Sequence Sn Research Articles

Note on the Multifractal Formalism of Covering Number on the Galton-Watson Tree

We consider, for t in the boundary of Galton-Watson tree (∂T), the covering number Nn(t) by cylinder of generation n. For a suitable set I and a sequence (sn,γ), we establish almost surely, and uniformly on γ, the Hausdorff and packing dimensions of the set {t ∈ ∂T : Nn(t) − nb ∼ sn,γ} for b ∈ I.

Divisor sequences of atoms in Krull monoids

The divisor sequence of an irreducible element (atom) a of a reduced monoid H is the sequence (sn)n∈ℕ where, for each positive integer n, sn denotes the number of distinct irreducible divisors of an. We investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying nonunique direct-sum decompositions of modules over local Noetherian rings for which the Krull–Remak–Schmidt property fails.

Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤ q

Abstract Let q be a positive integer and 𝒮 = { x 0 , x 1 , ⋯ , x T − 1 } ⊆ ℤ q = { 0 , 1 , … , q − 1 } {\scr S} = \{{x_0},{x_1}, \cdots ,{x_{T - 1}}\}\subseteq {{\rm{\mathbb Z}}_q} = \{0,1, \ldots ,q - 1\} with 0 ≤ x 0 < x 1 < ⋯ < x T − 1 ≤ q − 1. 0 \le {x_0} < {x_1} <\cdots< {x_{T - 1}} \le q - 1. . We derive from S three (finite) sequences: (1) For an integer M ≥ 2let (sn )be the M-ary sequence defined by sn ≡ xn +1 − xn mod M, n =0, 1,...,T − 2. (2) For an integer m ≥ 2let (tn ) be the binary sequence defined by s n ≡ x n + 1 − x n mod M , n = 0 , 1 , ⋯ , T − 2. \matrix{{{s_n} \equiv {x_{n + 1}} - {x_n}\,\bmod \,M,} & {n = 0,1, \cdots ,T - 2.}\cr} n =0, 1,...,T − 2. (3) Let (un ) be the characteristic sequence of S, t n = { 1 if 1 ≤ x n + 1 − x n ≤ m − 1 , 0 , otherwise , n = 0 , 1 , … , T − 2. \matrix{{{t_n} = \left\{{\matrix{1 \hfill & {{\rm{if}}\,1 \le {x_{n + 1}} - {x_n} \le m - 1,} \hfill\cr{0,} \hfill & {{\rm{otherwise}},} \hfill\cr}} \right.} & {n = 0,1, \ldots ,T - 2.}\cr} n =0, 1,...,q − 1. We study the balance and pattern distribution of the sequences (sn ), (tn )and (un ). For sets S with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: (1) The sequence (sn ) is (asymptotically) balanced and has uniform pattern distribution if T is of smaller order of magnitude than q. (2) The sequence (tn ) is balanced and has uniform pattern distribution if T is approximately u n = { 1 if n ∈ 𝒮 , 0 , otherwise , n = 0 , 1 , … , q − 1. \matrix{{{u_n} = \left\{{\matrix{1 \hfill & {{\rm{if}}\,n \in {\scr S},} \hfill\cr{0,} \hfill & {{\rm{otherwise}},} \hfill\cr}} \right.} & {n = 0,1, \ldots ,q - 1.}\cr} . (3) The sequence (un ) is balanced and has uniform pattern distribution if T is approximately q 2. These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.

Synthesis, crystal structure, and electrochemical hydrogenation of the La2Mg17-xMx (M = Ni, Sn, Sb) solid solutions

The crystal structure of La2Mg17-xSnx solid solution was determined by single crystal X-ray diffraction for the first time. This phase crystallizes in hexagonal symmetry with space group P63/mmc (a = 10.3911(3), c = 10.2702(3) Å, V = 960.36(6) Å3, R1 = 0.0180, wR2 = 0.0443 for the composition La3.65Mg30Sn1.10) and is related to the structure of CeMg10.3 and Th2Ni17-types which are derivative from the CaCu5-type. A series of isotypical solid solutions La2Mg17-xMx (M = Ni, Sn, Sb, x ~0.8) was synthesized and studied by X-ray powder diffraction, energy dispersive X-ray spectroscopy and fluorescent X-ray spectroscopy. All solid solutions crystallize with the structure related to the Th2Ni17-type. The electrochemical hydrogenation confirmed the similar electrochemical behavior of all studied alloys. The amount of deintercalated hydrogen depends on the physical and chemical characteristics of doping elements and increases in the sequence Sn < Mg < Sb < Ni. The most geometrically advantageous sites are octahedral voids 6h of the initial structure, thus a coordination polyhedron for H-atom is an octahedron [HLa2(Mg,M)4].

