New Class Of Sequences Research Articles

Uniform convergence of trigonometric series with $p$-bounded variation coefficients

In the present paper we introduce a new class of sequences called $GMS\left(p, \beta ,r\right) ,$ which is the generalization of a class considered by Tikhonov in [9] and Szal in [10]. Moreover, we obtained in this note sufficient and necessary conditions for the uniform convergence of sine and cosine series with $\left(p,\beta ,r\right) -$ general monotone coefficients.

A New Class of Sequences Without Interferences for Cluster Tools With Tight Wafer Delay Constraints

Robotized cluster tools for semiconductor wafer fabrication may have a wafer wait within a processing chamber after processing there until the wafer is unloaded from the chamber by a robot. Such wafer delays cause wafer quality degradation or variability due to residual gases and heats within the chamber. There have been numerous works on characterizing wafer delays and scheduling under upper limits on wafer delays while presuming that the tools are operated by well-known simple robot task sequences such as swap or backward sequences. However, when the wafer delay constraints are tight, there may not be feasible schedules for such sequences. We wish to know whether there can be alternative robot task sequences which can satisfy such tight wafer delay constraints. In this paper, we identify a new class of robot task sequences that can better satisfy tight wafer delay constraints than the conventional swap or backward sequences while keeping the same minimum tool cycle time. By examining the circuits of timed event graph (TEG) models for the tool operation behaviors in many different robot task sequences, we identify that such robot task sequences do not make interferences between the work cycles of the resources such as the robot and chambers. The resource interference can cause delays in the work cycles, and hence increase the wafer delays or the tool cycle time. To prove this, we examine circuits in an extended TEG model, a negative event graph, which incorporates time constraints as negative places and tokens. From this, we derive closed-form conditions for which such sequences are feasible against given wafer delay constraints. By experiments, we show that the proposed new sequences have shorter wafer delays, and hence better satisfy tight wafer delay constraints than conventional sequences. Note to Practitioners —As circuit widths shrink down to several nanometers, cluster tools for semiconductor fabrication require extreme process quality control. Even wafer delays within processing chambers of cluster tools can cause wafer quality degradation and variability. Therefore, it is desirable for cluster tools to have much shorter wafer delays. However, conventional sequences such as the swap and backward sequences, which are being prevalently used in practice, may not satisfy tight wafer delay constraints. Our proposed sequences, which are as simple as the conventional sequences, have much shorter wafer delays while keeping the same minimum tool cycle time.

Full Dimensional Sets of Reals Whose Sums of Partial Quotients Increase in Certain Speed

For a real x∈(0,1)∖Q, let x=[a1(x),a2(x),⋯] be its continued fraction expansion. Let sn(x)=∑j=1naj(x). The Hausdorff dimensions of the level setsEφ(n),α:={x∈(0,1):limn→∞⁡sn(x)φ(n)=α} for α≥0 and a non-decreasing sequence {φ(n)}n=1∞ have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams et al. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on φ(n) under which one can expect a 1-dimensional set Eφ(n),α. We give certain upper and lower bounds on the increasing speed of φ(n) when Eφ(n),α is of Hausdorff dimension 1 and a new class of sequences {φ(n)}n=1∞ such that Eφ(n),α is of full dimension. For an irregular sequence {φ(n)}n=1∞, a full dimensional set Eφ(n),α is impossible.

A New Class of Sequences and L-Convergence of Some Complex Modified Trigonometric Sums

We show that the new complex modified trigonometric sums, introduced by others, converge in the space L under a new class of sequences.

Discrepancy of Generalized LS-Sequences

Abstract The LS-sequences are a parametric family of sequences of points in the unit interval. They were introduced by Carbone [4], who also proved that under an appropriate choice of the parameters L and S, such sequences are lowdiscrepancy. The aim of the present paper is to provide explicit constants in the bounds of the discrepancy of LS-sequences. Further, we generalize the construction of Carbone [4] and construct a new class of sequences of points in the unit interval, the generalized LS-sequences.

