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We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. 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. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters. [ABSTRACT FROM AUTHOR]

In this paper, we study nonnegative weak solutions of the quasilinear elliptic equation div(A(x,u,∇u))=B(x,u, ∇u), in a bounded open set Ω, whose coefficients belong to a generalized Morrey space. We show that log(u + δ), for u a nonnegative solution and δ an arbitrary positive real number, belongs to BMO(B), where B is an open ball contained in Ω. As a consequence, this equation has the strong unique continuation property. For the main proof, we use approximation by smooth functions to the weak solutions to handle the weak gradient of the composite function which involves the weak solutions and then apply Fefferman's inequality in generalized Morrey spaces, recently proved by Tumalun et al. [1]. [ABSTRACT FROM AUTHOR]

In this article, we aim to introduce and explore a new class of preinvex functions called npolynomial m-preinvex functions, while also presenting algebraic properties to enhance their numerical significance. We investigate novel variations of Pachpatte and Hermite-Hadamard integral inequalities pertaining to the concept of preinvex functions within the framework of the Caputo-Fabrizio fractional integral operator. By utilizing this direction, we establish a novel fractional integral identity that relates to preinvex functions for differentiable mappings of first-order. Furthermore, we derive some novel refinements for Hermite-Hadamard type inequalities for functions whose first-order derivatives are polynomial preinvex in the Caputo-Fabrizio fractional sense. To demonstrate the practical utility of our findings, we present several inequalities using specific real number means. Overall, our investigation sheds light on convex analysis within the context of fractional calculus. [ABSTRACT FROM AUTHOR]

In this study, the concepts of lacunary (H , 1) − summability of order α , lacunary strongly harmonic summability of order α , lacunary statistical logarithmic convergence of order α of sequences of real numbers are introduced and relations between these concepts are investigated. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a simple connected graph with vertex set V = { 1 , 2 , ... , n } and edge set E. The normalized Laplacian matrix of G is L = D − 1 / 2 (D − A) D − 1 / 2 , where D is the degree-diagonal matrix of G and A is the adjacency matrix of G. Let λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n − 1 ≥ λ n = 0 be the eigenvalues of L , for any real number α , the topological index s α ∗ (G) of G is defined as s α ∗ (G) = ∑ i = 1 n − 1 λ i α. Let Ω i j denote the resistance distance between vertices i and j , the degree-Kirchhoff index K f ∗ (G) of G is defined as K f ∗ (G) = ∑ i < j d i d j Ω i j , where d i is the degree of the vertex i. In this paper, we first express s α ∗ (G) in terms of Ω i j 's explicitly for any real number α. Then as an application, we give a new proof for the classical Foster's k th formula of resistance distances. Finally, we generalize the well-known relation 2 | E | s − 1 ∗ (G) = K f ∗ (G) to any integer k ≥ − 1 , which provides some upper bounds for s k ∗ (G). [ABSTRACT FROM AUTHOR]

In our research work generalized Hyers-Ulam stability of the following functional inequalities is analyzed by using fixed point approach: 0.1 ∥ f (2 x + y) + f (2 x − y) − 2 f (x + y) − 2 f (x − y) − 12 f (x) − ρ (4 f (x + y 2) + 4 (f (x − y 2) − f (x + y) − f (x − y)) − 6 f (x) , r) ∥ ≥ r r + φ (x , y) and 0.2 ∥ f (2 x + y) + f (2 x − y) − 4 f (x + y) − 4 f (x − y) − 24 f (x) + 6 f (y) − ρ (8 f (x + y 2) + 8 (f (x − y 2) − 2 f (x + y) − 2 f (x − y)) − 12 f (x) + 3 f (y) , r) ∥ ≥ r r + φ (x , y) in the setting of fuzzy matrix, where ρ ≠ 2 is a real number. We also discussed Hyers-Ulam stability from the application point of view. [ABSTRACT FROM AUTHOR]

