Binary Recurrence Research Articles

Random balancing-like sequences

AbstractA balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

Bounds on the number of squares in recurrence sequences

We investigate the number of squares in a very broad family of binary recurrence sequences with u0=1. We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares.

Near-squares in binary recurrence sequences

We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. We show that for a given [Formula: see text], there is at most one [Formula: see text] such that [Formula: see text] is a near-square. With the exceptions of [Formula: see text] and [Formula: see text], any such [Formula: see text] can be a near-square only if [Formula: see text], [Formula: see text] is prime and [Formula: see text]. This is a part of a more general phenomenon regarding near-squares in nondegenerate recurrence sequences defined for the integers [Formula: see text] and [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. This arises from a novel Aurifeuillean-type factorization of [Formula: see text] we have found.

The Solution of Fermat’s Two Squares Equation and Its Generalization In Lucas Sequences

As it is well known, there are an infinite number of primes in special forms such as Fermat's two squares form, p=x^2+y^2 or its generalization, p=x^2+y^4, where the unknowns x, y, and p represent integers. The main goal of this paper is to see if these forms still have an infinite number of solutions when the unknowns are derived from sequences with an infinite number of prime numbers in their terms. This paper focuses on the solutions to these forms where the unknowns represent terms in certain binary linear recurrence sequences known as the Lucas sequences of the first and second types.

A prognostic model for ovarian neoplasms established by an integrated analysis of 1580 transcriptomic profiles

Even after debulking surgery combined with chemotherapy or new adjuvant chemotherapy paired with internal surgery, the average year of disease free survival in advanced ovarian cancer was approximately 1.7 years1. The development of a molecular predictor of early recurrence would allow for the identification of ovarian cancer (OC) patients with high risk of relapse. The Ovarian Cancer Disease Free Survival Predictor (ODFSP), a predictive model constructed from a special set of 1580 OC tumors in which gene expression was assessed using both microarray and sequencing platforms, was created by our team. To construct gene expression barcodes that were resistant to biases caused by disparate profiling platforms and batch effects, we employed a meta-analysis methodology that was based on the binary gene pair technique. We demonstrate that ODFSP is a reliable single-sample predictor of early recurrence (1 year or less) using the largest pool of OC transcriptome data sets available to date. The ODFSP model showed significantly high prognostic value for binary recurrence prediction unaffected by clinicopathologic factors, with a meta-estimate of the area under the receiver operating curve of 0.64 (P = 4.6E-05) and a D-index (robust hazard ratio) of 1.67 (P = 9.2E-06), respectively. GO analysis of ODFSP’s 2040 gene pairs (collapsed to 886 distinct genes) revealed the involvement in small molecular catabolic process, sulfur compound metabolic process, organic acid catabolic process, sulfur compound biosynthetic process, glycosaminoglycan metabolic process and aminometabolic process. Kyoto encyclopedia of genes and genomes pathway analysis of ODFSP’s signature genes identified prominent pathways that included cAMP signaling pathway and FoxO signaling pathway. By identifying individuals who might benefit from a more aggressive treatment plan or enrolment in a clinical trial but who will not benefit from standard surgery or chemotherapy, ODFSP could help with treatment decisions.

On solutions of the Diophantine equation $L_n\pm L_m=p^a$

Lucas sequence is one of the most studied binary recurrence sequence defined by the relation $L_{n+2}=L_{n+1}+L_n;~L_0=2, L_1=1$. In this paper, we investigate all the sums and differences of two Lucas numbers that are powers of a odd prime $p$ satisfying $p<10^3$.

Additive Diophantine Equations with Binary Recurrences, {{mathcal {S}}}-Units and Several Factorials

There are many results in the literature concerning linear combinations of factorials among terms of linear recurrence sequences. Recently, Grossman and Luca provided effective bounds for such terms of binary recurrence sequences. In this paper we show that under certain conditions, even the greatest prime divisor of u_n-a_1m_1!-dots -a_km_k! tends to infinity, in an effective way. We give some applications of this result, as well.

