Sequence Pn Research Articles

Total variation convergence preserves conditional independence

This note establishes that if a sequence Pn,n=1,… of probability measures converges in total variation to the limiting probability measure P, and σ-algebras A and B are conditionally independent given H with respect to Pn for all n, then they are also conditionally independent with respect to the limiting measure P. As a corollary, this also extends to pointwise convergence of densities to a density.

Electronic structure and complexation behavior of methyl substituted phosphonate ligands with uranyl nitrate

Quantum chemical calculations were applied to investigate the electronic structures and complexation behaviour of methyl-substituted phosphonic acids with uranyl (VI) nitrate. The Natural Bond Orbital (NBO) analysis indicated that the methyl groups in substituted phosphonic acids exhibit electron-withdrawing properties. Additionally, the methyl groups directly bonded to the phosphoryl P atom prefer a gauche orientation relative to the P=O group in compounds such as MPn, DMMP and DMPn. Weak to moderate H-bonding interactions provided extra stability to Pn, MHPn, MPn and DMPn complexes with U(VI). The ν(P=O) values were red-shifted in all the complexes, implying a moderate weakening of the P=O bond upon complexation. Furthermore, the complexation energy values indicated an increasing complexation tendency in 1:2 complexes, particularly when the methyl groups were introduced. This trend was observed in the sequence Pn → MHPn → MPn and DMHP → DMMP → DMPn.

Limit laws for the norms of extremal samples

Denote Sn(p)=kn−1∑i=1knlog(Xn+1−i,n∕Xn−kn,n)p, where p>0, kn≤n is a sequence of integers such that kn→∞ and kn∕n→0, and X1,n≤⋯≤Xn,n are the order statistics of iid random variables X1,…,Xn with regularly varying upper tail of index 1∕γ. The estimator γ̂(n)=(Sn(p)∕Γ(p+1))1∕p is an extension of the Hill estimator. We investigate the asymptotic properties of Sn(p) and γ̂(n) both for fixed p>0 and for p=pn→∞. We prove consistency for γ̂(n) and limit theorem for γ̂(n)−γ under appropriate assumptions. We obtain both Gaussian and non-Gaussian (stable) limit depending on the growth rate of the power sequence pn. Applied to real data we find that for larger p the estimator is less sensitive to the change in kn than the Hill estimator.

Carbon and nitrogen in forest floor and mineral soil under four forest species in the Mediterranean region

The organic and mineral horizons of soils are of great importance in C and N storage in forest areas. However, knowledge of the effects of forest species on the stocks of these elements is still scarce, especially in Portugal. In order to contribute to this knowledge, a study was carried out in forest stands of <em>Pinus pinaster</em> Aiton (PP), <em>Pinus nigra</em> Arnold (PN), <em>Pseudotsuga menziesii</em> (PM) and <em>Castanea sativa</em> Miller (CS), installed in the 1950s in northern Portugal. Sampling areas with similar topography, lithology and climate were selected, in order to better identify hypothesized differences in C and N storage due to forest species effect. In each stand, 15 sites were selected randomly and the forest floor (organic layers) was collected in a 0.49 m<sup>2</sup> area. The layers H, L and F of the forest floor were identified and, for L and F, their components were separated in leaves, pine cones/chestnut husks and branches. At the same sites, soil samples were also collected at 0-10 and 10-20 cm depth. At these depths, undisturbed samples were also collected for bulk density determination. The concentrations of C and N were determined in forest floor and mineral components of the soil, and converted in mass per unit area. The quantity of C storage per unit area followed the sequence PN > PM > CS > PP, while for N the sequence was CS > PM > PN > PP, OM and PP keeping the same relative position in the sequence in both C and N concentrations. The PM and CS species store similar amounts of C and N, and about 90% of these elements is found in the upper 20 cm of the mineral soil. In PN and PP species, the contribution of forest floor to the storage of these elements is more expressive than in the other species, but lower than 30% in all cases.

Growth of the Sudler product of sines at the golden rotation number

We study the growth at the golden rotation number ω=(5−1)/2 of the function sequence Pn(ω)=∏r=1n|2sin⁡πrω|. This sequence has been variously studied elsewhere as a skew product of sines, Birkhoff sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence Pn, namely the sub-sequence Qn=|∏r=1Fn2sin⁡πrω| for Fibonacci numbers Fn, and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of Pn(ω) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (2011) [10].

Convex polynomial approximation in [formula omitted] with Freud weights

We show that for multivariate Freud-type weights Wα(x)=exp(−|x|α), α>1, any convex function f on Rd satisfying fWα∈Lp(Rd) if 1≤p<∞, or lim|x|→∞f(x)Wα(x)=0 if p=∞, can be approximated in the weighted norm by a sequence Pn of algebraic polynomials convex on Rd such that ‖(f−Pn)Wα‖Lp(Rd)→0 as n→∞. This extends the previously known result for d=1 and p=∞ obtained by the first author to higher dimensions and integral norms using a completely different approach.