An alternative new method for solving an epistemic uncertainty such as fuzzy-based linear programming problem with equality constraints has been proposed here. In this approach, the optimal solution of the considered fuzzy problem has been achieved by solving a related crisp problem. Moreover, the objective function of the crisp problem has been obtained by taking the linear combination of the fuzzified version of the original uncertain objective function. The coefficients of the linear combination are considered as strictly positive real numbers. And the constraints of the crisp problem are obtained equivalently from the equality constraints of the fuzzy problem. Triangular fuzzy numbers are considered for this analysis. Comparisons have been made with the results of Kumar et al. (Appl Math Model 35:817–823, 2011), Najafi and Edalatpanah (Appl Math Model 37:7865–7867, 2013), Ezzati et al. (Appl Math Model 39:3183-3193, 2015) and Ghoushchi et al. (Mathematics 9:2937, 2021. https://doi.org/10.3390/math9222937). [ABSTRACT FROM AUTHOR]

The new modular approach to constrained control of higher-order processes with dominant first-order dynamics using generalized controllers with automatic resets (ARCs) is addressed. The controller design is based on the multiple real dominant pole (MRDP) method for the integrator plus dead time (IPDT) process models. The controller output constraints are taken into account by inserting the smallest numerator time constant of the controller transfer function into the positive feedback loop representing the automatic reset (integral) term. In the series realization of the proportional–integral–derivative–acceleration (PIDA) controller (and other controllers with even derivative degree), the time constant mentioned is complex, so only the real part of the time constant has been used so far. Other possible conversions of a complex number to a real number, such as the absolute value (modulus), can be covered by introducing a tuning parameter that modifies the calculated real time constant and generalizes the mentioned conversion when designing controllers with constraints. In this article, the impact of the tuning parameter on the overall dynamics of the control loop is studied by simulation. In addition, an evaluation of the stability of the closed-loop control system is performed using the circle criterion in the frequency domain. The analysis has shown that the approximation of the complex zero by its real part and modulus leads to a near optimal response to the set point tracking. The disturbance rejection can be significantly improved by increasing the tuning parameter by nearly 50%. In general, the tuning parameter can be used to find a compromise between servo and regulatory control. The robustness and applicability of the proposed controller is evaluated using a time-delayed process with first-order dominant dynamics when the actual transfer function is much more complicated than the IPDT model. A comparison of the proposed MRDP-PIDA controller with series PI, PID and PIDA controllers based on a modified SIMC method has shown that the MRDP-PIDA controller performs better than the SIMC method, although the SIMC uses a more complex process model. [ABSTRACT FROM AUTHOR]

Let α ≥ 2 , m ≥ 2 be integers, p be an odd prime with p ∤ m (m + 1) , 0 < λ 1 , λ 2 ≤ 1 be real numbers, q = p α > max { [ 1 λ 1 ] , [ 1 λ 2 ] } . For any integer n with (n , q) = 1 and a nonnegative integer k, we define M λ 1 , λ 2 (m , n , k ; q) = ∑ ′ a = 1 q ∑ ′ b = 1 [ λ 1 q ] ∑ ′ c = 1 [ λ 2 q ] a b ≡ 1 (mod q) c ≡ a m (mod q) n ∤ b + c (b − c) 2 k. In this paper, we study the arithmetic properties of these generalized Kloosterman sums and give an upper bound estimation for it. By using the upper bound estimation, we discuss the properties of M λ 1 , λ 2 (m , n , k ; q) and obtain an asymptotic formula. [ABSTRACT FROM AUTHOR]

There are various kinds of fuzzy inner products. It was shown (Byun et al. in Mathematics 8:571, 2020) that a fuzzy inner product on a vector space, producing a fuzzy real number as its value, is merely the embedding of a real-valued inner product into the fuzzy number systems over the set of tuples consisting of elements in the space because of its linear property. Thus, the linear property is needed to be weakened. In this paper, a new concept, called a fuzzy real weak inner product on a vector space, is introduced and some nontrivial examples are given. Then, a fuzzy real weak inner product in a suitable condition is deeply studied, which may be either almost positive-definite or positive-definite. Furthermore, it satisfies both Cauchy inequality and Parallelogram law even though it is not linear. And some properties of a general type of a fuzzy real weak inner product, especially the approximation related to the Parallelogram law, are investigated in analytic viewpoint. [ABSTRACT FROM AUTHOR]