More on variants of complete metric spaces

Some classes of metric spaces satisfying properties stronger than completeness but weaker than compactness have been studied by many authors over the years. One such significant family consists of those metric spaces on which every real-valued continuous function is uniformly continuous, which are widely known as Atsuji spaces or UC spaces. Recently in 2014, two new kinds of complete metric spaces are introduced, namely Bourbaki-complete and cofinally Bourbaki-complete metric spaces, whose idea has come from some new classes of sequences acting as generalizations of Cauchy sequences. Our major goal is to give several new equivalent conditions for metric spaces whose completions are one of the aforesaid spaces, especially in terms of some functions, sequences and geometric functionals.

Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials

AbstractAssuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constantCsuch that for any elliptic curveE/ℚ and non-torsion pointP∈E(ℚ), there is at most one integral multiple [n]Psuch that n >C. The proof is a modification of a proof of Ingram giving an unconditional, but not uniform, bound. The new ingredient is a collection of explicit formulæ for the sequencev(Ψn) of valuations of the division polynomials. ForPof non-singular reduction, such sequences are already well described in most cases, but forPof singular reduction, we are led to define a new class of sequences calledelliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on ĥ(P)/h(E) for integer points having two large integral multiples.

Spatial specificity in spatiotemporal encoding and Fourier imaging

PurposeUltrafast imaging techniques based on spatiotemporal encoding (SPEN), such as RASER (rapid acquisition with sequential excitation and refocusing), is a promising new class of sequences since they are largely insensitive to magnetic field variations which cause signal loss and geometric distortion in EPI. So far, attempts to theoretically describe the point-spread-function (PSF) for the original SPEN-imaging techniques have yielded limited success. To fill this gap a novel definition for an apparent PSF is proposed. TheorySpatial resolution in SPEN-imaging is determined by the spatial phase dispersion imprinted on the acquired signal by a frequency-swept excitation or refocusing pulse. The resulting signal attenuation increases with larger distance from the vertex of the quadratic phase profile. MethodsBloch simulations and experiments were performed to validate theoretical derivations. ResultsThe apparent PSF quantifies the fractional contribution of magnetization to a voxel's signal as a function of distance to the voxel. In contrast, the conventional PSF represents the signal intensity at various locations. ConclusionThe definition of the conventional PSF fails for SPEN-imaging since only the phase of isochromats, but not the amplitude of the signal varies. The concept of the apparent PSF is shown to be generalizable to conventional Fourier-imaging techniques.

Pair correlation of fractional parts derived from rational valued sequences, II

A rational valued vector sequence x→, for some fixed k and r∈N, is a map x→:Nk→Qr. In the present paper, we complement the results of [3] with a discussion on rational valued vector sequences. We investigate the pair correlation function for the fractional parts of sequences t→⋅x→, where x→ is a rational valued vector sequence and t→∈Rr. We offer a new class of sequences x→ whose pair correlation function behaves as that of random sequences for almost all real vectors t→, namely, injective sequences of the formx→(m,n)=(p1mp2nb→(m,n),p3mp4nb→(m,n)) for p1, p2, p3, and p4 primes and where b→:N2→N satisfies a certain growth condition.

Upper bounds for the distances between consecutive zeros of solutions of first order delay differential equations

By introducing a new class of sequences involving iterates of the delay, we are able to derive a sharper estimate on the ratio x(τ(t))/x(t). Based on this, we get some new upper bounds for the distance between consecutive zeros of solutions of the first order delay differential equationx′(t)+p(t)x(τ(t))=0. In particular, our results are not covered by previously known results. Some examples and a table are given to illustrate our results.

Mellin transforms of generalized fractional integrals and derivatives

We obtain the Mellin transforms of the generalized fractional integrals and derivatives that generalize the Riemann–Liouville and the Hadamard fractional integrals and derivatives. We also obtain interesting results, which combine generalized δr,m operators with generalized Stirling numbers and Lah numbers. For example, we show that δ1,1 corresponds to the Stirling numbers of the 2nd kind and δ2,1 corresponds to the unsigned Lah numbers. Further, we show that the two operators δr,m and δm,r,r,m∈N, generate the same sequence given by the recurrence relation.S(n,k)=∑i=0rm+(m-r)(n-2)+k-i-1r-iriS(n-1,k-i),0<k⩽n,with S(0,0)=1 and S(n,0)=S(n,k)=0 for n>0 and 1+min{r,m}(n-1)<k or k⩽0. Finally, we define a new class of sequences for r∈13,14,15,16,… and in turn show that δ12,1 corresponds to the generalized Laguerre polynomials.

