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We show that completeness at higher levels of the theory of the reals is a robust notion (under changing the signature and bounding the domain of the quantifiers). This mends recognized gaps in the hierarchy, and leads to stronger completeness results for various computational problems. We exhibit several families of complete problems which can be used for future completeness results in the real hierarchy. As an application we sharpen some results by Bürgisser and Cucker on the complexity of properties of semialgebraic sets, including the Hausdorff distance problem also studied by Jungeblut, Kleist, and Miltzow. [ABSTRACT FROM AUTHOR]

Kleene's computability theory based on the S1–S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental distinction in Kleene's framework is between normal and non-normal functionals where the former compute the associated Kleene quantifier ∃ n and the latter do not. Historically, the focus was on normal functionals, but recently new non-normal functionals have been studied based on well-known theorems, the weakest among which seems to be the uncountability of the reals. These new non-normal functionals are fundamentally different from historical examples like Tait's fan functional: the latter is computable from ∃ 2 , while the former are computable in ∃ 3 but not in weaker oracles. Of course, there is a great divide or abyss separating ∃ 2 and ∃ 3 and we identify slight variations of our new non-normal functionals that are again computable in ∃ 2 , i.e. fall on different sides of this abyss. Our examples are based on mainstream mathematical notions, like quasi-continuity, Baire classes, bounded variation, and semi-continuity from real analysis. [ABSTRACT FROM AUTHOR]

Although different social crises may eventually favor undemocratic and authoritarian forms of governance, at some point, such antidemocratic practices require the support of a significant part of the population to be implemented. The present research investigates how and whether the COVID‐19 pandemic might have favoured greater support for antidemocratic governmental practices, on the premise of regaining control and security. Using data from 17 countries (N = 4364) and national‐level indicators (i.e., real number of contagions and deaths, and sociopolitical indicators), we test how the risk of contagion and death from COVID‐19, along with personal orientations (i.e., social dominance orientation [SDO], right‐wing authoritarianism [RWA], and perceived anomie) motivate authoritarian and antidemocratic practices. Results from multilevel models indicate that risk perception and perceptions of political instability predict a wish for stronger leadership, agreement with martial law, and support for a controlling government especially when SDO and RWA are high, while more egalitarian and less conservative people agree less with these authoritarian measures in spite of the levels of risk perception. We discuss the implications for these findings for future research on similar but also dissimilar external events (natural disasters, war, or terror incidents) and the consequences for societies with higher authoritarian tendencies. [ABSTRACT FROM AUTHOR]

Denote by τ k (n) , ω (n) and μ 2 (n) the number of representations of n as a product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let [ t ] be the integral part of real number t. For f = ω , 2 ω , μ 2 , τ k , we prove that ∑ n ≤ x f x n = x ∑ d ≥ 1 f (d) d (d + 1) + O (x f + ) for x → ∞ , where ω = 5 3 1 1 0 , 2 ω = 9 1 9 , μ 2 = 2 5 , τ k = 5 k − 1 1 0 k − 1 and > 0 is an arbitrarily small positive number. These improve the corresponding results of Bordellès. [ABSTRACT FROM AUTHOR]

Based on an extension of Riemann sums, Moore and Yang have defined an integral notion for the context of continuous inclusion monotonic interval functions in which the limits of integration are real numbers. This integral notion generalizes the usual one for real-valued functions based on Riemann sums. In this paper we extend this approach by considering intervals as limits of integration and abolishing the inclusion monotonic restriction of the integrable interval functions. Also, such a new integration notion is used to define interval probability density functions and use it in interval probability distribution functions. [ABSTRACT FROM AUTHOR]