Outlier detection is an important topic in data mining. An information system (IS) is a database that shows relationships between objects and attributes. A real-valued information system (RVIS) is an IS whose information values are real numbers. People often encounter missing values during data processing. A RVIS with the miss values is an incomplete real-valued information system (IRVIS). Due to the presence of the missing values, the distance between two information values is difficult to determine, so the existing outlier detection rarely considered an IS with the miss values. This paper investigates outlier detection for an IRVIS via rough set theory and granular computing. Firstly, the distance between two information values on each attribute of an IRVIS is introduced, and the parameter λ to control the distance is given. Then, the tolerance relation on the object set is defined according to the distance, and the tolerance class is obtained, which is regarded as an information granule. After then, λ-lower and λ-upper approximations in an IRVIS are put forward. Next, the outlier factor of every object in an IRVIS is presented. Finally, outlier detection method for IRVIS via rough set theory and granular computing is proposed, and the corresponding algorithms is designed. Through the experiments, the proposed method is compared with other methods. The experimental results show that the designed algorithm is more effective than some existing algorithms in an IRVIS. It is worth mentioning that for comprehensive comparison, ROC curve and AUC value are used to illustrate the advantages of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study the Γ-limit, as p → 1 , of the functional J p ⁢ (u) = ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p , where Ω is a smooth bounded open set in ℝ N , p > 1 and β is a real number. Among our results, for β > - 1 , we derive an isoperimetric inequality for Λ ⁢ (Ω , β) = inf u ∈ BV ⁡ (Ω) , u ≢ 0 ⁡ | D ⁢ u | ⁢ (Ω) + min ⁡ (β , 1) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | which is the limit as p → 1 + of λ ⁢ (Ω , p , β) = min u ∈ W 1 , p ⁢ (Ω) ⁡ J p ⁢ (u) . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ⁢ (Ω , β) when β ∈ (- 1 , 0) and minimizes Λ ⁢ (Ω , β) when β ∈ [ 0 , ∞) . [ABSTRACT FROM AUTHOR]

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $ -function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$ , respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points. [ABSTRACT FROM AUTHOR]

Let $[t]$ be the integral part of the real number t and let $\mathbb {1}_{{\mathbb P}}$ be the characteristic function of the primes. Denote by $\pi _{\mathcal {S}}(x)$ the number of primes in the floor function set $\mathcal {S}(x) := \{[{x}/{n}] : 1\leqslant n\leqslant x\}$ and by $S_{\mathbb {1}_{{\mathbb P}}}(x)$ the number of primes in the sequence $\{[{x}/{n}]\}_{n\geqslant 1}$. Improving a result of Heyman ['Primes in floor function sets', Integers 22 (2022), Article no. A59], we show $$ \begin{align*} \pi_{\mathcal{S}}(x) = \int_2^{\sqrt{x}} \frac{d t}{\log t} + \int_2^{\sqrt{x}} \frac{d t}{\log(x/t)} + O(\sqrt{x}\,\mathrm{e}^{-c(\log x)^{3/5}(\log\log x)^{-1/5}}) \quad\mbox{and}\quad S_{\mathbb{1}_{{\mathbb P}}}(x) = C_{\mathbb{1}_{{\mathbb P}}} x + O_{\varepsilon}(x^{9/19+\varepsilon}) \end{align*} $$ for $x\to \infty $ , where $C_{\mathbb {1}_{{\mathbb P}}} := \sum _{p} {1}/{p(p+1)}$ , $c>0$ is a positive constant and $\varepsilon $ is an arbitrarily small positive number. [ABSTRACT FROM AUTHOR]

Polarimetric synthetic aperture radar (PolSAR) image classification has been an important area of research due to its wide range of applications. Traditional machine learning methods were insufficient in achieving satisfactory results before the advent of deep learning. Results have significantly improved with the widespread use of deep learning in PolSAR image classification. However, the challenge of reconciling the complex-valued inputs of PolSAR images with the real-valued models of deep learning remains unsolved. Current complex-valued deep learning models treat complex numbers as two distinct real numbers, providing limited assistance in PolSAR image classification results. This paper proposes a novel, complex-valued deep learning approach for PolSAR image classification to address this issue. The approach includes amplitude-based max pooling, complex-valued nonlinear activation, and a cross-entropy loss function based on complex-valued probability. Amplitude-based max pooling reduces computational effort while preserving the most valuable complex-valued features. Complex-valued nonlinear activation maps feature into a high-dimensional complex-domain space, producing the most discriminative features. The complex-valued cross-entropy loss function computes the classification loss using the complex-valued model output and dataset labels, resulting in more accurate and robust classification results. The proposed method was applied to a shallow CNN, deep CNN, FCN, and SegNet, and its effectiveness was verified on three public datasets. The results showed that the method achieved optimal classification results on any model and dataset. [ABSTRACT FROM AUTHOR]

