and differential scattering cross sections well. Corresponding differential energy cross sections are less accurate. An extended fit to (just) the total cross sections, over the temperature and energy range 0–5 MeV, is also described.

Waring’s problem for finite intervals

Let f ( n , k , s ) f(n,k,s) denote the cardinality of the smallest set A A of nonnegative k k -th powers such that every integer in [ 0 , n ] [0,n] is a sum of s s elements of A A , and let β ( k , s ) = lim su p n → ∞ log ⁡ f ( n , k , s ) / log ⁡ n \beta (k,s) = {\text {lim su}}{{\text {p}}_{n \to \infty }}\log f(n,k,s)/\log n . Clearly, β ( k , s ) ⩾ 1 / s \beta (k,s) \geqslant 1/s . In this paper it is proved that f ( n , k , s ) &gt; c n 1 / ( s − g ( k ) + k ) f(n,k,s){\text { &gt; }}c{n^{1/(s - g(k) + k)}} for all n ⩾ n 1 ( k , s ) n \geqslant {n_1}(k,s) , where g ( k ) g(k) is defined as in Waring’s problem, and β ( k , s ) ∼ 1 / s \beta (k,s) \sim 1/s as s → ∞ s \to \infty .

Divisibility properties of additive bases

Let h ⩾ 2 h \geqslant 2 . There exists an asymptotic basis A A of order h h such that ( a 1 , … , a k ) &gt; 1 ({a_1}, \ldots ,{a_k}){\text { &gt; 1}} for all a 1 … , a k ∈ A {a_1} \ldots ,{a_k} \in A if and only if k &gt; h k{\text { &gt; }}h . If k ⩾ h k \geqslant h , the sumset h A hA contains only composite numbers. For h = k h = k , there exists a set A A of nonnegative integers with ( a 1 , … , a h ) &gt; 1 ({a_1}, \ldots ,{a_h}){\text { &gt; }}1 for all a 1 , … , a h ∈ A {a_1}, \ldots ,{a_h} \in A such that for every prime p p the sumset h A hA contains all sufficiently large multiples of p p .

Testing for symmetry and independence in a bivariate exponential distribution

Several bivariate exponential distributions have been proposed in the literature. A common problem for independent exponentials is to test the quality of the two distributions. The analogous problem for bivariate exponentials is to test for symmetry. For the bivariate exponential model of Freund (1961, Journal of the American Statistical Association 56, 971–977), tests of symmetry and independence are derived and the small sample distributions of the test statistics are found. The power function of the tests are calculated. The efficiency of the tests is found to be high on both an asymptotic and small sample basis.

Cofinite subsets of asymptotic bases for the positive integers

Let A be an infinite set of integers containing at most finitely many negative terms. Let hA denote the set of all integers n such that n is a sum of h elements of A. Let F be a finite subset of A. Theorem. If hA contains an infinite arithmetic progression with difference d, and if gcd{a−a′¦a,a′∈A\\F} = d , then there exists q such that q( A\\ F) contains an infinite arithmetic progression. In particular, if A is an asymptotic basis, then A\\ F is an asymptotic basis if and only if gcd{ a− a′| a, a′∈ A\\ F} = 1. Theorem. If A is an asymptotic basis of order h, and if F⊆ A, card( F) = k, and A\\ F is an asymptotic basis, then the exact order of A\\ F is O( h k+1 ).

Minimal asymptotic bases for the natural numbers

The sequence A of nonnegative integers is an asymptotic basis of order h if every sufficiently large integer can be written as the sum of h elements of A. Let M h A denote the set of elements that have more than one representation as a sum of h elements of A. It is proved that there exists an asymptotic basis A such that M h A(x) = O(x 1−1 h+ϵ ) for every ϵ > 0. An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. It is proved that there does not exist a sequence A that is simultaneously a minimal basis of orders 2, 3, and 4. Several open problems concerning minimal bases are also discussed.

Asymptotically robust quantization for detection

The problem of designing quantizer-detectors whose performance is insensitive to small deviations in noise statistics is considered. The problem is approached on a small-signal asymptotic basis using the Huber-Tukey [1] contaminated density class to model the noise. By applying a formulation originally noted by Martin and Schwartz, it is shown that the proposed robust quantizer has properties exhibited by established robust procedures. As an example, results are presented for the case of contaminated Gaussian noise for various degrees of contamination.

