Sequence Sn Research Articles

On a sequence transformation with integral coefficients for Euler’s constant

Let γ \gamma denote Euler’s constant, and let \[ s n = ( 1 + 1 2 + ⋯ + 1 n − 1 ) − log ⁡ n ( n ≥ 2 ) . {s_n} = \left ( {1 + \frac {1}{2} + \cdots + \frac {1}{{n - 1}}} \right ) - \log n\quad (n \geq 2). \] We prove by Ser’s formula for the remainder γ − s n \gamma - {s_n} that for all integers n ≥ 1 n \geq 1 and τ ≥ 2 \tau \geq 2 there are integers μ n , 0 , μ n , 1 , … , μ n , n {\mu _{n,0,}}{\mu _{n,1}}, \ldots ,{\mu _{n,n}} such that \[ μ n , 0 s τ + μ n , 1 s τ + 1 + ⋯ + μ n , n s τ + n = γ + O τ ( ( n ( n + 1 ) ( n + 2 ) ∙ ⋯ ∙ ( n + τ ) ) − 1 ) , {\mu _{n,0}}{s_\tau } + {\mu _{n,1}}{s_{\tau + 1}} + \cdots + {\mu _{n,n}}{s_{\tau + n}} = \gamma + {O_\tau }({(n(n + 1)(n + 2) \bullet \cdots \bullet (n + \tau ))^{ - 1}}), \] where the constant in O τ {O_\tau } depends only on τ \tau . The coefficients μ n , k {\mu _{n,k}} are explicitly given and are bounded by 2 3 n + τ − 1 {2^{3n + \tau - 1}} .

Interspersions and dispersions

An array A = ( a i j ) A = ({a_{ij}}) of all the positive integers is an interspersion if the terms of any two rows, from some point on, alternate in size, and a dispersion if, for a suitable sequence ( s n ) ({s_n}) , the recurrence a j = s a j − 1 {a_j} = {s_{{a_{j - 1}}}} holds for each entry a j {a_j} of each row of A A , for j ⩾ 2 j \geqslant 2 . An array is proved here to be an interspersion if and only if it is a dispersion. Such arrays whose rows satisfy certain recurrences are considered.

A Tauberian Theorem for the General Euler-Borel Summability Method

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

From Discrete‐ to Continuous‐Time Finance: Weak Convergence of the Financial Gain Process1

Conditions suitable for applications in finance are given for the weak convergence (or convergence in probability) of stochastic integrals. For example, consider a sequence Sn of security price processes converging in distribution to S and a sequence θn of trading strategies converging in distribution to θ. We survey conditions under which the financial gain process θn dSn converges in distribution to θ dS. Examples include convergence from discrete‐ to continuous‐time settings and, in particular, generalizations of the convergence of binomial option replication models to the Black‐Scholes model. Counterexamples are also provided.

Large deviations for functionals of stationary processes

Let (Sn) be a sequence ofRd-valued random variables adapted to the internal history of a stationary sequence of random elements (Xn). We formulate conditions under which the principle of large deviations holds true for the sequence (Sn).

A Convergence Acceleration Method based on a good Estimation of the Absolute value of the Error

Suppose a sequence (Sn)n, converges to a limit S. In this paper we construct, from a good estimate (Dn)n of (∣Sn−S∣), an acceleration method for (Sn)n based on: (a) an automatic selection between two transformations if (Dn)n converges logarithmically; (b) composite sequence transformations if (Dn)n converges linearly.

On the luminescence of Te 4+ in A 2ZrCl 6 (A = Cs, Rb) and A 2ZrCl 6 (A = Cs, Rb, K)

The luminescence spectra and decay curves of Te 4+-doped A 2ZrCl 6 (A = Cs, Rb) and A 2ZrCl 6 (A = Cs, Rb, K) are reported. In these lattices the Te 4+ ion is octahedrally coordinated by the chloride ions. The emission consists of one band. For the compositions Cs 2ZrCl 6:Te 4+, Cs 2SnCl 6Te 4+ and Rb 2SnCl 6:Te 4+ vibrational structure is observed in the emission band at low temperatures. The excitation spectra can be interpreted by assuming a dynamical Jahn-Teller effect in the excited state of the Te 4+ ion. The vibrational frequency derived from the temperature dependence of the A band splitting is equal to the one obtained from the vibrational progression in the emission band and corresponds to the frequency of the e g mode of the TeCl 2 6− octahedron. Vibrational structure is also observed in the B excitation band of Cs 2SnCl 6: Te 4+ at low temperatures. Finally, a comparison is made with the luminescence of Sn 2+ in perovskite lattices and the luminescence of Sb 3+ in elpasolite lattices. The Stokes shift decreases in the sequence Sn 2+−Sb 3+−Te 4+ which can be explained by a decreasing contribution of the Jahn-Teller effect to the Stokes shift due to an increasing spin-orbit coupling.