Recently, a new paradigm for uncertain information has been proposed that can effectively handle various types of uncertainty in decision-making problems. This approach utilizes a certainty degree, which is represented by a real number indicating the level of certainty associated with input values. However, just like intuitionistic fuzzy information can handle more problems that cannot be well modeled by fuzzy information, the certainty degree in basic uncertain information can also be intuitionistic fuzzy granule, which allows it to handle more uncertainty involved decision making situations. In this paper, we introduce the concept of intuitionistic fuzzy type basic uncertain information and explain its parameters. We also define a weighted arithmetic mean for aggregating this type of information and discuss different approaches for allocating induced weights based on trust preferred preference from four perspectives: (i) preference for higher certainty degrees; (ii) aversion to higher levels of uncertainty; (iii) preference for greater differences in certainty degrees; and (iv) preference for intuitionistic fuzzy certainties. Additionally, we explore trichotomic rules-based decision making using intuitionistic fuzzy type basic uncertain information. Finally, we present an objective-subjective evaluation numerical example utilizing these methods. [ABSTRACT FROM AUTHOR]

A list Λ = {λ1, λ2, ..., λn} of complex numbers is said to be realizable if it is the spectrum of a nonnegative matrix. Λ is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ. In this paper, using companion matrices and applying a procedure by Šmigoc, we provide sufficient conditions for the universal realizability of left half-plane spectra, that is, spectra Λ = {λ1, ..., λn} with λ1 > 0, Re λi ≤ 0, i = 2, ..., n. It is also shown how the effect of adding a negative real number to a not UR left half-plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized. [ABSTRACT FROM AUTHOR]

Let (X, τ) be a topological space, we will denote by |X|,|X|K, |X|τ and |X|δ, the cardinalities of X; the family of compacts in X; the family of closed in X, and the family of Gδ-closed in X, respectively. The purpose of this work is to establish relationships between these four numbers and conditions under which two of them coincide or one of them is ≤ c, where c denotes, as usual, the cardinality of the set of real numbers R. We will use the Stone-Weierstrass theorem to prove that: Let (X, τ) be a compact Hausdorff topological space, then .... [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a more generalized class of convex stochastic processes and explore some of their algebraic properties. This new class of stochastic processes is called the n -polynomial s -type convex stochastic process. We demonstrate that this new class of stochastic processes leads to the discovery of novel Hermite–Hadamard type inequalities. These inequalities provide upper bounds on the integral of a convex function over an interval in terms of the moments of the stochastic process and the convexity parameter s. To compare the effectiveness of the newly discovered Hermite–Hadamard type inequalities, we also consider other commonly used integral inequalities, such as Hölder, Hölder–Ïşcan, and power-mean, as well as improved power-mean integral inequalities. We show that the Hölder–Ïşcan and improved power-mean integral inequalities provide a better approach for the n -polynomial s -type convex stochastic process than the other integral inequalities. Finally, we provide some applications of the Hermite–Hadamard type inequalities to special means of real numbers. Our findings provide a useful tool for the analysis of stochastic processes in various fields, including finance, economics, and engineering. [ABSTRACT FROM AUTHOR]

This paper addresses the internal rate of return (IRR) assessment of subsidised production. The difference between the result of IRR calculation and the reality can be largely attributed to uncertainty about the future market price and demand. Thus, in the IRR model the intervals of possible input values instead of uncertain point values should be taken into account. Within the intervals of the input values there is no relevant reason to prefer one value over another. The fuzzy approach is applied to decide whether or not the system of subsidies is adequate in terms of IRR. It leans on the "fuzzy numbers", which are generalisations of real numbers within the meaning of not referring to a single value but rather to intervals of possible values. The results are then compared with the conventional statistical approach of IRR single-value evaluation. The analogy between the conventional statistical and fuzzy approach is shown on the theoretical level. The practical contribution of this paper lies in the IRR assessment of a subsidised electric car production project. Two scenarios of possible project development are considered. In the study, we derive the algorithm to calculate the average probability of the appearance of the subjectively expected IRR, which is compared with the minimum required profitability. We show that the IRR evaluation of the subsidised production should not be underestimated. The need to verify the adequacy of a subsidised project by a proper tool is given by the fact that many investors mistakenly believe that subsidies will ensure the required profitability. The use of statistical methods in a state of uncertainty can lead to misleading results. The fuzzy analysis offers the investors involving the subjective risk in their decision making more benefits and brings more information compared to statistical approach. [ABSTRACT FROM AUTHOR]

We consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane e−N|z|2∏j=1ν|z−aj|2cjdA(z)$$\begin{equation*}{{\mathrm{e}}}^{ - N|z{|}^2}\prod\limits_{j = 1}^\nu {|z - {a}_j{|}^{2{c}_j}} {\mathrm{d}}A(z)\end{equation*}$$where dA is the Lebesgue measure of the plane, N is a positive constant, {c1, ..., cν} are nonzero real numbers greater than −1 and {a1,...,aν}⊂D∖{0}$\{ {a}_1, \ldots ,{a}_\nu \} \subset \mathbb{D}\backslash \{ 0\} $ are distinct points inside the unit disk. In the scaling limit when n/N = 1 and n → ∞ we obtain the strong asymptotics of the polynomial pn(z). We show that the support of the roots converges to what we call the "multiple Szegő curve," a certain connected curve having ν + 1 components in its complement. We apply the nonlinear steepest descent method [9,10] on the matrix Riemann‐Hilbert problem of size (ν + 1) × (ν + 1) posed in [22]. © 2023 Wiley Periodicals, LLC. [ABSTRACT FROM AUTHOR]

This paper is concerned with a generalization of the atom-bond sum-connectivity (ABS) index, devised recently in [A. Ali, B. Furtula, I. Redžepovíc, I. Gutman, Atom-bond sum-connectivity index, J. Math. Chem., 60 (2022), 2081--2093]. For a connected graph G with an order greater than 2, the general atom-bond sum-connectivity index is represented as ABS γ(G) and is defined as the sum of the quantities (1-2(dx +dy)-1)γ over all edges xy of the graph G, where dx and dy represent the degrees of the vertices x and y of G, respectively, and γ is any real number. For -10 ≤ γ ≤ 10, the significance of ABSγ is examined on the data set of octane isomers for predicting six selected physicochemical properties of the mentioned compounds; promising results are obtained when the approximated value of γ belongs to the set {-3, 1, 3}. The effect of the addition of an edge between two non-adjacent vertices of a graph under ABS γ is also investigated. Moreover, the graphs possessing the maximum value of ABS γ, with γ > 0, are characterized from the set of all connected graphs of a fixed order and a fixed (i) vertex connectivity not greater than a given number or (ii) matching number. [ABSTRACT FROM AUTHOR]

With the advancement of combat equipment technology and combat concepts, new requirements have been put forward for air defense operations during a group target attack. To achieve high-efficiency and lowloss defensive operations, a reasonable air defense weapon assignment strategy is a key step. In this paper, a multi-objective and multi-constraints weapon target assignment (WTA) model is established that aims to minimize the defensive resource loss, minimize total weapon consumption, and minimize the target residual effectiveness. An optimization framework of air defense weapon mission scheduling based on the multiobjective artificial bee colony (MOABC) algorithm is proposed. The solution for point-to-point saturated attack targets at different operational scales is achieved by encoding the nectar with real numbers. Simulations are performed for an imagined air defense scenario, where air defense weapons are saturated. The non-dominated solution sets are obtained by the MOABC algorithm to meet the operational demand. In the case where there are more weapons than targets, more diverse assignment schemes can be selected. According to the inverse generation distance (IGD) index, the convergence and diversity for the solutions of the non-dominated sorting genetic algorithm III (NSGA-III) algorithm and the MOABC algorithm are compared and analyzed. The results prove that the MOABC algorithm has better convergence and the solutions are more evenly distributed among the solution space. [ABSTRACT FROM AUTHOR]