Trapezoidal positive/negative fuzzy numbers have no single definition; instead, various authors define them in relation to different concepts. This means that arithmetic operations for trapezoidal fuzzy numbers also differ. For the operations of addition, subtraction, and scalar multiplication, there are not many differences; for multiplication, however, there are many differences. In general, multiplication is divided into various cases. For the inverse operation, there is not much to define; in general, for any trapezoidal fuzzy number u ~ ,   u ~ ⊗ 1 u ~ = i ~ = (1 , 1 , 0 , 0) does not necessarily apply. As a result of the different arithmetic operations for multiplication and division employed by various authors, several researchers have tackled the same problem and reached different solutions, meaning that the application will also produce different results. To date, many authors have proposed various alternatives for the algebra of the trapezoidal fuzzy number. In this paper, using the parametric form approach to trapezoidal fuzzy numbers, an alternative to multiplication with only one formula is constructed for various cases. Furthermore, based on the definition of multiplication for any trapezoidal fuzzy number, u ~ is constructed 1 u ~ so that u ~ ⊗ 1 u ~ = i ~ = (1 , 1 , 0 , 0) . Based on these conditions, we show that various properties that apply to real numbers also apply to any trapezoidal fuzzy number. Furthermore, we modify the elementary row operational steps for the trapezoidal fuzzy number matrix, which can be used to determine the inverse of a trapezoidal fuzzy number matrix with the order m × m . We also give the steps and examples necessary to determine the general inverse for a trapezoidal fuzzy number matrix of the order m × n with m   ≠ n . This ability to easily determine the inverse and general inverse of a trapezoidal fuzzy number matrix has a number of applications, such as solving fully trapezoidal fuzzy number linear systems and fuzzy transportation problems, especially in applications in fields outside of mathematics; for example, the application of triangular fuzzy numbers in medical problems is a topic currently receiving a significant amount of attention. [ABSTRACT FROM AUTHOR]

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui–Hà–Jayanthan–Thomas [Collect. Math. 72 (2021), pp. 605–614]. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers a and b, we consider the set Res(a,b) of possible values of the resurgence of I+J where I and J are ideals in disjoint sets of variables having resurgence a and b, respectively. Some questions and partial results about Res(a,b) are discussed. [ABSTRACT FROM AUTHOR]

Let (t (m)) m ≥ 0 be Thue-Morse sequence and b > 2 be an integer. In this paper, we prove that the real numbers 1 , ∑ m = 0 ∞ t (m 2) b m + 1 and ∑ m = 0 ∞ t (m 3) b m + 1 are linearly independent over Q . [ABSTRACT FROM AUTHOR]

Let be an array of martingale difference random vectors and be an array of m × d matrices of real numbers. In this paper, the Marcinkiewicz–Zygmund type weak law of large numbers for maximal weighted sums of martingale difference random vectors is obtained with not necessarily finite p-th (1 < p < 2) moments. Moreover, the complete convergence and strong law of large numbers are established under some mild conditions. An application to multivariate simple linear regression model is also provided. [ABSTRACT FROM AUTHOR]

We consider the Vilenkin orthonormal system on a Vilenkin group G and the Vilenkin-Fourier coefficients f̂(n), n ∈ N, of functions f∈ LP(G) for some 1 < p ⩽ 2 We obtain certain sufficient conditions for the finiteness of the series sum n = 1∑∞ an |f̂(n)|r where {an} is a given sequence of positive real numbers satisfying a mild and 0 < r < 2 We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of f and give multiplicative analogue of some results due to Móricz (2010), Móricz and Veres (2011), Golubov and Volosivets (2012), and Volosivets and Kuznetsova (2020). [ABSTRACT FROM AUTHOR]

TT□C is a generic family of effectful, extensional type theories with a forcing interpretation parameterized by modalities. This paper identifies a subclass of TT□C theories that internally realizes continuity principles through stateful computations, such as reference cells. The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. In this paper, the modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled internally in the theory. [ABSTRACT FROM AUTHOR]

We propose a simple imperative programming language, ERC, that features arbitrary real numbers as primitive data type, exactly. Equipped with a denotational semantics, ERC provides a formal programming language-theoretic foundation to the algorithmic processing of real numbers. In order to capture multi-valuedness, which is well-known to be essential to real number computation, we use a Plotkin powerdomain and make our programming language semantics computable and complete: all and only real functions computable in computable analysis can be realized in ERC. The base programming language supports real arithmetic as well as implicit limits; expansions support additional primitive operations (such as a user-defined exponential function). By restricting integers to Presburger arithmetic and real coercion to the ‘precision’ embedding Z ∋ p 7→ 2p ∈ R, we arrive at a first-order theory which we prove to be decidable and model-complete. Based on said logic as specification language for preconditions and postconditions, we extend Hoare logic to a sound (w.r.t. the denotational semantics) and expressive system for deriving correct total correctness specifications. Various examples demonstrate the practicality and convenience of our language and the extended Hoare logic. [ABSTRACT FROM AUTHOR]