A Proof of the Lucas-Lehmer Test

(1988). A Proof of the Lucas-Lehmer Test. The American Mathematical Monthly: Vol. 95, No. 9, pp. 855-856.

On Cauchy's Notion of Infinitesimal

The historical literature of nineteenth century mathematical analysis contains much discussion of Cauchy's conception of the continuum, and of the nature of his infinitesimals in particular. In a contribution to this debate, John Cleave [19 72], following Lakatos [19 78], suggests that Cauchy's 'dynamic' conception of infinitesimals can be accommodated within models of the nonstandard real numbers. Cleave interprets Cauchy's infinitesimals as values of null sequences at infinite integers: an infinitesimal x e *R is accessible if there is a null sequence (sn) (i.e. Sn-+O) such that x = SM for some infinite M. (See Cleave [1972] for the basic definitions). Cleave states (Theorem 5) that there are non-standard models *R in which every infinitesimal is accessible; in his proof he claims that this is the case whenever *R is the ultrapower RN/D, with D a non-principal ultrafilter on N. In this note we show that this claim is correct if and only if D is a special kind of ultrafilter-known as a P-point (defined below). Given the continuum hypothesis (CH) or even Martin's Axiom (MA) P-points can be shown to exist; but not all ultrafilters are P-points. Thus, given CH (or MA), Cleave's Theorem is true, but requires a more careful proof. On the other hand, it is consistent with the axioms of set theory (ZFC) that there are no P-points. These matters are discussed in Jech [1978], pp. 25 7-8. Thus it cannot be shown in ZFC that there are models RN/D in which all infinitesimals are accessible.

On an improved Erdő‐Rényi‐type law for increments of partial sums

AbstractLet X1, X2,… be an independently and identically distributed sequence with ξX1 = 0, ξ exp (tX1 < ∞ (t ≧ 0) and partial sums Sn = X1 + … + Xn. Consider the maximum increment D1 (N, K) = max0≤n ≤ N ‐ K (Sn + K ‐ Sn)of the sequence (Sn) in (0, N) over a time K = KN, 1 ≦ K ≦ N. Under appropriate conditions on (KN) it is shown that in the case KN/log N → 0, but KN/(log N)1/2 → ∞, there exists a sequence (αN) such that K‐1/2 D1 (N, K) ‐ αN converges to 0 w. p. 1. This result provides a small increment analogue to the improved Erdős‐Rényi‐type laws stated by Csörgő and Steinebach (1981).

A note on the strong regularity of Nrlund means

Let (pn), (qn) and (un) be sequences of real or complex numbers withThe sequence (sn) is strongly generalized Nrlund summable with index 0, to s, or s or snsN, p, Q ifand pnv=pnvpnv1, with p10. Strong Nrlund summability N, p was first studied by Borweing and Cass (1), and its generalization N, p, Q by Thorp (6). We shall say that (sn) is strongly generalized convergent of index 0, to s, and write snsC, 0, Q if sns and where sn=a0+a1++an. When qn all n, this definition reduces to strong convergence of index , introduced by Hyslop (4). If as n, the sequence (sn) is summable (, q) to s sns(, q).

The Weak Basis Theorem Fails in Non-Locally Convex F-Spaces

W. J. Stiles showed in [10, Corollary 4.5] that Banach's weak basis theorem fails in the spaces lp, 0 < p < 1. Then, J. H. Shapiro [9] indicated certain general classes of non-locally convex F-spaces with the same property, and asked whether the weak basis theorem fails in every non-locally convex F-space with a weak basis. Our purpose is to answer this question in the affirmative. In [3] we observed that, essentially, the only case that remained open is that of an F-space with irregular basis (en), i.e. such that snen →0 for any scalar sequence (sn).

Dopant and Alloying Effects in Gamma Irradiated GaAs Light Emitters

Techniques for producing radiation hardening in GaAs light emitters by use of donor dopants and isovalent anion substitutional alloying are reported. Six different donor dopants, within the concentration range from 1 to 20 x 1017 cm-3, were added to GaAs substrates. P-N light emitters were then fabricated and exposed to gamma irradiation from a 60Co source in steps to a total absorbed dose of 108 rads(Si). Results of the study are reported in terms of the damage parameter K? ?? where K? is the damage coefficient and ?? is the preirradiated minority carrier lifetime. Selenium (Se) was the most effective dopant for hardening GaAs light emitters followed in order of decreasing effectiveness by Sn, Si (donor), Te, Ge (amphoteric), and Si (amphoteric). Two additional observations in this portion of the investigation also seemed to favor the use of Se for hardening: (1) The recovery of the emitters following isochronal annealing occurs in the sequence Se, Sn, Si, Te, and Ge, with the Se-doped samples showing the greatest annealing (to 60% of original quantum efficiency). (2) The shift of the peak emitting wavelength with irradiation (always toward shorter wavelengths) is found to be least for the Se-doped samples. The second technique for hardening involved substituting the isovalent anion phosphorus (P) into the GaAs lattice in place of As. This action resulted in a lowering of the damage coefficient, K? , by several orders of magnitude as the P content increased to about 50%.