At present, China is actively piloting the construction of "zero-waste city". Aiming at the decision-making problem in the selection of pilot cities for the construction of "zero-waste city", this paper proposed a multi-source heterogeneous multi-attribute decision-making model based on a two-tuple grey relational bidirectional projection (TT-GRBP) method. Firstly, based on the theoretical support of sustainable development, a comprehensive evaluation index system for "zero-waste city" was constructed with four aspects, i.e. economic level, environmental pollution, resource consumption, and waste utilization which includes 13 secondary indexes. Secondly, considering the index values have the characteristics of multi-source heterogeneity (the real numbers, fuzzy language variables, and interval numbers coexist simultaneously), the multi-source heterogeneous data were uniformly converted into 2 tuples. Considering the variability and correlation of indexes, a two-tuple criteria importance though intercriteria correlation (TT-CRITIC) method was proposed to determine the index weights. Thirdly, combining the grey relational analysis with projection ideas, a TT-GRBP method was proposed to make the selection decision-making of "zero-waste city". Finally, an empirical analysis was carried out on the "zero-waste city" selection for the 39 cities of Yangtze River Economic Zone in China. According to the ranking results, some suggestions and reform directions for the construction and development of "zero-waste city" were provided from economic, environmental, and social aspects. [ABSTRACT FROM AUTHOR]

We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale on a computational tree, which leads to explicit and interpretable bounds with nonlinear terms controlled explicitly rather than by big-O terms. Two probability parameters strengthen the precision-awareness of our bounds: one parameter controls the first order terms in the summation error, while the second one is designed for controlling higher order terms in low precision or extreme-scale problem dimensions. Our systematic approach yields new deterministic and probabilistic error bounds for three classes of mono-precision algorithms: general summation, shifted general summation, and compensated (sequential) summation. Extension of our systematic error analysis to mixed-precision summation algorithms that allow any number of precisions yields the first probabilistic bounds for the mixed-precision FABsum algorithm. Numerical experiments illustrate that the probabilistic bounds are accurate, and that among the three classes of mono-precision algorithms, compensated summation is generally the most accurate. As for mixed precision algorithms, our recommendation is to minimize the magnitude of intermediate partial sums relative to the precision in which they are computed. [ABSTRACT FROM AUTHOR]

This paper introduces some new Hermite-Hadamard type integral inequalities for s-convex function via Caputo-Fabrizio fractional integral. At the end, applicability of obtained results to special means of real numbers has been discussed. [ABSTRACT FROM AUTHOR]

In many emerging scientific applications, the input of random initial time is often required for predicting and controlling the dynamics of real-order systems that include or does not include time-delays. In this work, we provide a new design for a class of random initial-time non-autonomous linear Caputo-type real-order time-delay systems that involves constant discrete delays. We first introduce a new comparison principle and a linear comparison lemma for Caputo derivative. By using the idea of comparison methodology and generalized Laplace transform, we develop some new elementary asymptotic theories and establish order-dependent and delay-independent conditions that give convergence of nontrivial solutions to such a class of system. One of the major challenges encountered in the investigation is discovering some simpler classes of autonomous linear real-order systems in bounding the coefficient matrices of a designed class of systems above by constant matrices in our assumptions. These assumptions including newly introduced matrices Λ , Θ k and G provide some fundamental key tools for an effective asymptotic analysis of newly designed class of systems. A typical theorem that we develop says that if all the diagonal entries of Λ become negative and every root of the characteristic equation associated with the matrix Λ + ∑ k = 1 m Θ k G lies in the open left-half complex plane, then it is possible to predict the limiting behavior of the designed class of systems. For a practical application of interest, we show how to synchronize the possible chaotic dynamics of the coupled nonlinear Duffing oscillators if the system starts at a negative initial time through some proposed results. [ABSTRACT FROM AUTHOR]

A topological index is a real number that is obtained from a chemical graph's structure. Determining the physiochemical and biological characteristics of a variety of medications is useful since it more accurately represents the theoretical characteristics of organic molecules. This is accomplished using degree-based topological indices. The QSPR research has improved the structural understanding of the physiochemical properties of fungicides. Thirteen fungicides are examined for some of their physiochemical properties, and a QSPR model is built using nine of the drugs' topological indices. Here, we examine the degree to which the topological indices and physiochemical attributes are connected. To do this, we create networks connecting each of the topological indices to the properties of fungicides and computationally construct topological indices of the drugs mentioned above. According to this QSPR model, the melting point, boiling point, flash point, complexity, surface tension, etc. of fungicides are strongly connected. It was discovered that the topological indices (TIs) applied to the fungicides more accurately represent their theoretical features and show a strong correlation with their physical attributes. [ABSTRACT FROM AUTHOR]

