On a two-dimensional nonlinear system of difference equations close to the bilinear system by Stevo Stević, Durhasan Turgut Tollu

“…We consider the two-dimensional nonlinear system of difference equations $ x_n = x_{n-k}\frac{ay_{n-l}+by_{n-(k+l)}}{cy_{n-l}+dy_{n-(k+l)}},\quad y_n = y_{n-k}\frac{{\alpha} x_{n-l}+{\beta} x_{n-(k+l)}}{{\gamma} x_{n-l}+{\delta} x_{n-(k+l)}}, $ for $ n\in{\mathbb N}_0, $ where the delays $ k $ and $ l $ are two natural numbers, and the initial values $ x_{-j}, y_{-j} $, $ 1\le j\le k+l $, and the parameters $ a, b, c, d, {\alpha}, {\beta}, {\gamma}, {\delta} $ are real numbers. We show that the system of difference equations is solvable by presenting a method for finding its general solution in detail. …”
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Spectrophores as one-dimensional descriptors calculated from three-dimensional atomic properties: applications ranging from scaffold hopping to multi-target virtual screening by Rafaela Gladysz, Fabio Mendes Dos Santos, Wilfried Langenaeker, Gert Thijs, Koen Augustyns, Hans De Winter

“…A typical spectrophore consists of a vector of 48 real numbers. This makes it highly suitable for the calculation of a wide range of similarity measures for use in virtual screening and for the investigation of quantitative structure–activity relationships in combination with advanced statistical approaches such as self-organizing maps, support vector machines and neural networks. …”
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On a higher-order system of difference equations by Stevo Stevic, Mohammed Alghamdi, Abdullah Alotaibi, N. Shahzad

“…Here we study the following system of difference equations \begin{align} x_n&=f^{-1}\bigg(\frac{c_nf(x_{n-2k})}{a_n+b_n\prod_{i=1}^kg(y_{n-(2i-1)})f(x_{n-2i})}\bigg),\nonumber\\ y_n&=g^{-1}\bigg(\frac{\gamma_n g(y_{n-2k})}{\alpha_n+\beta_n \prod_{i=1}^kf(x_{n-(2i-1)})g(y_{n-2i})}\bigg),\nonumber \end{align} $n\in\mathbb{N}_0,$ where $f$ and $g$ are increasing real functions such that $f(0)=g(0)=0$, and coefficients $a_n,$ $b_n$, $c_n$, $\alpha_n$, $\beta_n$, $\gamma_n$, $n\in\mathbb{N}_0$, and initial values $x_{-i}$, $y_{-i}$, $i\in\{1,2,\ldots,2k\}$ are real numbers. We show that the system is solvable in closed form, and study asymptotic behavior of its solutions.…”
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Anomaly Detection Paradigm for Multivariate Time Series Data Mining for Healthcare by Abdul Razaque, Marzhan Abenova, Munif Alotaibi, Bandar Alotaibi, Hamoud Alshammari, Salim Hariri, Aziz Alotaibi

“…Time series data are significant, and are derived from temporal data, which involve real numbers representing values collected regularly over time. …”
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THE TWO-SIDED ESTIMATES OF THE FREDHOLM RADIUS AND COMPACTNESS CONDITIONS FOR THE OPERATOR ASSOCIATED WITH A SECOND-ORDER DIFFERENTIAL EQUATION by K. N. Ospanov, A. N. Yesbayev

“…The singularity of the specified differential equation means that it is defined in a noncompact domain - on the whole set of real numbers, and its coefficients are unbounded functions. …”
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Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products by Arnon Ploymukda, Pattrawut Chansangiam

“…When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. …”
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A Low-Latency, Low-Area Hardware Oblivious RAM Controller by Fletcher, Christopher Wardlaw, Ren, Ling, Kwon, Young Hyun, van Dijk, Marten, Stefanov, Emil, Serpanos, Dimitrios, Devadas, Srinivas

“…Tiny ORAM is also the first design to implement and report real numbers for the cost of symmetric encryption in hardware ORAM constructions. …”
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Membership function model for defining optimality of vapor pressure deficit in closed-field cultivation of tomato by Shamshiri, Ramin, Che Man, Hasfalina, Zakaria, Abd Jamil, Beveren, Peter van, Wan Ismail, Wan Ishak, Ahmad, Desa

“…Results were validated with three published literatures and were shown to be capable of exploring optimal levels of VPD by means of real numbers between 0 and 1. This study can contribute to knowledge-based information and decision support systems in greenhouse climate control and management by quantifying comfort level of microclimate.…”
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Algebraic Perspective of Cubic Multi-Polar Structures on BCK/BCI-Algebras by Anas Al-Masarwah, Halimah Alshehri

“…Cubic multipolar structure with finite degree (briefly, cubic <i>k</i>-polar (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula>) structure) is a new hybrid extension of both <i>k</i>-polar fuzzy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula>) structure and cubic structure in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>k</mi></msub><mi>P</mi></mrow></semantics></math></inline-formula> structure consists of two parts; the first one is an interval-valued <i>k</i>-polar fuzzy (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msub><mi>V</mi><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula>) structure acting as a membership grade extended from the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">P</mi><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mi>k</mi></msup></mrow></semantics></math></inline-formula> (i.e., from interval-valued of real numbers to the <i>k</i>-tuple interval-valued of real numbers), and the second one is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow></mrow><mi>k</mi></msub><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula> structure acting as a nonmembership grade extended from the interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mi>k</mi></msup></semantics></math></inline-formula> (i.e., from real numbers to the <i>k</i>-tuple of real numbers). …”
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Pre-Calculus / by Sterling, Mary Jane, author