Modular Divisor Functions and Quadratic Reciprocity

It is a well-known result by B. Riemann that the terms of a conditionally convergent series of real numbers can be rearranged in a permutation such that the resulting series converges to any prescribed sum s: add p1 consecutive positive terms until their sum is greater than s; then subtract q1 consecutive negative terms until the sum drops below s, and so on. For the alternating harmonic series, with the aid of a computer program, it can be noticed that there are some fascinating patterns in the sequences pn and qn. For example, if s = log 2 + (1/2) log (38/5) the sequence pn is 5, 7, 8, 7, 8, 7, 8, 8, 7, 8, 7, 8, . . . in which we notice the repetition of the pattern 8, 7, 8, 7, 8, while if s = log 2+ (1/2) log (37/5) the sequence pn is 5, 7, 7, 7, 8, 7, 8, 7, 7, 8, 7, 8, . . . in which the pattern is 7, 7, 8, 7, 8. Where do these patterns come from? Let us observe that 38/5 = 7 + 3/5 and 37/5 = 7 + 2/5. The length of the repeating pattern is the denominator 5, the values of pn, at least from some n on, are 7 and 8, and the number 8 appears 3 times in the pattern of the first example, and 2 times in that of the second one. These are not coincidences: we explain them in this paper.

On Riemann's Rearrangement Theorem for the Alternating Harmonic Series

Let pn be the number of consecutive positive summands and let qn be the number of consecutive negative summands that appear in the classical Riemann’s rearrangement of the alternating harmonic series to sum a prescribed real number s. Assume that s > log2 and let x = (1=4)e 2s . It is shown that the sequence qn is constant equal to 1, and that the values of pn become stabilized: Eventually pn = bxc or pn =bxc + 1. Moreover, it is shown that x is rational if and only if the sequence pn is eventually periodic. The sequence pn is eventually constant if and only if x is integer, in which case pn =bxc for n big enough. Similar results are also true for s < log2.

The Berman conjecture is true for nilpotent extensions of regular semigroups

Let S be any semigroup. A term operation of S is an n-ary operation of S which is induced by some nonempty word w by substitution of letters. Such an operation is essentially n-ary if it depends on all of its variables. For n ≥ 1, we denote the set of all essentially n-ary term operations of S by ES n , while ES 0 is the set of all constant unary term operations of S. Let pn(S) = |ES n | for all n ≥ 0. It is easy to see that the sequence pn(S), n ≥ 0, consists entirely of finite cardinals if and only if S generates a locally finite semigroup variety. Of course, all these concepts can be generalized for arbitrary algebras, cf. the survey paper by Gratzer and Kisielewicz [7]. The theory of pn-sequence of general algebras was founded by E. Marczewski and his ‘Wroc law School’ back in the sixties (see [9]). They identified as the main goal of this theory to characterize all sequences of non-negative integers which are representable as the pn-sequence of some algebra (or some particular kind of algebra). Later on, most of the research was limited to finite algebras. One of the most intriguing hypotheses in this direction was given by J. Berman, who conjectured in [1] that the pn-sequence of any finite algebra is either bounded above by a constant, or eventually strictly increasing. While this assertion, referred to as the Berman conjecture, is easily shown to be true for finite monoids, groups, rings, modules, lattices, Boolean algebras, etc., R. Willard refuted this conjecture in [10] by constructing a very nice counterexample, a 4-element algebra with finitely many basic operations whose pn-sequence is (0, 3, 2, 5, 4, 7, 6, 9, 8, . . .). On the other hand, there are several papers (co-authored by the first named author of this note) investigating the (still open) question whether the Berman conjecture holds in the class of all finite semigroups. Namely, in [5] all finite semigroups whose pn-sequences have constant bounds are determined. In [6],

Growing random sequences

We consider the random sequence xn = xn−1 + γxq, with γ > 0, where q = 0, 1, ..., n − 1 is chosen randomly from a probability distribution Pn(q). When all q are chosen with equal probability, i.e. Pn(q) = 1/n, we obtain an exact solution for the mean xn and the divergence of the second moment x2n as functions of n and γ. For γ = 1 we examine the divergence of the mean value of xn, as a function of n, for the random sequences generated by power law and exponential Pn(q) and for the non-random sequence Pn(q) = δq,β(n−1).

Orthonormal bases of polynomials in one complex variable

Let a sequence (Pn) of polynomials in one complex variable satisfy a recurrence relation with length growing more slowly than linearly. It is shown that (Pn) is an orthonormal basis in L2μ for some measure μ on C, if and only if the recurrence is a 3-term relation with special coefficients. The support of μ lies on a straight line. This result is achieved by the analysis of a formally normal irreducible Hessenberg operator with only finitely many nonzero entries in every row. It generalizes the classical Favard's Theorem and the Representation Theorem.