On a variant of Pillai’s problem with binary recurrences and S-units

Let [Formula: see text] be a fixed binary recurrence with real characteristic roots [Formula: see text] satisfying [Formula: see text] and let [Formula: see text] be fixed distinct prime numbers. In this paper, we show that there exist effectively computable, positive constants [Formula: see text] and [Formula: see text] such that the Diophantine equation [Formula: see text] has at most [Formula: see text] solutions [Formula: see text] if [Formula: see text] and at most [Formula: see text] solutions [Formula: see text] if [Formula: see text]. In order to demonstrate the strength of our method we show that for the binary recurrence sequence [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text], the Diophantine equation [Formula: see text] has at most six solutions [Formula: see text] unless [Formula: see text] in which case it has exactly seven solutions.

On prime factors of terms of binary recurrence sequences

We establish estimates from below for the greatest prime factor of the $n$th term of a non-degenerate binary recurrence sequence when the sequence belongs to a class of sequences which includes the Lucas sequences.

Machine Learning-Based Desmoplastic Signatures Predict Patient Outcomes in Pancreatic Ductal Adenocarcinoma

Introduction: Pancreatic ductal adenocarcinoma (PDAC) is characterized by desmoplasia, a fibrotic matrix response with unclear impact on prognosis. We profiled clinical PDAC using two novel technologies: a machine learning algorithm to quantify matrix patterning in 436 patients and a spatial protein/cell phenotyping algorithm using CO-Detection by indEXing (CODEX). We integrated these modalities with patient metadata to define a unified, machine learning-based histologic signature for PDAC prognosis. Methods: Trichrome-stained clinical specimens were analyzed by an unsupervised algorithm for ultrastructural analysis (>13,000 images, 147 local/global fiber features), which were processed by uniform manifold approximation and projection (UMAP) and fitted to a minimum spanning tree-based pseudotime model. A custom 31-plex CODEX panel was used to visualize tumor- and stroma-associated cells by UMAP of protein expression. Cell interactions were quantified based on spatial co-localization (k=20 nearest neighbors). Binary classifiers for Poor/Good survivorship and recurrence (two- and one-year thresholds, respectively) were trained in MATLAB. Results: Matrix patterning comprised of two architectural groups (L,R), with group-R patients experiencing earlier median death and relapse by 461d and 95d, respectively. Poorer outcomes were primarily driven by three survival-negative interaction “nodes”: Type-I stromal fibroblasts, endothelial cells, and proliferating tumor cells, collectively forming two cell-pattern signatures (A,B) in which signature-B patients experienced earlier median death by 858d. A unified machine learning model successfully classified binary survivorship and recurrence with area under the curve of 0.943-0.946. Conclusion: Fibrotic architecture and cell patterning are powerful independent predictors for PDAC outcomes, with specific desmoplasia-associated cell phenotypes largely driving differential prognoses.

Diophantine triples with distinct binary recurrences

In this paper, we look at Diophantine triples with values in three different binary recurrence sequences. These are the Fibonacci and Pell sequences and the sequence of one more of powers of a given prime $p$. The novelty of the article is the appearance of three different sequences, as up to now the analogous problem had been investigated only for one sequence.

The Molecular Twin platform: a novel machine learning tool for democratization of precision cancer medicine.