Essential Convexity and Complexity of Semi-Algebraic Constraints by Manuel Bodirsky, Peter Jonsson, Timo von Oertzen

“…Let \Gamma be a structure with a finite relational signature and a first-order definition in (R;*,+) with parameters from R, that is, a relational structure over the real numbers where all relations are semi-algebraic sets. …”
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Positive periodic solution for indefinite singular Liénard equation with p-Laplacian by Tiantian Zhou, Bo Du, Haiqing Du

“…Here ϕp(s)=|s|p−2s $\phi _{p}(s)=|s|^{p-2}s$, p>1 $p>1$, α1,α2∈L([0,T],R) $\alpha _{1},\alpha _{2}\in L([0,T],{R}) $, f∈C(R+,R) $f\in C({R}_{+},{R})$ ( R+ ${R} _{+}$ stands for positive real numbers) with a singularity at x=0 $x=0$, g(x) $g(x)$ is continuous on (0;+∞) $(0;+\infty )$, μ is a constant with μ>0 $\mu >0$, the signs of α1 $\alpha _{1}$ and α2 $\alpha _{2} $ are allowed to change. …”
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On a Non-Newtonian Calculus of Variations by Delfim F. M. Torres

“…As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. …”
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An Interactive Decision-Making Method for Third-Party Logistics Provider Selection under Hybrid Multi-Criteria by Yumin Liu, Peng Zhou, Liyuan Li, Feng Zhu

“…In this paper, a novel interactive decision-making method to deal with the 3PL provider selection problem of hesitant fuzzy sets, intuitionistic fuzzy sets and real numbers is developed. We first investigate the positive and negative ideal solutions of the alternative and the satisfaction degree of the DMs under hybrid multi-criteria circumstances. …”
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Software implementation of protection against unauthorized access by Albina S. Ismagilova, Nikita D. Lushnikov

“…Based on the generated data set, a method for synthesizing the parameters of a mathematical model of a convolutional neural network, presented as an array of real numbers, which are unique identifiers of a personal computer user, has been developed and proposed. …”
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An ensemble novel architecture for Bangla Mathematical Entity Recognition (MER) using transformer based learning by Tanjim Taharat Aurpa, Md Shoaib Ahmed

“…Here, we identify the mathematical operator, operands as numbers, and popular mathematical terms (complex numbers, real numbers, prime numbers, etc.). In this work, we recognize Bangla Mathematical Entity Recognition (MER) utilizing the ensemble architecture of deep neural networks known as Bidirectional Encoder Representations from Transformers (BERT). …”
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Exact inequalities involving power mean, arithmetic mean and identric mean by Yu-ming Chu, Ming-yu Shi, Yue-ping Jiang

“…For \(p\in \mathbb{R}\), the power mean \(M_{p}(a,b)\) of order \(p\), identric mean \(I(a,b)\) and arithmetic mean \(A(a,b)\) of two positive real numbers \(a\) and \(b\) are defined by \begin{equation*} M_{p}(a,b)= \begin{cases} \displaystyle\left(\tfrac{a^{p}+b^{p}}{2}\right)^{1/p}, & p\neq 0,\\ \sqrt{ab}, & p=0, \end{cases} \quad I(a,b)= \begin{cases} \displaystyle\tfrac{1}{\rm {e}}\left(b^{b}/a^{a}\right)^{1/(b-a)}, & a\neq b,\\ \displaystyle a, & a=b, \end{cases} \end{equation*} and \(A(a,b)=(a+b)/2\), respectively.   …”
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A flexible dose-response modeling framework based on continuous toxicity outcomes in phase I cancer clinical trials by Se Yoon Lee

“…A continuous toxicity response pertains to a quantitative measure represented by real numbers. A higher value corresponds not only to an elevated likelihood of side effects for patients but also to an increased probability of treatment efficacy. …”
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From Boolean Valued Analysis to Quantum Set Theory: Mathematical Worldview of Gaisi Takeuti by Masanao Ozawa

“…While it is known that the distributive law does not apply to quantum logic, and the equality axiom turns out not to hold in quantum set theory, he showed that the real numbers in quantum set theory are in one-to-one correspondence with the self-adjoint operators on a Hilbert space, or equivalently the physical quantities of the corresponding quantum system. …”
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A new chaos function development through the combination of Circle map and MS map by Mursidah Ichsani, Mt Suryadi, Madenda Sarifuddin

“…Lyapunov exponent has a non-negative value at x0 = 0.4, r = 3.8, Ω = 0.5, λ = 2.1, K = 4 which is the domain xn ∈ (0, 1) and parameter values r, Ω, λ and K are any real numbers. The results of the NIST randomness level test show that the MSI-Circle map function passed all the randomness test of 16 NIST tests.…”
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