Sets of natural numbers with no minimal asymptotic bases

The set A of natural numbers is an asymptotic basis for S if the sets S and 2A eventually coincide. An asymptotic basis A for S is minimal if no proper subset of A is a basis for S. Sets S are constructed which possess infinitely many asymptotic bases, but no minimal asymptotic basis.

A theory on transition of ignition phase of coal particles

A century-long held notion was that the ignition of coal particles always occurs in the gas phase, as in a totally pyrolysing material. Recent experimental results disprove such a sequence, at least for small bituminous coal particles. Coals undergo only partial pyrolysis and hence ignition could occur in one of two possible modes: (i) exothermic gasification due to heterogeneous oxidation (regime I), and (ii) endothermic pyrolysis and subsequent oxidation of the pyrolysate in the gas phase (regime II). A steady-state ignition theory is developed for partially pyrolysing solid particles. The theory is formulated on an asymptotic basis: regime I is assumed to proceed independent of regime II. For regime I, Semenov's thermal theory is used to obtain the heterogeneous ignition temperature (HIT) and the solutions reveal that an increase in particle size and oxygen concentration results in decrease of HIT. For regime II, an adiabatic criterion is invoked to define the gas ignition temperature (GIT). The solutions reveal that a decrease in particle size and an increase in oxygen concentration increases the GIT. Superimposing the results for HIT and GIT, the conditions for transition of ignition phase (TIP) are found. The Pittsburgh seam bituminous particles of size below 350 μm may not undergo gas phase ignition, confirming recent experimental results.

Asymptotic nonbases which are not subsets of maximal asymptotic nonbases

Let A be a set of positive integers. If all but a finite number of the positive integers can be written as a sum of h elements of A, then A is called an asymptotic basis of order h. Otherwise, A is called an asymptotic nonbasis of order h. For each h ⩾ 2 h \geqslant 2 , we construct an asymptotic nonbasis of order h which is not a subset of a maximal asymptotic nonbasis of order h.

Maximal asymptotic nonbases

Let $A$ be a set of nonnegative integers. If all but a finite number of positive integers can be written as a sum of $h$ elements of $A$, then $A$ is an asymptotic basis of order $h$. Otherwise, $A$ is an asymptotic nonbasis of order $h$. A class of maximal asymptotic nonbases is constructed, and it is proved that any asymptotic nonbasis of order 2 that satisfies a certain finiteness condition is a subset of a maximal asymptotic nonbasis of order 2.

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis A of order h is minimal if no proper subset of A is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal. If B is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of B is a basis (resp. asymptotic basis) of order h, then B is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.

The method of generating functions for the calculation of matrix elements in the deformed-harmonic oscillator basis

The method of generating functions which was previously only employed for the spherical basis of harmonic-oscillator single-particle wave functions is here generalized to the deformed (=cylindrical = asymptotic) basis. One-center and two-center matrix elements which are important in fission or heavy-ion scattering theories are obtained for various operators and potentials, i.e., spin-orbit, l-squared, Gaussian, or Gaussian multiplied by a polynomial, etc. If they cannot be calculated explicitely, recurrence formulae are evaluated. The method circumvents integrating the matrix elements by the introduction of generating parameters and looking for the coefficient of some power of the generating parameter in the expansion of the generating function. The method is further simplified by transforming the operators acting on the wave functions into operators which act onto the generating function or into functions to be multiplied by the generating integral.

Single-particle energy levels in a two-centre shell model

A two-centre shell model, which behaves correctly at the two limits of one or two fragments, is formulated. The parameters of the model are adjusted so that equilibrium level orders are approximately reproduced. A classical Thomas-Fermi prescription is used for the volume conservation condition. Asymptotic basis states are employed and found adequate also for very constricted shapes.

Skyrme's interaction in the asymptotic basis

Skyrme's interaction in the asymptotic basis

Nilsson's Hamiltonian in the asymptotic basis

In this work, the set of eigenvectors common to the axially symmetrical deformed harmonic oscillator and to the third component of angular momentum, is used as a basis of representation (asymptotic basis). In this basis, the matrix elements of Nilsson's Hamiltonian are calculated by means of creation and destruction operators. A rather simple calculation gives, after a diagonalisation, level diagrams as a function of deformation. The influence of various corrective terms, used instead of l 2 and l · s (particularly l T Nilsson's term) is discussed. Within the framework of these approaches and in order to test them, we calculate total energy as a function of deformation for some nuclei. Such a more precise calculation allows also a study of pseudocrossings of levels associated with ΔN = ± 2.