In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: (− Δ) α / 2 u (x) = f (u (x) , v (x)) , x ∈ R n , (− Δ) α / 2 v (x) = g (u (x) , v (x)) , x ∈ R n. Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang ("A direct method of moving spheres on fractional order equations," J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu ("Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space," Potential Anal., vol. 46, pp. 569–588, 2017). [ABSTRACT FROM AUTHOR]

In this paper, we focus on obtaining the closed-form solution for the following bidimensional system of higher-order rational difference equations: u(1) n+1 = 1 3 - 2u(2) n-m, u(2) n+1 = 1 3 - 2u(1) n-m, n,m N0, where the initial values u(1) -j and u(2) -j, j {0, 1, ...,m} are real numbers not equal to 3/2. We show that the solutions of this system are associated with Mersenne numbers and/or Mersenne-Lucas numbers. It is shown that the global stability of positive solutions of this system holds. Finally, we provide numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]

We define and study the notion of the golden ratio of order k = 0, denoted ϕk, as a generalized form of the golden ratio ϕ for any real number k = 0. We show that similar to the special case of ϕ and its conjugate ϕ, ϕk and ϕk are the two distinct roots of a quadratic polynomial for any fixed real k = 0. We express some numerical and algebraic properties of ϕk and ϕk and write their relations to ϕ and ϕ, respectively, with some examples for some special values of k. In particular, it is shown that ϕk = ϕ and ϕk = ϕ if and only if k = 0. We show that Z[(k + 1)ϕk] is a subring of the ring Z[ϕ] for any nonnegative integer k. We will define the golden rectangle of order k (or k-golden rectangle for short) with a class of examples for all k = 0. We also discuss some cases of two Fibonacci numbers in connection to the golden ratio. We will show that the ratio of height to width of the pages of the Gutenberg Bible is the golden ratio of order k ϕ= 0. Actually, some erroneous ideas and examples of disputed observations related to the golden ratio are good reasons to apply ϕk to improve the measurements regarding ?ϕ for some k ϕ= 0. Finally, we end the paper by posing a question related to the Penrose tiling and quasicrystals in connection to the golden ratio of order k > 0. [ABSTRACT FROM AUTHOR]

Context. In many areas of science and technology, the numerical solution of problems is not enough for the further development of the implementation of the obtained results. Among the existing information visualization approaches, the one that allows you to effectively reveal unstructured actionable ideas, generalize or simplify the analysis of the received data is chosen. The results of visualization of generalized structural diagrams based on the textual description of the algorithm clearly reflect the interaction of its parts, which is important at the system engineering stage of computer design. Objective of the study is the analysis and software implementation of structure visualization using the example of discrete harmonic transformation calculators obtained as a result of the synthesis of an algorithm based on cyclic convolutions with the possibility of extending the structure visualization to other computational algorithms. Method. The generalized scheme of the synthesis of algorithms of fast harmonic transformations in the form of a set of cyclic convolution operations on the combined sequences of input data and the coefficients of the harmonic transformation function with their visualization in the form of a generalized structural diagram of the calculator. The results. The result of the work is a software implementation of the visualization of generalized structural diagrams for the synthesized algorithms of cosine and Hartley transformations, which visually reflect the interaction of the main blocks of the computer. The software implementation of computer structure visualization is made in TypeScript using the Phaser 3 framework. Conclusions. The work considers and analyzes the developed software implementation of visualization of the general structure of the calculator for fast algorithms of discrete harmonic transformations in the domain of real numbers, obtained as a result of the synthesis of the algorithm based on cyclic convolutions. The results of visualization of variants of structural schemes of computers clearly and clearly reflect the interaction of its parts and allow to evaluate one or another variant of the computing algorithm in the design process [ABSTRACT FROM AUTHOR]

