We prove that Anderson's conjecture on symmetric sequencings and Bailey's conjecture on 2-sequencings hold for sufficiently large groups. In addition, we discuss extensions of partial harmonious sequences and partial R-sequencings. Several further results on double sequencings are presented, both in the context of abelian groups and for sufficiently large non-abelian groups.

The \textit{order of appearance} $ z(n) $ of a positive integer $ n $ in the Fibonacci sequence is defined as the smallest positive integer $ j $ such that $ n $ divides the $ j $-th Fibonacci number. A \textit{fixed point} arises when, for a positive integer $ n $, we have that the $ n^{\text{th}} $ Fibonacci number is the smallest Fibonacci that $ n $ divides. In other words, $ z(n) = n $. In 2012, Marques proved that fixed points occur only when $ n $ is of the form $ 5^{k} $ or $ 12\cdot5^{k} $ for all non-negative integers $ k $. It immediately follows that there are infinitely many fixed points in the Fibonacci sequence. We prove that there are infinitely many integers that iterate to a fixed point in exactly $ k $ steps. In addition, we construct infinite families of integers that go to each fixed point of the form $12 \cdot 5^{k}$. We conclude by providing an alternate proof that all positive integers $n$ reach a fixed point after a finite number of iterations.

We show that any weighted geometric mean of Chebyshev polynomials is bounded from above by another Chebyshev polynomial. We also study a related homogeneous cyclic inequality $$ \left (\sum_{i=1}^n x_i^{(a+b+1)/2} \right )^2 \geq \sum_{i=1}^n x_i \sum_{i=1}^n x_i^a x_{i+1}^b,$$ where $a,b,x_1,\ldots, x_n$ (with $x_{n+1}=x_1$) are nonnegative. In particular, we prove that the inequality holds when $a=b=1$ and $n\leq 8$ for all nonnegative numbers $x_1,\ldots, x_n$.

Given a sequence g : g 0 , horizontal ellipsis , g m ${\bf{g}}:{g}_{0},{\rm{\ldots }},{g}_{m}$ in a finite group G $G$ with g 0 = 1 G ${g}_{0}={1}_{G}$, let g over bar : g over bar 0 , horizontal ellipsis , g over bar m $\bar{{\bf{g}}}:{\bar{g}}_{0},{\rm{\ldots }},{\bar{g}}_{m}$ be the sequence of consecutive quotients of g ${\bf{g}}$ defined by g over bar 0 = 1 G ${\bar{g}}_{0}={1}_{G}$ and g over bar i = gi - 1- 1 g i ${\bar{g}}_{i}={g}_{i-1}<^>{-1}{g}_{i}$ for 1 <= i <= m $1\le i\le m$. We say that G $G$ is doubly sequenceable if there exists a sequence g ${\bf{g}}$ in G $G$ such that every element of G $G$ appears exactly twice in each of g ${\bf{g}}$ and g over bar $\bar{{\bf{g}}}$. We show that if a group is abelian, odd, sequenceable, R-sequenceable, or terraceable, then it is doubly sequenceable. We also show that if H $H$ is an odd or sequenceable group and K $K$ is an abelian group, then H x K $H\times K$ is doubly sequenceable.

We show that vertical bar Omega(p)vertical bar < 3(p + 1)/4 + root p/2 for all primes p, where Omega p is the set of Fibonacci numbers modulo prime p. In the case of maximal Pisano periods, we determine the exact value of vertical bar Omega(p)vertical bar .

Finding more energy-efficient and lower carbon emitting ways to develop heavy oil reservoirs is key for economically unlocking more than 50% of the remaining oil reserves worldwide. Continuous steam injection (CSI) is one of the most widely used yet energy-intensive heavy oil enhanced oil recovery techniques. One of the proposed methods to reduce the carbon footprint of heavy oil developments is using cyclic steam development (CSD) rather than continuous steam injection where applicable. The study has two objectives. First, we developed a CSD conceptual model which describes the physical principles, represents the full life cycle of a CSD and focuses on key elements. Second, a thermal simulation model (based on the conceptual model) for CSD was calibrated to field data. Using thismodel, we conducted a parametric study to select the most impactful input parameters for machine learning model (MLM) training. We introduced a point-based regression method to train the MLM at every single time step for time series prediction of selected properties such as oil production, reservoir pressure, and so on. To manage large amounts of trained MLMs, we also conducted dimension reduction on prediction matrix using Principal Component Analysis (PCA). Once the MLM was trained with the field and simulated data, a web-based fast predictive tool was developed to provide consistent and robust predictions within seconds. This makes it possible to quickly evaluate various scenarios, assess uncertainties and optimize injection job parameters. Furthermore, the tool generates type-curves to identify the inflection point where increasing injected steam volume (per cycle) will not lead to improved recovery due to longer injection period or down-time for CSD wells. The presented workflow can also be widely applied to different recovery techniques, and asset classes.

We give a new proof of the Shah-Bruckner theorem, which states that vertical bar Omega(p)vertical bar < p for every prime p > 7, where Omega(p) is the set of Fibonacci numbers modulo p. We also show that lim inf (p -> 8) vertical bar Omega(p)vertical bar/p = 0; in other words, for every epsilon > 0, there exists a prime p such vertical bar Omega(p)vertical bar < p epsilon.

A finite group of order $n$ is said to have the distinct divisor pro-perty (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i

We find necessary and sufficient conditions for a set closed under the cross product to be dense in . We also show that, up to orthogonal transformations, the vertices of a regular octahedron and its center at the origin provide the only nontrivial example of a finite set closed under the cross product.

We study generalized Fibonacci sequences $F_{n+1}=PF_n-QF_{n-1}$ with initial values $F_0=0$ and $F_1=1$. Let $P,Q$ be nonzero integers such that $P^2-4Q$ is not a perfect square. We show that if $Q=\pm 1$ then the sequence $\{F_n\}_{n=0}^\infty$ misses a congruence class modulo every prime large enough. On the other hand, if $Q \neq \pm 1$, we prove that (under GRH) the sequence $\{F_n\}_{n=0}^\infty$ hits every congruence class modulo infinitely many primes.

SummaryHorizontal steam injectors can improve the efficiency of thermal operations relative to vertical injectors. However, effective in-well and reservoir surveillance are needed to understand steam conformance. Uniform steam-chest development improves the steam/oil ratio in continuous steam injection and accelerates recovery in cyclic steam injection. The conformance of the injected steam can be achieved by flow control devices (FCDs) deployed on either tubing or liner. A new liner-deployed FCD was used in a horizontal steam injector in the Kern River field. The liner-deployed FCD is intended to replace the tubing-deployed FCDs while reducing capital costs, surveillance costs, and well intervention costs for conformance control.Fiber optics was used for surveillance, which is the most promising method in horizontal steam injectors considering reliability, accuracy, and cost. Fiber optic data enables monitoring the performance of liner-deployed FCDs as well as estimating the flow profile along the lateral length. Multimode distributed temperature sensing (DTS) optical fibers and single-mode distributed acoustic sensing (DAS) optical fibers were installed in the well for these objectives. Algorithms for interpreting DTS were improved to include a new technique, shape language modeling (SLM), and a probabilistic approach. The configuration of the FCDs was changed during the first well intervention, and it was monitored by DTS and DAS. Data from both DTS and DAS confirms the open/closed position of the sliding sleeve of FCDs initially and after the intervention. The probabilistic estimates of steam outflow in several FCD configurations match well with the theoretical outflow that is expected from the critical flow of steam through chokes installed in the FCDs.

We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational coefficient other than the constant term. In the case of even degree, we prove a stronger result: $\lambda(P)$ intersects every congruence class modulo every integer if and only if $P(x)$ has at least one irrational coefficient other than the constant term.

Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $P_m$ and $G_m/P_m$ have infinite rank for all $m>1$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.

We show that if a homeomorphism of a separable locally compact metric space has a unique fixed point that is attracting or repelling, then its corresponding composition operator is cyclic. On the real line, we show that a composition operator is cyclic if and only if its symbol has at most one fixed point. Other results on the real line and circle are also discussed.

A loxodrome is a curve that makes a constant angle with the meridians.We use conformal maps and the notion of parallel transport in differential geometry to investigate loxodromes on hypersurfaces of revolution and their spiral behavior near a pole.

Let 𝕂=ℝ or ℂ, and T_n(𝕂) be the set of n× n lower triangular matrices with entries in 𝕂. We show that T_n(𝕂) has dense subsemigroups that are generated by n+1 matrices.

Given infinite-dimensional real vector spaces $V,W$ with $|W| \leq |V|$, it is shown that there exists a collection of subspaces of $V$ that are isomorphic to $W$, mutually intersect only at 0, and altogether cover $V$.

Let A is an element of M-n(C). We prove that if W-c(A), the correlation numerical range introduced in [2], is a subset of [0,infinity), then A = P + D where P is positive semidefinite and D is a diagonal matrix such that Tr(D) = 0. This answers a question of D. Hadwin and D. Han. Additionally, we explore a few properties of W-c(A) and W-uc(A), another numerical range introduced in [2] that is closely related to Connes's Embedding Conjecture.

In this paper, we study the existence of cycle double covers for infinite planar graphs. We show that every infinite locally finite bridgeless k-indivisible graph with a 2-basis admits a cycle double cover.