The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the architecture and building blocks of complex-valued neural networks. In this paper, we generalize models by considering other types of hypercomplex numbers of the second order: dual and double numbers. We developed basic operators for these algebras, such as convolution, activation functions, and batch normalization, and rebuilt several real-valued networks to use them with these new algebras. We developed a general methodology for dual and double-valued gradient calculations based on Wirtinger derivatives for complex-valued functions. For classical computer vision (CIFAR-10, CIFAR-100, SVHN) and signal processing (G2Net, MusicNet) classification problems, our benchmarks show that the transition to the hypercomplex domain can be helpful in reaching higher values of metrics, compared to the original real-valued models. [ABSTRACT FROM AUTHOR]

For any real number α ∈ [ 0 , 1 ] , by the A α -matrix of a graph G we mean the matrix A α (G) = α D (G) + (1 − α) A (G) , where A (G) and D (G) are the adjacency matrix and the diagonal matrix of vertex degrees of G , respectively. The largest eigenvalue of A α (G) is called the A α -index of G. In this paper, we settle the problem of characterizing graphs which attain the maximum A α -index over G (n , n + k) , the class of graphs with n vertices and n + k edges, for − 1 ≤ k ≤ n − 3 and 1 2 ≤ α < 1. The following result is obtained: for − 1 ≤ k ≤ n − 3 , when 1 2 ≤ α < 1 , H n , k is the unique graph in G (n , n + k) that maximizes the A α -index, except when (n , k) = (4 , − 1) , (n , 2) or (7 , 3) and α = 1 2 , or (n , k) = (5 , 1) and α ∈ [ 1 2 , 35 − 409 24 ]. When (n , k , α) = (4 , − 1 , 1 2) , the optimal graphs are H 4 , − 1 and K 3 ∪ K 1 ; when (n , k , α) = (n , 2 , 1 2) , the optimal graphs are H n , 2 and G n , 2 ; when (n , k , α) = (5 , 1 , 35 − 409 24) , the optimal graphs are H 5 , 1 and K 4 ∪ K 1 ; when (n , k , α) = (7 , 3 , 1 2) , the optimal graphs are H 7 , 3 and K 5 ∪ 2 K 1 ; when (n , k) = (5 , 1) and 1 2 ≤ α < 35 − 409 24 , K 4 ∪ K 1 is the unique graph that maximizes the A α -index. Our work completes the corresponding work of Chang and Tam (2010) and Zhai et al. (2022) for the special case α = 1 2. As a by-product, we provide a new proof for the known result that for any positive integer m and any real number α ∈ [ 1 2 , 1) , if (m , α) ≠ (3 , 1 2) , then a graph maximizes the A α -index over all graphs with m edges if and only if it is the union of K 1 , m with a (possibly empty) null graph; a graph maximizes the A 1 2 -index over all graphs with three edges if and only if it is the union of K 1 , 3 or K 3 with a (possibly empty) null graph. Some open questions are also posed. [ABSTRACT FROM AUTHOR]

The Analytic hierarchy process (AHP) is a widely used multi-criteria decision theory, and most AHP relies on the judgments of experts to derive priority scales. However, the judgments of experts may be subjective. Using machine learning algorithms for decision-making can be more objective, but machine learning algorithms are strongly related to the collected data and not being flexible enough. This paper tries to combine experts' judgments with algorithmic judgments to improve the bias of experts' judgments while still making decision-making flexible. In this paper, the authors introduce the ordered pair of normalized real numbers (OPNs) into the AHP method for the first time and propose the fuzzy analytic hierarchy process with the OPNs (OFAHP). The OFAHP uses OPNs to combine experts' judgments with those of machine learning algorithms and then make decisions by OPNs. Experiments on real data sets show that the proposed method can get reasonable decision results. Moreover, when the experts' judgments are wrong or invalid, the judgments given by the machine learning algorithm can correct the experts' judgments to obtain a reasonable decision-making result. [ABSTRACT FROM AUTHOR]