On the almost sure approximation of self-adjoint operators in L2 (0, 1)

There are several important theorems concerning the almost sure convergence of (monotone) sequences of orthogonal projections in L2-spaces. Let us mention here the martingale convergence theorems or the results on the developments of functions with respect to orthogonal systems. On the other hand every self-adjoint operator with the spectrum on the interval [0, 1] is a limit of some sequence of orthogonal projections in the weak operator topology (see [1]). This paper is devoted to a problem of approximation of a self-adjoint operator A acting in L2 (0, 1) by a sequence Pn of orthogonal projections in the sense that

Invariant domains and singularities

AbstractLet U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pn∈P(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) &gt; 0. In this example, U is bounded by a Jordan curve.

A Lower Bound for Orthogonal Polynomials with an Application to Polynomial Hypergroups

We give a lower bound for solutions of linear recurrence relations of the form zan = ∑n+Nk=n−N αk, nak, whenever z is not in the lp-spectrum of the corresponding banded operator. In particular if Pn, are polynomials orthonormal with respect to a measure μ supported in a bounded interval the sequence Pn(x)2 + Pn+1(x)2 is bounded from below by (1 + ϵ)n, for x ∉ supp μ. We give an application to polynomial hypergroups.

Weak convergence of asymptotically homogeneous functions of measures

FUNCTIONS of measures have already been introduced and studied by Goffman and Serrin [8], Reschetnyak [lo] and Demengel and Temam [5]: especially for the convex case, some lower continuity results for the weak tolopology (see [5]) permit us to clarify and solve, from a mathematical viewpoint, some mechanical problems, namely in the theory of elastic plastic materials. More recently, in order to study weak convergence of solutions of semilinear hyperbolic systems, which arise from mechanical fluids, we have been led to answer the following question: on what conditions on a sequence pn of bounded measures have we f(~,) --* f(p) where P is a bounded measure and f is any “function of a measure” (not necessarily convex)? We answer this question in Section 1 when the functions f are homogeneous, and in Section 2 for sublinear functions; the general case of asymptotically homogeneous function is then a direct consequence of the two previous cases! An extension to x dependent functions is described in Section 4. In the one dimensional case, we give in proposition 3.3 a criteria which is very useful in practice, namely for the weak continuity of solutions measures of hyperbolic semilinear systems with respect to weakly convergent Cauchy data. Another application concerns a work in preparation [6) on measure-valued solutions for hyperbolic scalar equations. The results of this paper were announced in [2].

Orthogonality via Transforms

Let e(x, t) = ∑pn(x)tn be the generating function of a polynomial sequence, and the transform of multiplication by x relative to e(x, t). We show that the sequence pn(x) is orthogonal precisely when is a t‐variable, i.e., maps K[t] into itself and increases degree by 1. We also show how transform techniques can shed light on the recursion relations and differential equations for pn(x).

Optimization of Coded Spread Spectrum System Performance

Error correction coding techniques significantly improve performance of spread spectrum communication systems in environments containing jamming, multipath, and unregulated multiple access. This paper investigates the optimization of spread spectrum system performance for time-varying unknown interference. Noncoherent frequency hopping (FH) spread spectrum modulation, and hybrid FH-PN incorporating a direct sequence PN modulation on each hopped frequency are studied. For FH or FH-PN, the data modulations considered are differential phase-shift-keying (DPSK), differential quadriphase-quadriphase-shift-keying (DQPSK), and multiple-frequency-multiple-frequency-shift-keying (MFSK). Both block and convolutional error correction coding techniques are studied as a means of improving the spread spectrum performance.

Approximation by rational and meromorphic functions having a bounded number of free poles

then one can deduce certain properties of the function g(z) and the sequence Pn(Z). As a consequence of the Maximum Principle g(z) is the set of boundary values on r of a functionf(z) which is analytic in the interior D of P and continuous on the closed region D + P. It is also clear that the sequence pn(Z) converges to f(z) at each point of D. In addition, some continuity properties of g(z) on r may be deduced if an estimate is known on the rapidity of convergence of the sequence e,n. Indeed, the inequality En < A/n' + , where k is a nonnegative integer and 0 < a < 1, implies that the kth derivative of g(z) exists on r (in the one-dimensional sense) and satisfies there a Lipschitz condition of order a. In this paper we make the weaker assumption that the function g(z) is the uniform limit on r of a sequence of rational functions each having at most v free poles, and we establish analogues of the above mentioned conclusions. Specifically we shall deal with rational functions of type (n, v), i.e., rational functions of the form

The extremal values of Legendre polynomials and of certain related functions

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show thatwhere jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.

On a theorem of Erdös and Turán

changes its sign an infinite number of times. The aim of the present paper is to prove an improvement of this theorem. It shall be proved that the total curvature of the polygonal line having the vertices n+i log Pn (n = 1, 2, 3, ***) in the complex z-plane is infinite. This implies that Pnu?lPn-1 -p2 changes its sign an infinite number of times, but our result also contains some quantitative information regarding the oscillations of the sequence Pn+IPPn-lp.2 The total curvature G of a finite polygonal line situated in the complex z-plane may be defined as follows: