Existence Of Such Sequences Research Articles

Almost p-ary sequences

In this paper we study almost p-ary sequences and their autocorrelation coefficients. We first study the number l of distinct out-of-phase autocorrelation coefficients for an almost p-ary sequence of period n + s with s consecutive zero-symbols. We prove an upper bound and a lower bound on l. It is shown that l can not be less than $\min \limits \{s,p,n\}$ . In particular, it is shown that a nearly perfect sequence with at least two consecutive zero symbols does not exist. Next we define a new difference set, partial direct product difference set (PDPDS), and we prove the connection between an almost p-ary nearly perfect sequence of type (γ1, γ2) and period n + 2 with two consecutive zero-symbols and a cyclic $(n+2,p,n,\frac {n-\gamma _{2} - 2}{p}+\gamma _{2},0,\frac {n-\gamma _{1} -1}{p}+\gamma _{1},\frac {n-\gamma _{2} - 2}{p},\frac {n-\gamma _{1} -1}{p})$ PDPDS for arbitrary integers γ1 and γ2. Then we prove a necessary condition on γ2 for the existence of such sequences. In particular, we show that they do not exist for γ2 ≤ − 3.

Giant tufas of Lake Van record lake-level fluctuations and climatic changes in eastern Anatolia,Turkey

Lake Van, the largest soda lake in the world, is a key area where climatic changes are recorded by well-preserved sedimentary successions. In spite of the existence of such sequences, the ancient lake levels are still under debate. Here, we present U/Th ages of tufa layers exposed along the northern margin of the lake near the town of Adilcevaz. Tufa are interpreted to form during humid and temperate climate conditions in the contact zone between the highly alkaline lake water – rich in bicarbonate and carbonate ions – and calcium-rich groundwater. These contact zones most likely appear near shore. Thus, U/Th ages and altitude of the tufa present minimum ancient lakestands. The tufas obtained in our study are dated between 112.7 and 19.3 ka. They are interpreted to record two transgressional intervals, the first starting at 1701 meters above sea level (masl) altitude at 112.7 ka reaching 1706 masl at 72.5 ka (based on three U/Th ages). The second transgression starts at the present Lake Van level of 1646 masl at 30.1 ka, reaching 1725 masl at 19.3 ka (based on 15 U/Th ages). Comparison with climate proxies reveals that the first transgression was caused by intensified precipitation accompanied by warming, while the second transgression was triggered mainly by warming that initiated melting of glaciers. The lake-level fluctuations described here are almost in line with the ones gathered from dated lake terraces but contradict lake-level reconstruction based on pore water salinity.

Superoscillating Sequences Towards Approximation in $${\mathscr {S}}$$ S or $${\mathscr {S}}'$$ S ′ -Type Spaces and Extrapolation

Aharonov–Berry superoscillations are band-limited sequences of functions that happen to oscillate asymptotically faster than their fastest Fourier component. In this paper we analyze in what sense functions in the Schwartz space \(\mathscr {S}(\mathbb {R},\mathbb {C})\) or in some of its subspaces, tempered distributions or also ultra-distributions, could be approximated over compact sets or relatively compact open sets (depending on the context) by such superoscillating sequences. We also show how one can profit of the existence of such sequences in order to extrapolate band-limited signals with finite energy from a given segment of the real line.

On quasi-orthogonal polynomials of order r

ABSTRACTThe sequences of quasi-orthogonal polynomials of order r are defined for non-quasi-definite moment functionals. Properties concerning the existence of such sequences, and relations between a quasi-orthogonal polynomial of order r and a set of orthogonal polynomials are proved. Two determinantal expressions of quasi-orthogonal polynomials of order r are given. At last it is proved that three consecutive polynomials of a sequence of quasi-orthogonal polynomials of order r satisfy a three term recurrence relation.

Early-type galaxies in the Antlia Cluster: a deep look into scaling relations

We present the first large-scale study of the photometric and structural relations followed by early-type galaxies (ETGs) in the Antlia cluster. Antlia is the third nearest populous galaxy cluster after Fornax and Virgo (d $\sim 35$ Mpc). A photographic catalog of its galaxy content was built by Ferguson & Sandage in 1990 (FS90). Afterwards, we performed further analysis of the ETG population located at the cluster centre. Now, we extend our study covering an area four times larger, calculating new total magnitudes and colours, instead of isophotal photometry, as well as structural parameters obtained through S\'ersic model fits extrapolated to infinity. Our present work involves a total of 177 ETGs, out of them 56 per cent have been cataloged by FS90 while the rest (77 galaxies) are newly discovered ones. Medium-resolution GEMINI and VLT spectra are used to confirm membership when available. Including radial velocities from the literature, 59 ETGs are confirmed as Antlia members. Antlia scaling relations mainly support the existence of unique functions (linear and curved) that join bright and dwarf ETGs, excluding compact ellipticals (cEs). Lenticular galaxies are outliers only with respect to the curved relation derived for effective surface brightness versus absolute magnitude. The small number of bright ellipticals and cEs present in Antlia, prevents us from testing if the same data can be fitted with two different linear sequences, for bright and dwarf ETGs. However, adding data from other clusters and groups, the existence of such sequences is also noticeable in the same scaling relations.