A New Difference Sequence Set of Orderαand Its Geometrical Properties

We introduce a new class of sequences named asmαΔr,ϕ,pand, for this space, we study some inclusion relations, topological properties, and geometrical properties such as order continuous, the Fatou property, and the Banach-Saks property of typep.

A new class of fuzzy sequences related to the ℓp space defined by Orlicz function

In this paper we introduce the new classes of sequences $\ell _\infty ^F$ M, A, φ, p, Δr, mFM, A, φ, p, Δr and CFM, A, p, Δr of fuzzy numbers defined by Orlicz function. We study their different properties and established some inclusion relation involving these classes of sequences.

New Classes of Sequences for Encryption Procedures in Symmetric Cryptography

New Classes of Sequences for Encryption Procedures in Symmetric Cryptography

Qualitative and quantitative ultrashort‐TE MRI of cortical bone

Osteoporosis causes over 1.5 million fractures per year, costing about $15 billion annually in the USA. Current guidelines utilize bone mineral density (BMD) to assess fracture risk; however, BMD alone only accounts for 30-50% of fractures. The other two major components of bone, organic matrix and water, contribute significantly to bone mechanical properties, but cannot be assessed with conventional imaging techniques in spite of the fact that they make up about 57% of cortical bone by volume. Conventional clinical MRI usually detects signals from water in tissues without difficulty, but cannot detect the water bound to the organic matrix, or the free water in the microscopic pores of the Haversian and the lacunar-canalicular system of cortical bone, because of their very short apparent transverse relaxation times (T2 *). In recent years, a new class of sequences, ultrashort-TE (UTE) sequences, with nominal TEs of less than 100 µs, which are much shorter than the TEs available with conventional sequences, have received increasing interest. These sequences can detect water signals from within cortical bone and provide an opportunity to study disease of this tissue in a new way. This review summarizes the recent developments in qualitative UTE imaging (techniques and contrast mechanisms to produce bone images with high contrast) and quantitative UTE imaging (techniques to quantify the MR properties, including T1 , T2 * and the magnetization transfer ratio, and tissue properties, including bone perfusion, as well as total, bound and free water content) of cortical bone in vitro and in vivo. The limitations of the current techniques for clinical applications and future directions are also discussed.

New Inequalities for Convex Sequences with Applications

In this paper, we will show some new inequalities for convex sequences, and we will also make a connection between them and Chebyshev’s inequality, which implies the existence of new class of sequences satisfying Chebyshev’s inequality. We give also some applications and generalization of Haber and Mercer’s inequalities.

On L 1-convergence of sine series

Our aim is to find the source why the logarithm sequences play the crucial role in the L1-convergence of sine series. We define three new classes of sequences; one of them has the character of the logarithm sequences, the other two are the extensions of the class defined by Zhou and named Logarithm Rest Bounded Variation Sequences. In terms of these classes, extended analogues of Zhou’s theorems are proved.

A new class of numerical sequences and its applications to uniform convergence of sine series

AbstractIn the present paper we introduce a new class of sequences called GM(β, r), which is a generalization of the class considered by Tikhonov in 13. Moreover, we obtain sufficient and necessary conditions for uniform convergence of sine series with (β, r)‐general monotone coefficients.

A New Class of Sequences Related to the Spaces Defined by Sequences of Orlicz Functions

We introduce new sequence space defined by combining an Orlicz function, seminorms, and -sequences. We study its different properties and obtain some inclusion relation involving the space Inclusion relation between statistical convergent sequence spaces and Cesaro statistical convergent sequence spaces is also given.

Remarks on the uniform and L 1-convergence of trigonometric series

Chaundry and Jolliffe [1] proved that if ak is a nonnegative sequence tending monotonically to zero, then a necessary and sufficient condition for the uniform convergence of the series Σk=1∞ak sin kx is limk→∞kak = 0. Lately, S. P. Zhou, P. Zhou and D. S. Yu [4] generalized this classical result. In this paper we propose new classes of sequences for which we get the extended version of their results. Moreover, we generalize the results of S. Tikhonov [2] on the L1-convergence of Fourier series.