On three problems for sequences

If ξ∈ (0,1) and A=an, nɛℕ is a sequence of real numbers define Sn(ξ,A)∶=Σ{ak∶:k=[nξ]+1 to n}, nɛℕ, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence Sn (ξ,A), nɛℕ, for all ξ in an appropriate set K of real numbers, that the sequence an, nɛℕ, converges to zero. It was shown that such a conclusion is possible if K={ξ,1−ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper: 1) does the convergence of Sn (ξ,A), nɛℕ, for a single irrational number ξ imply an→0. 2) does the convergence of Sn(ξ,A), nɛℕ, for finitely many rational numbers ξ∈ (0, 1) imply an→0. 3) does the convergence of Sn (ξ,A), nɛℕ, for all rational numbers ξ∈ (0,1) imply an→0?

A Note on Logarithmic Summability (L)

It is well known that two mutually related summability methods for a sequence sn(n = 0, 1, 2, …) are any Cesaro method (C, γ) of order γ > 0 and the Abel method (A). The notation used in this statement is that of Hardy ((1), pp. 96-7, 71) and the statement itself can be amplified as follows. Summability (C, γ), γ > 0, of sn implies (i) sn = o(nγ), (ii) summability (A) of sn. Also, as a conditional converse of this result, we have Offord's result ((3), first part of Theorem 2), that hypothesis (i), and hypothesis (ii) suitably strengthened, together imply summability (C, γ), γ > 0, of sn. It is the object of this note to bring to light a second pair of summability methods mutually related like the methods (C, γ) and (A), by following an argument which is essentially similar to Offord's but differs sufficiently from Offord's in details to justify a separate and self-contained treatment of the second pair of methods.

The Distribution of Sequences and Summability

We suppose that 0 <sn≤ 1 for everyn, and denote byn(α, β) the number ofS0, S1,S2, . . . ,Snwhich fall in the interval 0 ≤ α <x≤ β ≤ 1. If there exists a functiong(t), 0 ≤t≤ 1, such thatfor every interval (α, β] with 0 ≤ β — α ≤ 1, the sequence (Sn) is said to have a distribution functiong(t), 0 ≤t≤ 1, in the interval [0, 1], (see 9, p. 87).

Knopp’s core theorem and subsequences of a bounded sequence

Let sn, be a sequence in a p-dimensional Euclidean space EP. Let Kn=K(sn, Sn+l, ) be the convex hull of Sn Sn+l, and Kn its closure. The core of Sn is defined as n ln =Kn. Knopp's core theorem states that if A = (aij) is an infinite regular mnatrix with nonnegative elements, then the core of the A-transform of Sn is contained in the core of sn. In particular if Sn is bounded, every A-limit of a subsequence of sn is contained in the convex hull of limit points of Sn. With certain restrictions on A, the converse is also true; i.e., for any element t in the convex hull of limit points of sn, there is a subsequence of Sn which is A-limitable to t. The main objective of this paper is to show that for any t in the convex hull of limit points of a bounded sequence Sn, there is a subsequence of sn which is C1and E1-limitable to t. The following is Knopp's core theorem in EP.

On convergence fields of Nörlund means

If 0n = s+o(l) when nw, the sequence { s, } (of complex numbers) is said to be limitable N(p) to the value s. If 0n =o(1), we shall write snGo(N(p)), denoting by o(N(p)) the set of all the sequences limitable N(p) to zero. If o-n=a(l) when ncc',2 the sequence {s4I is said to be absolutely limitable N(p), and we shall write snCa(N(p)), denoting by a(N(p)) the set of all the sequences absolutely limitable N(p). A N6rlund mean is called regular if any convergent sequence sn-)s is transformed by (1) into a convergent sequence 0n)-s. Necessary and sufficient conditions in terms of the sequence { pn } (of complex numbers) in order that (1) be regular are

Ergodic Property of the Brownian Motion Process

In a previous paper (“On the Equidistribution of Sums of Independent Random Variables,” to appear elsewhere; an abstract will appear in Bull. Am. Math. Soc., 59, (May, 1953); we shall refer to this paper as [1]) we considered some properties of the sequence Sn of partial sums of independent and identically distributed random variables or random vectors in two dimensions. Here we show in Theorems 1-4 that the results of [1] carry over to the Brownian motion process in one and two dimensions.

Direct Theorems on Methods of Summability

1.1. A regular Toeplitz method of summability is given by a transformation m = 0, 1, 2 , … of the sequence sn into the sequence σm. According to the definition of regularity, every such method sums a convergent sequence sn to the value lim sn. The question naturally arises, whether there are more extensive classes of sequences summable by all regular methods 1.1(1) or at least by all such methods subject to some simple additional conditions. Questions of this kind have been treated by the author (Lorentz [2], [5]) and, from another point of view, by R. P. Agnew [2] [3] ; in this paper we wish to discuss the problem systematically.