Monotone iterative sequences for nonlinear integro-differential equations of second order

This paper presents an efficient algorithm based on a monotone method for the solution of a class of nonlinear integro-differential equations of second order. This method is applied to derive two monotone sequences of upper and lower solutions which are uniformly convergent. Theorems which list the conditions for the existence of such sequences are presented. The numerical results demonstrate reliability and efficiency of the proposed algorithm.

The distribution of values of harmonic functions in the unit disk

In this paper we study the distribution of values of harmonic functions in non-Euclidean disks. We introduce the notion of a P′-sequence, which enables us to characterize the class of normal harmonic functions defined in the unit disk. Moreover, we obtain sufficient conditions for the existence of such sequences and give examples which show that these conditions are essential in the stated theorems.

Small-molecule-mediated G-quadruplex isolation from human cells

Nucleic acids containing stretches of tandem guanines can fold into four-stranded structures called G-quadruplexes. The existence of such sequences in genomic DNA suggests the occurrence of these motifs in cells, with potential implications in a number of biological processes relevant to cancer. Small molecules have proven to be valuable tools to dissect cell circuitry. Here, we describe a synthetic small molecule derived from an N,N'-bis(2-quinolinyl)pyridine-2,6-dicarboxamide, which is designed to mediate the selective isolation of G-quadruplex nucleic acids. The methodology was successfully applied to a range of DNA and RNA G-quadruplexes in vitro. We demonstrate the general applicability of the method by isolating telomeric DNA-containing G-quadruplex motifs from cells. We show that telomeres are targets for the probe, providing further evidence of the formation of G-quadruplexes in human cells.

On non-existence of perfect and nearly perfect sequences

We study the complex p-ary perfect and nearly perfect sequences where p is an odd prime and show that the existence of such sequences is equivalent to the existence of certain kinds of difference sets. Using results from difference sets, there are no p-ary perfect sequences of length p3 for s ≥ 3, 2p3 for s≥1, and pq for prime q > p. Also, there are no ternary perfect sequences of length 3q1q2 for primes q1, q2 where 3< q1 < q2. Using the standard 'self-conjugate' conditions, more non-existence results on perfect and nearly perfect sequences are obtained. To summarise the works, tables of existence and non-existence of perfect and nearly perfect sequences of length n are listed for 2 ≤n≤50

On powers of words occurring in binary codings of rotations

We discuss combinatorial properties of a class of binary sequences generalizing Sturmian sequences and obtained as a coding of an irrational rotation on the circle with respect to a partition in two intervals. We give a characterization of those having a finite index in terms of a two-dimensional continued fraction like algorithm, the so-called D -expansion. Then, we discuss powers occurring at the beginning of these words and we prove, contrary to the Sturmian case, the existence of such sequences without any non-trivial asymptotic initial repetition. We also show that any characteristic sequence (that is, obtained as the coding of the orbit of the origin) has non-trivial repetitions not too far from the beginning and we apply this property to obtain the transcendence of the continued fractions whose partial quotients arises from such sequences when the two letters are replaced by distinct positive integers.

On the weak convergence of sequences of circuit processes: a probabilistic approach

The asymptotic behaviour of sequences of Markov processes whose finite distributions depend upon the sample paths ω of a positive recurrent Markov chain ξ is studied. The existence of such sequences depends upon the existence of a unique class of directed weighted circuits having a probabilistic interpretation in terms of the directed circuits occurring along the sample paths of ξ. An application to multiple Markov chains is given.

On the weak convergence of sequences of circuit processes: a probabilistic approach

The asymptotic behaviour of sequences of Markov processes whose finite distributions depend upon the sample paths ω of a positive recurrent Markov chain ξ is studied. The existence of such sequences depends upon the existence of a unique class of directed weighted circuits having a probabilistic interpretation in terms of the directed circuits occurring along the sample paths of ξ. An application to multiple Markov chains is given.

Homogeneously distributed sequences and Poincaré sequences of integers of sublacunary growth

The sequence of integersn1<n2<n3<... is said to be homogeneously distributed if\(\mathop {\lim }\limits_{m \to + \infty } (1/m)\sum\limits_{k = 1}^m {\exp (2\pi in_k \alpha )} = 0\) for all non-integral real α. The existence of such sequences with a prescribed subexponential growth is shown, the recurrent properties of these sequences are discussed.