When the adaptive filter length is increased, the calculation complexity increases rapidly because the relationship between the calculation and the adaptive filter length N contains a power function with no secondary path identification algorithm. Under the basic premise of unreduced noise reduction, herein, a simplified frequency-domain feedback active noise control algorithm is proposed. To reduce the computation complexity, the total delay is adopted as the estimated secondary path; the filtered reference signal is produced in the frequency domain by using multiplication to replace convolution calculation in the time domain and then updating the adaptive filter coefficients in the frequency domain. Therefore, the computational complexity has a logarithmic function with the increased adaptive filter length in the proposed algorithm. If the adaptive filter length is 512, the existing WSMANC algorithm's calculation is 271,360 real number multiplications, while that of the proposed algorithm is only 38,912 real number multiplications. To verify the proposed algorithm's stability, convergence speed, and noise reduction, the single-frequency noise, narrowband white noise, and narrowband pink noise, respectively, are used as the primary noise types in the simulations. The results show that (1) the proposed SFDFBANC algorithm can obtain similar noise reduction performance to existing algorithm, (2) the convergence rate is faster than existing algorithm, and (3) if the adaptive filter length is more than 64, the proposed algorithm exhibits a lower computational complexity. [ABSTRACT FROM AUTHOR]

Ostrowski's iterative method is a classical method for solving systems of nonlinear equations. However, it is not stable enough. In order to obtain a more stable Ostrowski-type method, this paper presented a new family of fourth-order single-parameter Ostrowski-type methods for solving nonlinear systems. As a generalization of the Ostrowski's methods, the Ostrowski's methods are a special case of the new family. It was proved that the order of convergence of the new iterative family was always fourth-order when the parameters take any real number. Finally, the dynamical behavior of the family was briefly analyzed using real dynamical tools. The new iterative method can be applied to solve a wide range of nonlinear equations, and it was used in numerical experiments to solve the Hammerstein equation, boundary value problem, and nonlinear system. These numerical results supported the theoretical results. [ABSTRACT FROM AUTHOR]

Let N denote a sufficiently large real number. In this paper, we prove that for 1 < c < 104349/77419, c ≠ 43, for almost all real numbers T ∈ (N, 2N] (in the sense of Lebesgue measure), the Diophantine inequality |p1c + p2c - T| < T-9/10c (104349/77419 -c) is solvable in primes p1, p2. In addition, it is proved that the Diophantine inequality |p1c + p2c + p3c + p4c - N| < N-9/10c(104349/77419 -c) is solvable in primes p1, p2, p3, p4. This result constitutes a refinement upon that of Li and Cai. [ABSTRACT FROM AUTHOR]

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials. [ABSTRACT FROM AUTHOR]

The general sum-connectivity and Platt indices of a graph G are defined by SCa(G) = Σxy∈E(G)(dx +dy)a and Pla(G) = Σxy∈E(G)(dx +dy -2)a, respectively, where a is a real number, E(G) indicates the edge set of G, and dv indicates the degree of a vertex v of G. The cyclomatic number of G is the least number of edges required to be deleted from G to make it acyclic. If the maximum degree of G is less than 5, then G is referred to as a molecular graph. In this paper, the problem of determining graphs possessing the minimum values of the indices SCa and Pla among all connected (molecular) graphs of order n and cyclomatic number t is solved for 0 < a < 1 and n ≥ 2(t - 1) ≥ 2 with n ≤ 4. It is proved that the difference between the maximum and minimum degrees of the aforementioned extremal graphs is at most 1. [ABSTRACT FROM AUTHOR]

We study some properties of spaces obtained by interpolation of the Sobolev spaces Wk,1(Rd) and Wl,∞(Rd), where l and r are nonnegative integers, and d≥2. We are concerned with the standard real and complex methods of interpolation. In the case of the real method, an old result of De Vore and Scherer (1979) gives that (Wl,1 (Rd),Wl,∞(Rd))θ,pθ​​=Wl,pθ​(Rd), where θ∈(0,1) and 1/pθ​=1−θ. We complement this result by considering the case l ≠ r. We prove that, when l ≠ r, (Wl,1(Rd),Wr,∞(Rd))θ,q​=Bqσ,q, ​(Rd),(⋆) where σ:=(1−θ)l+θr and 1/q=1−θ, if and only if l−r∈R∖[1,d]. Also, we prove a similar fact when Wl,¹ is replaced in (⋆) by a space Ws,p where s ≠ r is a real number and p∈(1,∞). Several other problems like the boundedness of the Riesz transforms on interpolation spaces are also considered. In the case of the complex method, it was proved by M. Milman (1983) that, for any 1l,1(Rd),Wl,p(Rd))θ​=Wl,pθ​(Rd),(⋆⋆) where 1/pθ​=(1−θ)+θ/p. We show by simple arguments that (⋆⋆) fails when p=∞ and l≥1, answering a question of P. W. Jones (1984). As an immediate consequence of these arguments, we show that certain closed subspaces of (C(Td))N (with N∈N∗) that are described by Fourier multipliers are not complemented in (C(Td))N. [ABSTRACT FROM AUTHOR]