Study on Generalized Prefer-Opposite (GPO) Algorithm for Constructing the De Bruijn Binary Sequence

De Bruijn binary sequence Sn of order n is a string of bits si ∈ {0,1} = {s1, …, s2 n}, so that each string with length n, {a 1, …, a n } ∈ {0,1}n appears exactly once. De Bruijn binary sequence can be built using the Generalized Prefer-Opposite (GPO) algorithm. The GPO algorithm is based on the Feedback Shift Register (FSR), which is a clock-regulated circuit with n sequential storage units. In running the algorithm, we need input consisting of the feedback function f(x 0,x 1, …, x n−1) and initial state b =b 0,b 1, …, b n−1 . There are many combinations of feedback functions and initial states that can be used, then grouped into three families namely family one, family two and family three. The purpose of this research is studying the GPO algorithm which use the feedback function from family two and initial state b =1100 in constructing the de Bruijn binary sequence. The result of this study indicate that the GPO algorithm with the feedback function and the initial state from the family two construct a de Bruijn binary sequence.

Potential hazards of metal-contaminated soils in an estuarine impoundment

A recreational impoundment was constructed in the mid-nineteenth century on the mudflats reclaimed from the Plym Estuary (SW England) following salt marsh removal and infilling with waste soils from local catchments. Restoration of the salt marsh was attempted about 25 years ago when a regulated tidal exchange system was installed in the embankment separating the impoundment from the estuary. Currently, the embankment is disintegrating with the potential loss of the impounded soils, of unknown composition, to the estuary. Cores were obtained from the impoundment and the adjoining estuary, sectioned, dried and analysed. The geochronology of the soils, and estuarine sediments, was established using gamma-ray spectroscopy to determine the activities of fallout radionuclides, 137Cs and 210Pb. The concentrations of As, Co, Cr Cu, Fe, Mn, Ni, Pb, Rb, Sn, W and Zn in the core sections were determined by quantitative X-ray fluorescence spectrometry. Below a shallow surface layer (> 5 cm and post-1963), metal concentrations were high with several exceeding soil quality indices, and enrichment factors (EFs) were elevated, in the sequence Sn > W≈As > Cu > Pb. Estimates of the total masses of particulate Sn, Pb, As and Cu available for down-estuary migration were significant. Given the ecotoxicological implications resulting from a loss of metal-contaminated soils into the estuary, a strategy for the future management of the impoundment is required. The conditions at this site are compared with ageing estuarine impoundments at other locations, where polluted sediments, or soils, could be vulnerable to release.

Experimental measurement and thermodynamic model predictions of the distributions of Cu, As, Sb and Sn between liquid lead and PbO–FeO–Fe2O3–SiO2 slag

AbstractDue to the increasing complexity of materials processed in primary and secondary lead smelting, better control of impurity elements is required. In the present study, distributions of Cu, As, Sb and Sn between PbO–FeO–Fe2O3–SiO2 slag and Pb metal are characterized experimentally and analyzed using thermodynamic calculations. Experimental methodology involved closed-system equilibration of sample mixtures at high temperature followed by rapid quenching. The compositions of phases were measured using electron probe X-ray microanalysis and laser ablation inductively coupled plasma mass spectrometry. Thermodynamic calculations were performed using the FactSage software coupled with an internal thermodynamic database. Experimentally obtained distribution coefficients wt.% in slag/wt.% in metal at 1 200 °C (1 473 K) follow the sequence Sn >> Cu > As ≈ Sb at P(O2) < 10−9.5 atm and Sn >> As ≈ Sb > Cu at P(O2) > 10−8.5 atm. Model predictions are in good agreement with the experiment.

On the multifractal analysis of the covering number on the Galton–Watson tree

AbstractWe consider, for t in the boundary of a Galton–Watson tree $(\partial \textsf{T})$, the covering number $(\textsf{N}_n(t))$ by the generation-n cylinder. For a suitable set I and sequence (sn), we almost surely establish the Hausdorff dimension of the set $\{ t \in \partial {\textsf{T}}:{{\textsf{N}}_n}(t) - nb \ {\sim} \ {s_n}\} $ for b ∈ I.