e13546 Background: Pancreatic ductal adenocarcinoma (PDAC) is one of the most aggressive cancers. Contemporary analyses focused on a handful of molecular and clinical variables combined with machine learning algorithms (MLA) are unable to accurately predict therapy outcomes. Here, we use the Molecular Twin multi-omic analytical platform that evaluates tumor and host features extracted from 10 multi-omic analytes and provides an array of MLA, including a Parsimonious Biomarker Model that can predict survival and recurrence with limited analytic burden, while maintaining a high degree of fidelity. Methods: Retrospectively collected serum and tissue samples from 74 patients with Stage I/II resectable PDAC were subjected to targeted NGS DNA sequencing, whole transcriptome RNA sequencing, paired tissue proteomics, unpaired serum proteomics, lipidomics and computational pathology. Analytes including plasma proteins, RNA fusions, tissue proteins, plasma lipids, RNA gene expressions, CNVs, INDELS, SNVs and tumor nuclei characteristics, were processed to obtain a panel of 6363 features. 1024 single-omic and multi-omics feature combinations generated from this panel served as input for 7 different types of MLA to predict binary survival (SR) and disease recurrence (DR) outcomes. The resultant 70 single and 7098 multi-omic biomarker models were evaluated for positive predictive value (PPV) and accuracy (ACC) in predicting DR and SR, and feature proportions learned by each ML model using leave-one-patient-out cross-validation strategy. By recursively eliminating features with low importance, we developed progressively parsimonious biomarker models for predicting SR and DR. Results: Our top model was multi-omic and predicted the SR with ACC = 0.85, PPV = 0.87 and the DR with ACC = 0.90, and PPV = 0.91. It outperformed all models based only on one single analyte type including plasma protein, RNA fusion, tissue protein, plasma lipid, clinical, RNA gene expression, tumor nuclei characteristics, CNV, INDEL and SNV, in predicting the SR. This model contained predominantly plasma protein features. Interestingly, less accurate models contained a greater proportion of other features in addition to plasma proteins. Parsimonious feature reduction of the top model stabilized at 589 features yielding an ACC = 0.85, and PPV = 0.85, comparable to the intact model. Conclusions: This proof-of-concept of the Molecular Twin precision medicine platform applied in PDAC reveals the potential of our unique MLA to provide a novel parsimonious biomarker panel with similar fidelity as much larger biomarker panels. If these results are reproduced on larger datasets, across tumor types, the Molecular Twin platform would have significant potential to democratize precision cancer medicine by discovering smaller biomarker panels with the predictive performance of much larger ones thus reducing cost and simplifying assays.

On the least common multiple of binary linear recurrence sequences

We present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let P, Q, R0 and R1 be fixed integers, and let R=(Rn)n≥0 be the recurrence sequence defined by Rn+2=PRn+1−QRn for all n≥0. Under some conditions on the parameters, we determine a rational nontrivial divisor for Lk,n:= lcm(Rk,Rk+1,…,Rn), for all positive integers n and k, such that n≥k. As consequences, we derive nontrivial effective lower bounds for Lk,n, and we establish an asymptotic formula for log(Ln,n+m), where m is a fixed positive integer. Denoting by (Fn)n the usual Fibonacci sequence, we prove, for example, that for any m≥1, we have loglcm(Fn,Fn+1,…,Fn+m)∼n(m+1)logΦ( as n→+∞), where Φ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.

Studying the functional connectivity of the primary motor cortex with the binarized cross recurrence plot: The influence of Parkinson's disease.

Two new recurrence plot methods (the binary recurrence plot and binary cross recurrence plot) were introduced here to study the long-term dynamic of the primary motor cortex and its interaction with the primary somatosensory cortex, the anterior motor thalamus of the basal ganglia motor loop and the precuneous nucleus of the default mode network. These recurrence plot methods: 1. identify short-term transient interactions; 2. identify long-lasting delayed interactions that are common in complex systems; 3. work with non-stationary blood oxygen level dependent (BOLD) data; 4. may study the relationship of centers with non-linear functional interactions; 5 may compare different experimental groups performing different tasks. These methods were applied to BOLD time-series obtained in 20 control subjects and 20 Parkinson´s patients during the execution of motor activity and body posture tasks (task-block design). The binary recurrence plot showed the task-block BOLD response normally observed in the primary motor cortex with functional magnetic resonance imaging methods, but also shorter and longer BOLD-fluctuations than the task-block and which provided information about the long-term dynamic of this center. The binary cross recurrence plot showed short-lasting and long-lasting functional interactions between the primary motor cortex and the primary somatosensory cortex, anterior motor thalamus and precuneous nucleus, interactions which changed with the resting and motor tasks. Most of the interactions found in healthy controls were disrupted in Parkinson’s patients, and may be at the basis of some of the motor disorders and side-effects of dopaminergic drugs commonly observed in these patients.

Hyperelliptic curves and newform coefficients

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes ℓ<100 and all integers m≥1, we haveτ(n)≠±ℓ,±5m. For general newforms f with even integer weight 2k and integer coefficients, we prove for most integers j dividing 2k−1 and all ordinary primes p that af(p2) is never a j-th power. We prove a similar result for af(p4), conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.

Resolution of the equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$

Consider the diophantine equation $(3^{x_1}-1)(3^{x_2}-1)=(5^{y_1}-1)(5^{y_2}-1)$ in positive integers $x_1\le x_2$, and $y_1\le y_2$. Each side of the equation is a product of two terms of a given binary recurrence, respectively. In this paper, we prove that the only solution to the title equation is $(x_1,x_2,y_1,y_2)=(1,2,1,1)$. The main novelty of our result is that we allow products of two terms on both sides.