We consider a very weak chain condition for a poset, that is the absence of subsets which are order isomorphic to the set of real numbers in their natural ordering; we study generalised radical groups in which this finiteness condition is set on the poset of subgroups which do not have certain properties which are generalizations of normality. This completes many previous results which considered (apparently) stronger chain conditions. [ABSTRACT FROM AUTHOR]

We parameterize countable locally standard Boolean measure algebras by pairs of a Steinitz number and a real number greater or equal to 1. This is an analog of the theorems of Dixmier and Baranov. [ABSTRACT FROM AUTHOR]

Ptolemy's inequality is a classic relationship between the distances among four points in Euclidean space. Another relationship between six distances is the 4-point condition, an inequality satisfied by the lengths of the six paths that join any four points of a metric (or weighted) tree. The 4-point condition also characterizes when a finite metric space can be embedded in such a tree. The curious observer might realize that these inequalities have similar forms: if one replaces addition and multiplication in Ptolemy's inequality with maximum and addition, respectively, one obtains the 4-point condition. We show that this similarity is more than a coincidence. We identify a family of Ptolemaic inequalities in CAT-spaces parametrized by a real number and show that a certain limit involving these inequalities, as the parameter goes to negative infinity, yields the 4-point condition, giving an elementary proof that the latter is the tropicalization of Ptolemy's inequality. [ABSTRACT FROM AUTHOR]

Let p be a configuration of n points in R d for some n and some d ≥ 2 . Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that begins and ends at the same point. A path or loop, as a sequence of edges, also has a Euclidean length, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing p given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths. In this paper, we study the question of when p will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has already been used for the simpler problem of reconstruction using unlabeled edge lengths. This paper also provides a complete proof that this process must work in that edge-setting when given a sufficiently rich set of edge measurements and assuming that p is generic. [ABSTRACT FROM AUTHOR]

Chipman contended, in stark contrast to the conventional view, that, utility is not a real number but a vector, and that it is inherently lexicographic in nature. On the other hand, in recent years continuous multi-utility representations of a preorder on a topological space, which proved to be the best kind of continuous representation, have been deeply studied. In this paper, we first state a general result, which guarantees, for every preordered topological space, the existence of a lexicographic order-embedding of the Chipman type. Then, we combine the Chipman approach and the continuous multi-utility approach, by stating a theorem that guarantees, under certain general conditions, the coexistence of these two kinds of continuous representations. [ABSTRACT FROM AUTHOR]

Recently, the growth of the products of consecutive partial quotients a i (x) in the continued fraction expansion of a real number x was studied in connections with improvements to Dirichlet's theorem. In this paper, for a non-decreasing positive measurable function F (x 1 , ... , x m) and a function ϕ : ℕ → ℝ > 0 , we consider the set ℰ F (ϕ) = { x ∈ [ 0 , 1 ] : F (a n (x) , ... , a n + m − 1 (x)) ≥ ϕ (n)  for infinitely many  n ∈ ℕ } , and obtain its Lebesgue measure ℒ (ℰ F (ϕ)). As an application of our result, we reprove a theorem of Bakhtawar–Hussain–Kleinbock–Wang. We also consider the case when F (x 1 , ... , x m) = x 1 + ⋯ + x m . [ABSTRACT FROM AUTHOR]

The emergence of quantum computers is having an impact on almost every field of engineering. Electrical drives, in particular, can benefit from the power of quantum computing. This paper presents a quantum version of the pulse-width modulation (PWM) algorithm, which is used widely to control electrical motors. The proposed algorithm is implemented using a real number comparator, which is novel to the authors' knowledge. Unlike other quantum comparators in the literature, which provide quantum versions of binary-based comparators, the suggested comparator compares real numbers ranging from 0 to 100%, making it more usable in practice for many industrial applications, particularly control. When applied to DC motor speed control, the quantum PWM controller achieved similar speed precision to the classical version, but with lower energy consumption and less switching. It could therefore significantly increase the lifetime of power devices while achieving similar control performance at lower energy consumption. [ABSTRACT FROM AUTHOR]