Strong approximation and a central limit theorem for St. Petersburg sums

The St. Petersburg paradox (Bernoulli, 1738) concerns the fair entry fee in a game where the winnings are distributed as P(X=2k)=2−k,k=1,2,…. The tails of X are not regularly varying and the sequence Sn of accumulated gains has, suitably centered and normalized, a class of semistable laws as subsequential limit distributions (Martin-Löf, 1985; Csörgő and Dodunekova, 1991). This has led to a clarification of the paradox and an interesting and unusual asymptotic theory in past decades. In this paper we prove that Sn can be approximated by a semistable Lévy process {L(n),n≥1} with a.s. error O(n(logn)1+ε) and, surprisingly, the error term is asymptotically normal, exhibiting an unexpected central limit theorem in St. Petersburg theory.

Ergodic theorems for coset spaces

We study in this paper the validity of the Mean Ergodic Theorem along left Folner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Folner sequence in G, we provide examples where the Mean Ergodic Theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g., if G is virtually nilpotent, then the Mean Ergodic Theorem holds along any left Folner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a “sufficiently thin” subgroup, we show that the Mean Ergodic Theorem holds for the induced representation along any left Folner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Folner sequence (Fn) in L, there exists a sequence (sn) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (Fnsn) in G. Finally, we provide examples of countable (not necessarily amenable) groups G with proper, infinite-index subgroups H, so that the Pointwise Ergodic Theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset G/H.

Tauberian conditions with controlled oscillatory behavior for statistical convergence

We present new Tauberian conditions in terms of the general logarithmic control modulo of the oscillatory behavior of a real sequence (sn) to obtain lim n?? sn = ? from st - lim n?? sn = ?, where ? is a finite number. We also introduce the statistical (l,m) summability method and extend some Tauberian theorems to this method. The main results improve some well-known Tauberian theorems obtained for the statistical convergence.

Calculation of levels, transition rates, and lifetimes for the arsenic isoelectronic sequence Sn XVIII-Ba XXIV, W XLII

Multi-configuration Dirac–Fock (MCDF) calculations of energy levels, wavelengths, oscillator strengths, lifetimes, and electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), magnetic quadrupole (M2) transition rates are reported for the arsenic isoelectronic sequence Sn XVIII-Ba XXIV, W XLII. Results are presented among the 86 levels of the 4s24p3, 4s4p4, 4p5, 4s24p24d, and 4s4p34d configurations in each ion. The relativistic atomic structure package GRASP2K is adopted for the calculations, in which the contributions from the correlations within the n≤7 complexes, Breit interaction (BI) and quantum electrodynamics (QED) effects are taking into account. The many-body perturbation theory (MBPT) method is also employed as an independent calculation for comparison purposes, taking W XLII as an example. Calculated results are compared with data from other calculations and the observed values from the Atomic Spectra Database (ASD) of the National Institute of Standards and Technology (NIST). Good agreements are obtained. i.e, the accuracy of our energy levels is assessed to be better than 0.6%. These accurate theoretical data should be useful for diagnostics of hot plasmas in fusion devices.

Non-conflicting ordering cones and vector optimization in inductive limits

Let (E, ξ)= ind (En, ξn) be an inductive limit of a sequence (En, ξn)n∈ N of locally convex spaces and let every step (En, ξn) be endowed with a partial order by a pointed convex (solid) cone Sn. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence (Sn)n∈ N of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.

Some Diophantine properties of the sequence of S-units

Let S be a finite set of rational primes, and let sn denote the increasing sequence of the positive integers having all their prime factors in S. In this paper we develop a method to explicitly give the gaps in the sequence sn. In other words, for any term sn we can find both sn−1 and sn+1, at least in principle, without enumerating all terms of the sequence. In the case when S contains two fixed primes, we even give an efficient algorithm to find these terms explicitly. Further, we apply our results to prove some Diophantine properties of the sequence sn.

Sequential dependency computation via geometric data structures

We are given integers 0⩽G1⩽G2≠0 and a sequence Sn=〈a1,a2,…,an〉 of n integers. The goal is to compute the minimum number of insertions and deletions necessary to transform Sn into a valid sequence, where a sequence is valid if it is nonempty, all elements are integers, and all the differences between consecutive elements are between G1 and G2. For this problem from the database theory literature, previous dynamic programming algorithms have running times O(n2) and O(A⋅nlogn), for a parameter A unrelated to n. We use a geometric data structure to obtain a O(nlognloglogn) running time.