Asymptotically invariant sequences and an action of SL (2,Z) on the 2-sphere

LetG be a countable group which acts non-singularly and ergodically on a Lebesgue space (X, ȑ, μ). A sequence (Bn) in ℒ is calledasymptotically invariant in limn μ (BnΔgBn)=0 for everygeG. In this paper we show that the existence of such sequences can be characterized by certain simple assumptions on the cohomology of the action ofG onX. As an explicit example we prove that a natural action of SL (2,Z) on the 2-sphere has no asymptotically invariant sequences. The last section deals with a particular cocycle for this action which has an interpretation as a random walk on the integers with “time” in SL (2,Z).

Gleichverteilung auf Diskreten Halbgruppen

Let (μn) be a sequence of probability measures on a discrete semigroup S. In this paper we study the (left, right) uniform distribution of (μn), i.e. the vague or norm convergence of μn−δxμn→0 for every x in S δx means the point measure of x and δxμn the convolution of δx and μn). We shall give equivalent conditions for the (left, right) uniform distributed sequences and relate the existence of such sequences to the (left, right) amenable semigroups. In the last section we consider especially the uniform distribution of the convolution sequence μn of a probability measure.

Noncomplex sequences: characterizations and examples

We are concerned in this paper with infinite binary sequences which are noncomplex in the sense that their minimal-program complexity (i.e., the lengths of shortest programs for computing their finite initial segments), as a function of the lengths of their initial segments, grows arbitrarily (in an effective sense) slowly. Indeed, the existence of such sequences which are also nonrecursive raises some interesting questions concerning the notion of computability itself. Our first task is to give a characterization of these sequences which does not directly involve the notions of complexity theory. Although these characterizations do involve the primitives of recursive function theory, they do not involve the types of properties which one usually encounters there. While a closer connection is still hoped for, the lack of one is not entirely unexpected. For example, except for the trivial case of degree 0, there is no general correlation between program complexity and degrees of unsolvability. The reason, roughly, is this: even though the inequality deg (f) ≤ deg (g) expresses the relative information between f and g (in the sense that if one knows g then one can compute f), it is a qualitative sort of information. The minimal-program complexity is used as a measure of algorithmic information content and as such is quantitative.The second task of this paper is to present an example of one of these sequences which to its credit possesses a fairly long list of interesting properties, not the least attractive of which is that, though nonrecursive, it is very simple to describe. In fact this sequence has occurred previously in the literature. It is the characteristic sequence of the set of busy beaver numbers first studied by T. Rado [13]. Of particular interest to the discussion in the last section of this paper concerning the notion of computability is the fact that all the initial segments of this sequence are computable by arbitrarily short programs each of which is defined on all inputs and runs very quickly.

Noncomplex sequences: characterizations and examples

We are concerned in this paper with infinite binary sequences which are noncomplex in the sense that their minimal-program complexity (i.e., the lengths of shortest programs for computing their finite initial segments), as a function of the lengths of their initial segments, grows arbitrarily (in an effective sense) slowly. Indeed, the existence of such sequences which are also nonrecursive raises some interesting questions concerning the notion of computability itself. Our first task is to give a characterization of these sequences which does not directly involve the notions of complexity theory. Although these characterizations do involve the primitives of recursive function theory, they do not involve the types of properties which one usually encounters there. While a closer connection is still hoped for, the lack of one is not entirely unexpected. For example, except for the trivial case of degree 0, there is no general correlation between program complexity and degrees of unsolvability. The reason, roughly, is this: even though the inequality deg (f) ≤ deg (g) expresses the relative information between f and g (in the sense that if one knows g then one can compute f), it is a qualitative sort of information. The minimal-program complexity is used as a measure of algorithmic information content and as such is quantitative.The second task of this paper is to present an example of one of these sequences which to its credit possesses a fairly long list of interesting properties, not the least attractive of which is that, though nonrecursive, it is very simple to describe. In fact this sequence has occurred previously in the literature. It is the characteristic sequence of the set of busy beaver numbers first studied by T. Rado [13]. Of particular interest to the discussion in the last section of this paper concerning the notion of computability is the fact that all the initial segments of this sequence are computable by arbitrarily short programs each of which is defined on all inputs and runs very quickly.

General Shift-Register Sequences of Arbitrary Cycle Length

An r-ary shift-register sequence is desired that has arbitrary cycle length L≤rkfor arbitrary r and k, where k is the number of stages (degree) of the shift register. The existence of such sequences is established for cycle lengths L. Furthermore, existence of such sequences which are zero free for almost all cycle lengths L is proved.