For any real numbers x ∈ [ 0 , 1 ] and β > 1 , let r n (x , β) be the maximal length of consecutive 0's in the first n digits of the beta-expansion of x in base β. The run-length function r n (x , β) has been well studied for a fixed base β > 1 or a fixed real number x = 1. In this paper, we prove that for any x ∈ (0 , 1) , the set D x = { β > 1 : lim n → ∞ ⁡ r n (x , β) log β ⁡ n = 1 } is of full Lebesgue measure in (1 , + ∞). When the exceptional set is considered, we prove that for any real numbers 0 ≤ a ≤ b ≤ + ∞ , the set D x (a , b) = { β > 1 : lim inf n → ∞ r n (x , β) log β ⁡ n = a , lim sup n → ∞ r n (x , β) log β ⁡ n = b } is of full Hausdorff dimension. We also determine the Hausdorff dimension of the set F x (c , d) = { β > 1 : lim inf n → ∞ r n (x , β) n = c , lim sup n → ∞ r n (x , β) n = d } for any real numbers 0 ≤ c ≤ d ≤ 1. [ABSTRACT FROM AUTHOR]

Where does arithmetic come from, and why are addition and multiplication its fundamental operations? Although we know that arithmetic is true, no explanation that meets standards of scientific rigor is available from philosophy, mathematical logic, or the cognitive sciences. We propose a new approach based on the assumption that arithmetic has a biological origin: Many examples of adaptive behavior such as spatial navigation suggest that organisms can perform arithmetic-like operations on represented magnitudes. If so, these operations—nonsymbolic precursors of addition and multiplication—might be optimal due to evolution and thus identifiable according to an appropriate criterion. We frame this as a metamathematical question, and using an order-theoretic criterion, prove that four qualitative conditions—monotonicity, convexity, continuity, and isomorphism—are sufficient to identify addition and multiplication over the real numbers uniquely from the uncountably infinite class of possible operations. Our results show that numbers and algebraic structure emerge from purely qualitative conditions, and as a construction of arithmetic, provide a rigorous explanation for why addition and multiplication are its fundamental operations. We argue that these conditions are preverbal psychological intuitions or principles of perceptual organization that are biologically based and shape how humans and nonhumans alike perceive the world. This is a Kantian view and suggests that arithmetic need not be regarded as an immutable truth of the universe but rather as a natural consequence of our perception. Algebraic structure may be inherent in the representations of the world formed by our perceptual system. [ABSTRACT FROM AUTHOR]

In this paper, we study a special algebraic inequality containing a parameter, the sum of reciprocals and the product of positive real numbers whose sum is 1. Using a new optimization argument the best values of the parameter are determined. In the case of three numbers the algebraic inequality has some interesting geometric applications involving a generalization of Euler's inequality about the ratio of radii of circumscribed and inscribed circles of a triangle. [ABSTRACT FROM AUTHOR]

Let G be a family of subsets of an n-element set. The family G is called non-trivial 3-wise intersecting if the intersection of any three subsets in G is non-empty, but the intersection of all subsets is empty. For a real number p ∈ (0 , 1) we define the measure of the family by the sum of p | G | (1 - p) n - | G | over all G ∈ G . We determine the maximum measure of non-trivial 3-wise intersecting families. We also discuss the uniqueness and stability of the corresponding optimal structure. These results are obtained by solving linear programming problems. [ABSTRACT FROM AUTHOR]

In this paper, by using both the Hölder, power‐mean integral inequalities and Hölder‐İşcan, improved power‐mean integral inequalities that give approach better than Hölder and power‐mean inequalities respectively, some new Ostrowski type inequalities are derived for multiplicatively P‐function. The results obtained in both cases are compared and the results obtained using Hölder‐İşcan and improved power‐mean integral inequalities are proven to be better. Some applications to special means of real numbers are given. [ABSTRACT FROM AUTHOR]