On the l.c.m. of random terms of binary recurrence sequences

For every positive integer n and every δ∈[0,1], let B(n,δ) denote the probabilistic model in which a random set A⊆{1,…,n} is constructed by choosing independently every element of {1,…,n} with probability δ. Moreover, let (uk)k≥0 be an integer sequence satisfying uk=a1uk−1+a2uk−2, for every integer k≥2, where u0=0, u1≠0, and a1,a2 are fixed nonzero integers; and let α and β, with |α|≥|β|, be the two roots of the polynomial X2−a1X−a2. Also, assume that α/β is not a root of unity.We prove that, as δn/log⁡n→+∞, for every A in B(n,δ) we havelog⁡lcm(ua:a∈A)∼δLi2(1−δ)1−δ⋅3log⁡|α/(a12,a2)|π2⋅n2 with probability 1−o(1), where lcm denotes the lowest common multiple, Li2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ=1.This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ=1, and is motivated by an asymptotic formula for lcm(A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui.

On special extrema of polynomials with applications to Diophantine problems

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial fin mathbb {Q}[x] can have at most one shift f(x)-lambda (lambda in mathbb {C}) of the form u(g(x))^q(h(x))^k with uin mathbb {C}, g,hin mathbb {C}[x] and either deg (g)=2, k is even, q=k/2 or deg (g)le 1, kge 2, qge 1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

A novel 95-gene signature (Curebest 95GC Breast) that predicts recurrence-risk in patients with ER-positive, HER2-negative, node-negative, early-stage primary invasive breast cancer with an intermediate Oncotype DX Recurrence Score.

542 Background: The TAILORx trial demonstrated that adjuvant endocrine and chemoendocrine therapies had similar efficacy in patients with hormone receptor-positive, HER2-negative, node-negative breast cancer with an Oncotype DX recurrence score (RS) of 11-25. However, a predictive strategy is needed to identify patients with intermediate RS who may benefit from adjuvant chemoendocrine therapy. Curebest 95GC Breast (95GC) is a 95-gene signature that can stratify patients into two groups with high (95GC-H) and low (95GC-L) groups to predict the risk of recurrence. Our primary objective was to show that 95GC can classify patients with intermediate RS into binary recurrence risk groups. Methods: Patients with ER-positive, HER2-negative, node-negative invasive breast cancer and RS 11-30 who underwent definitive surgery and adjuvant endocrine therapy were included. RNA was derived from archived formalin-fixed, paraffin-embedded samples, and 95GC was calculated as reported previously. The Fisher exact and Brunner-Munzel tests were used to compare variables between 95GC groups. A Kaplan-Meier estimate with a log-rank test was used for recurrence-free survival (RFS) analysis. Results: The analysis included 178 patients from five institutions. The 5-year RFS rate in patients with RS 18-30 was higher in the 95GC-L group (n = 129, 96.3%) than in the 95GC-H group (n = 49, 90.9%; p = 0.002), which was consistent with results in an independent Japanese population (n = 224; p &lt; 0.001). RFS rates significantly differed between the groups among patients with RS 11-25 as well (95GC-L, 97.4%; 95GC-H, 87.1%; p = 0.001). RFS rates did not differ between patients with RS 18-25 (94.8%) and those with RS 26-30 (93.8%; p = 0.33). Conclusions: 95GC can predict recurrence risk in patients with ER-positive, HER2-negative, node-negative invasive breast cancer and intermediate RS. Further prospective retrospective studies in the TAILORx population are warranted to confirm that 95GC can identify patients who may benefit from adjuvant chemoendocrine therapy.

Somos 4 Dizilerinin Tamsayılık Özelliği Üzerine

In this paper, it is shown that if (hn) is a Somos 4 sequence associated to an elliptic curve then each hn can be expressed as elements of the ring R = ℤ[a1, a2, a3, a4, x, y, h–1±1, h0]. In particular, if h –1 = ±1, then the Somos 4 sequence consist entirely of integers for n ≥ 0. Also the general term of the Somos 4 sequence is given which satisfies a binary recurrence relation.