Finding structure in sequences of real numbers via graph theory: a problem list

For kge 3, a k-rollercoaster is a sequence of numbers whose every maximal contiguous subsequence, that is increasing or decreasing, has length at least k; 3-rollercoasters are called simply rollercoasters. Given a sequence of distinct real numbers, we are interested in computing its maximum-length (not necessarily contiguous) subsequence that is a k-rollercoaster. Biedl et al. (in: ICALP, volume 107 of LIPIcs. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, pp 18:1–18:15, 2018) have shown that each sequence of n distinct real numbers contains a rollercoaster of length at least lceil n/2rceil for n>7, and that a longest rollercoaster contained in such a sequence can be computed in O(nlog n)-time (or faster, in O(n log log n) time, when the input sequence is a permutation of {1,ldots ,n}). They have also shown that every sequence of ngeqslant (k-1)^2+1 distinct real numbers contains a k-rollercoaster of length at least frac{n}{2(k-1)}-frac{3k}{2}, and gave an O(nklog n)-time (respectively, O(n klog log n)-time) algorithm computing a longest k-rollercoaster in a sequence of length n (respectively, a permutation of {1,ldots ,n}). In this paper, we give an O(nk^2)-time algorithm computing the length of a longest k-rollercoaster contained in a sequence of n distinct real numbers; hence, for constant k, our algorithm computes the length of a longest k-rollercoaster in optimal linear time. The algorithm can be easily adapted to output the respective k-rollercoaster. In particular, this improves the results of Biedl et al. (2018), by showing that a longest rollercoaster can be computed in optimal linear time. We also present an algorithm computing the length of a longest k-rollercoaster in O(n log ^2 n)-time, that is, subquadratic even for large values of kle n. Again, the rollercoaster can be easily retrieved. Finally, we show an Omega (n log k) lower bound for the number of comparisons in any comparison-based algorithm computing the length of a longest k-rollercoaster.

Asymptotic behavior of solutions for neutral difference equations

Oscillation of second order half-linear difference equations (II)

The Fibonacci sequence has broad applications in mathematics, where its inherent patterns and properties are utilized to solve various problems. The sequence often emerges in areas involving growth patterns, series, and recursive relationships. It is known for its connection to the golden ratio, which appears in numerous natural phenomena and mathematical constructs. In this research paper, we introduce new concepts of convergence and summability for sequences of real and complex numbers by using Fibonacci sequences, called Δ-Fibonacci statistical convergence, strong Δ-Fibonacci summability, and Δ-Fibonacci statistical summability. And, these new concepts are supported by several significant theorems, properties, and relations in the study. Furthermore, for this type of convergence, we introduce one-sided Tauberian conditions for sequences of real numbers and two-sided Tauberian conditions for sequences of complex numbers.

Statistical convergence for sequences of fuzzy real numbers has been studied by various authors. In this paper we study the concept of statistical summability (C, 1) for fuzzy real numbers which for real numbers was introduced by Moricz [18]. We also construct some interesting examples.

Classes of sequences of real numbers, games and selection properties

This chapter collects problems on sequences of real numbers. The problems included are challenging and the topics covered vary in diversity: from limits of sequences involving special numerical terms, applications of Stolz–Cesaro Theorem, both ∞∕∞ and 0∕0 cases, Wolstenholme sequences, to the calculation of spectacular limits of integrals.

Chapter 5 - Generalized Convolutions

Oscillations of Delay Difference Equations with Oscillating Coefficients

The inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is a so-called Green's matrix. A Green's matrix is a symmetric matrix which is given by two sequences of real numbers {ui} and {vi} such that ci,j = uivj for $i \leq j$. A similar result holds for nonsymmetric matrices. An open problem on nonsingular sparse matrices is whether there exists a similar structure for their inverses as in the tridiagonal case. Here we positively answer this question for irreducible acyclic matrices, i.e., matrices whose undirected graphs are trees. We prove that the inverses of irreducible acyclic symmetric matrices are given as the Hadamard product of three matrices, a type D matrix, a flipped type D matrix, and a matrix of tree structure which is closely related to the graph of the original matrix itself. For nonsymmetric matrices we obtain a similar structure. Moreover, our results include the result for symmetric and nonsymmetric tridiagonal matrices.

Let be a doubly infinite sequence of identically distributed -mixing random variables, an absolutely summable sequence of real numbers. In this paper, we prove the complete convergence and Marcinkiewicz-Zygmund strong law of large numbers for the partial sums of the moving average processes .

A sequence (αk) of real numbers is called λ-statistically upward quasi-Cauchy if for every ε > 0 limn→∞1λn| { k∈In:αk−αk+1≥ε } |=0, where (λn) is a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn+1, λ1=1, and In=[n − λn+1, n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numbers is λ-statistically upward continuous if it preserves λ-statistical upward quasi-Cauchy sequences. It turns out that the set of λ-statistical upward continuous is a proper subset of the set of uniformly continuous functions.

0. It is well known and easy to prove that if a measurable subgroup G of n-dimensional Euclidean space Rn has positive n-dimensional Lebesgue measure, then G = Rn. In the first section we give an analogue of this result for subgroups of real-valued sequences, where the space RX of all such sequences is given product normalized Gaussian measure. In the second section we use this result to sharpen and extend a zero-one law of Cameron and Graves [1]. 1. In this section, R is the set of real numbers, Rn is n-dimensional Euclidean space, and RX is the set of all sequences (xi, x2, * * * ) of real numbers. 63 is the class of Borel subsets of R, and GBn (G?0) is the corresponding product o-field over Rn (R??). We will use ,u to denote the Gaussian measure on (G3 mentioned above. We say that a subset of RX is measurable if it is in the completion of (Bo with respect to I THEOREM. If G is a measurable subgroup of RX, then either ,u(G) = 0 or j(G) = 1. LEMMA. Suppose that f and g are bounded functions on Ro, each measurable with respect to the ,A-completion of (3X. For each xERn, let gz be the function on RX defined by

Data-centric transformations have been used in recent years to improve locality for several classes of applications. However, the existing work has applied these transformations for integer iteration spaces, i.e., the iteration spaces involving loop variables that take integer values between specified lower and upper bounds. In many applications, a loop could involve a loop variable which takes values from a sequence or set of real numbers, strings, or any other data type. We refer to such iteration spaces as non-integer iteration spaces. This paper focuses on the problem of applying data-centric transformations on applications with non-integer iteration spaces. We first present a general algorithm that uses a hash table. Then, we show how in many cases, we can exploit the repetitive nature of the dataset to avoid the overhead associated with such a table. Our algorithms have been implemented as part of a compiler for the XML query language XQuery, which supports processing over virtual XML. Our system also parallelizes the processing. We present experimental results from several applications to demonstrate the effectiveness of our transformations and parallel performance.

An Algorithm to Estimate the Hausdorff Dimension of Self-Affine Sets