In this paper, we study the following two-dimesional system of difference equations ... where the parameters a;b; c;d and the initial values x i; y i, i 2 f1;2;3;4;5;6g, are real numbers. We show that some subclasses of nonlinear two-dimensional system of difference equations are solvable in closed form. We also describe the forbidden set of solutions of the system of differ- ence equations. Some numerical examples are given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

Zilber's Exponential-Algebraic Closedness Conjecture states that algebraic varieties in C n × (C ×) n intersect the graph of complex exponentiation, unless that contradicts the algebraic and transcendence properties of exp . We establish a case of the conjecture, showing that it holds for varieties which split as the product of a linear subspace of the additive group and an algebraic subvariety of the multiplicative group. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required the linear subspace to either be defined over a generic subfield of the real numbers, or it to be any subspace defined over the reals assuming unproved conjectures from Diophantine geometry and transcendence theory. The proofs use the theory of amoebas and tropical geometry. [ABSTRACT FROM AUTHOR]

It is clear that I z SB k sb i has degree Graph HT ht and thus has the representation Graph HT ht , where each Graph HT ht is a positive integer. Compute Graph HT ht . Prove that Graph HT ht 1260. [Extracted from the article]

As a continuation of previous studies, we give some results about the neutrosophic integers theory. We first stated that the neutrosophic real numbers are not closed according to the division operation. Later, we gave divisibility properties of neutrosophic integers. We have given properties such as the greatest common divisor for two neutrosophic integers being positive and unique. Then, we gave the Euclid's Theorem, Bezout's Theorem for neutrosophic ingers set Z[I]. It is known that these concepts are important for number theory in integers set Z. Finally, it is defined the least common multiple for neutrosophic integers. Finally, a theorem is given which enables one to easily find the least common multiple of neutrosophic integers and after a conclusion about the sign of the product of two neutrosophic integers, a theorem is given that shows the relationship of between the greatest common divisor with the least common multiple. [ABSTRACT FROM AUTHOR]

Let (M , g , f) be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric + ∇ 2 ⁡ f = λ ⁢ g , where λ is a positive real number. We prove that if M has constant scalar curvature S = 2 ⁢ λ , it must be a quotient of 핊 2 × ℝ 2 . Together with the known results, this implies that a four-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton ℝ 4 , 핊 2 × ℝ 2 or 핊 3 × ℝ . [ABSTRACT FROM AUTHOR]

In this article, we investigate the Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group of the following form: { − (a + b ∫ Ω | ∇ H u | p d ξ) Δ H , p u − μ ϕ | u | p − 2 u = λ | u | q − 2 u + | u | Q ∗ − 2 u in Ω , − Δ H ϕ = | u | p in Ω , u = ϕ = 0 on ∂ Ω , where a , b are positive real numbers, Ω ⊂ H N is a bounded region with smooth boundary, 1 < p < Q , Q = 2 N + 2 is the homogeneous dimension of the Heisenberg group H N , Q ∗ = p Q Q − p , q ∈ (2 p , Q ∗) and Δ H , p u = div (| ∇ H u | p − 2 ∇ H u) is the p -horizontal Laplacian. Under some appropriate conditions for the parameters μ and λ , we establish existence and multiplicity results for the system above. To some extent, we generalize the results of An and Liu (Israel J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022). [ABSTRACT FROM AUTHOR]

Gijswijt's Sequence (A090822) For this sequence, it will be helpful to remember that chemists do not write HT ht for water; they write HT ht . So the sequence with a single term 99, say, can be written as HT ht , where I Y i is the number 99, and it has curling number 1. If you think of an animal with its head looking to the left, with a very curly tail, then I X i represents the head and body of the animal, and HT ht represents the curls in its tail. The case of an HT ht grid seems even harder. [Extracted from the article]

The analytic properties of fuzzy sedenion numbers and sedenion valued series are studied. For this aim, we first introduce fuzzy sedenion numbers. Then we present infimum, supremum, distance, limit of sequences and other analytical properties of fuzzy sedenion numbers. We also define interval sedenion numbers and fuzzy sedenion valued series. Finally, we obtain comparison criteria of the fuzzy sedenion valued series. [ABSTRACT FROM AUTHOR]