Computational Analysis of n→π* Back-Bonding in Metallylene–Isocyanide Complexes R2MCNR′ (M = Si, Ge, Sn; R =tBu, Ph; R′ = Me,tBu, Ph)

A detailed computational investigation of orbital interactions in metal–carbon bonds of metallylene–isocyanide adducts of the type R2MCNR′ (M = Si, Ge, Sn; R, R′ = alkyl, aryl) was performed using density functional theory and different methods based on energy decomposition analysis. Similar analyses have not been carried out before for metal complexes of isocyanides, even though the related carbonyl complexes have been under intense investigations throughout the years. The results of our work reveal that the relative importance of π-type back-bonding interactions in these systems increases in the sequence Sn < Ge ≪ Si, and in contrast to some earlier assumptions, the π-component cannot be neglected for any of the systems investigated. While the fundamental ligand properties of isocyanides are very similar to those of carbonyl, there are significant variations in the magnitudes of different effects observed. Most notably, on coordination to metals, both ligands can display positive or negative shifts in t...

Pascal's Prism

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

The intersection spectrum of Skolem sequences and its applications to [formula omitted]-fold cyclic triple systems

A Skolem sequence of order n is a sequence Sn=(s1,s2,…,s2n) of 2n integers containing each of the integers 1,2,…,n exactly twice, such that two occurrences of the integer j∈{1,2,…,n} are separated by exactly j−1 integers. We prove that the necessary conditions are sufficient for existence of two Skolem sequences of order n with 0,1,2,…,n−3 and n pairs in the same positions. Further, we apply this result to the fine structure of cyclic two-, three-, and four-fold triple systems, and also to the fine structure of λ-fold directed triple systems and λ-fold Mendelsohn triple systems.

Lower bounds for norms of products of polynomials via Bombieri inequality

In this paper we give a different interpretation of Bombieri’s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence Sn(P ) = supQn [PQn]2, where P is a fixed m−homogeneous polynomial and Qn runs over the unit ball of the Hilbert space of n−homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on C and we obtain sharp inequalities whenever the number of factors is no greater than N . In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set {zk}k=1 of unit vectors in a complex Hilbert space for which sup‖z‖=1 |〈z, z1〉 · · · 〈z, zn〉| is minimum must be an orthonormal system.

THE MULTI-EPOCH NEARBY CLUSTER SURVEY: TYPE Ia SUPERNOVA RATE MEASUREMENT INz∼ 0.1 CLUSTERS AND THE LATE-TIME DELAY TIME DISTRIBUTION

We describe the Multi-Epoch Nearby Cluster Survey (MENeaCS), designed to measure the cluster Type Ia supernova (SN Ia) rate in a sample of 57 X-ray selected galaxy clusters, with redshifts of 0.05 < z < 0.15. Utilizing our real time analysis pipeline, we spectroscopically confirmed twenty-three cluster SN Ia, four of which were intracluster events. Using our deep CFHT/Megacam imaging, we measured total stellar luminosities in each of our galaxy clusters, and we performed detailed supernova detection efficiency simulations. Bringing these ingredients together, we measure an overall cluster SN Ia rate within R_{200} (1 Mpc) of 0.042^{+0.012}_{-0.010}^{+0.010}_{-0.008} SNuM (0.049^{+0.016}_{-0.014}^{+0.005}_{-0.004} SNuM) and a SN Ia rate within red sequence galaxies of 0.041^{+0.015}_{-0.015}^{+0.005}_{-0.010} SNuM (0.041^{+0.019}_{-0.015}^{+0.005}_{-0.004} SNuM). The red sequence SN Ia rate is consistent with published rates in early type/elliptical galaxies in the `field'. Using our red sequence SN Ia rate, and other cluster SNe measurements in early type galaxies up to $z\sim1$, we derive the late time (>2 Gyr) delay time distribution (DTD) of SN Ia assuming a cluster early type galaxy star formation epoch of z_f=3. Assuming a power law form for the DTD, \Psi(t)\propto t^s, we find s=-1.62\pm0.54. This result is consistent with predictions for the double degenerate SN Ia progenitor scenario (s\sim-1), and is also in line with recent calculations for the double detonation explosion mechanism (s\sim-2). The most recent calculations of the single degenerate scenario delay time distribution predicts an order of magnitude drop off in SN Ia rate \sim 6-7 Gyr after stellar formation, and the observed cluster rates cannot rule this out.