We present three cut trees of graphs, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation R on the vertex set of the graph, which generalizes Gomory-Hu trees. Applying these cut trees, we prove the following: • A pair of vertices { v , w } of a graph G is pendant if λ (v , w) = min ⁡ { d (v) , d (w) }. Mader showed in 1974 that every simple graph with minimum degree δ contains at least δ (δ + 1) / 2 pendant pairs. We improve this lower bound to δ n / 24 for every simple graph G on n vertices with δ ≥ 5 or λ ≥ 4 or vertex connectivity κ ≥ 3 , and show that this is optimal up to a constant factor with regard to every parameter. • Every simple graph G satisfying δ > 0 has O (n / δ) δ -edge-connected components. Moreover, every simple graph G that satisfies 0 < λ < δ has O ((n / δ) 2) cuts of size less than min ⁡ { 3 2 λ , δ } , and O ((n / δ) ⌊ 2 α ⌋) cuts of size at most min ⁡ { α ⋅ λ , δ − 1 } for any given real number α ≥ 1. • A cut is trivial if it or its complement in V (G) is a singleton. We provide an alternative proof of the following recent result of Lo et al.: Given a simple graph G on n vertices that satisfies δ > 0 , we can compute vertex subsets of G in near-linear time such that contracting these vertex subsets separately preserves all non-trivial min-cuts of G and leaves a graph having O (n / δ) vertices and O (n) edges. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define new concepts, namely a homogenous system of difference equations x(n+1) = Bx(n) where B is a matrix of real numbers, which is called P-semi homogenous of order m if there exists a non-zero matrix A and integer number m such that the following equation holds: F (A(c)x(n)) = P(A(c)) mF(x(n)), Where F is a function, m and P are integer numbers and c is a real number. This definition is a generalization to the (3 × 3) -semi-homogeneous system of difference equations of order m. Special cases are studied of this definition and illustrative examples are given and some characterizations of this definition are also given. The necessary and sufficient conditions for a homogenous system of difference Equations to be P-semi homogenous of order one or greater than one as well as some examples and theorems about there are given. [ABSTRACT FROM AUTHOR]

We generalized the concepts in the probability of rough lacunary statistical analysis of order η by introducing the difference operator A of fractional order, where a is a proper fraction and y = (Ymnk) is any fixed sequence of nonzero real or complex numbers. We study some properties of this operator involving lacunary sequence and arbitrary sequence p = (prst) of strictly positive real numbers and investigate the topological structures of related with triple difference sequence spaces. The main focus of the present paper is to generalize rough lacunary statistical analysis of order n of triple difference sequence spaces and give the relations between statistical analysis and lacunary statistical analysis of order n. [ABSTRACT FROM AUTHOR]

This paper studies limit cycle bifurcation of an (n + 1) th order generalized Liénard differential system with an elliptic Hamiltonian H (x , y) = 1 2 x 2 + 1 2 y 2 - 1 4 x 4 , which bifurcates at least [ n 2 ] limit cycles near either a centre or a heteroclinic loop with two hyperbolic saddles for n ≤ 150 , where about two thirds of the limit cycles are large amplitude ones. To achieve this goal, we provide the precise expression of the third coefficient in the asymptotic expansion of the first order Melnikov function of an analytic near-Hamiltonian system near a polycycle with m (≥ 2) hyperbolic saddles, without any prescribed condition, whereas all the known results need. As well as we also characterize its higher order coefficients. Here [a] is the integer part function of a real number a. [ABSTRACT FROM AUTHOR]

The Hebrew Bible is often read tliat men once lived nearly a thousand years. Then, after a cataclysm in the form of a flood, their life spmis followed an exponential decline to contemporary values, but no explanation for this decline was offered. Recently, it lias been suggested tlmt the decline can be attributed to marriage of die descendants of Noah to n population of short-lived survivors of the flood. This journal has previously published a paper that argued that the numbers expressing these longevities have the properties of real numbers and none of the properties of artficial or made-up numbers, ami that the reasons for doubting the biblical numbers do not survive close examination. Moreover, recent work argues that humans may ultimately have biblicnl longevities. Tlze present article shows that the intermarriage explanation for the decay of longevities is plausible. [ABSTRACT FROM AUTHOR]

Lifting problem for universal quadratic forms asks for totally real number fields K$K$ that admit a positive definite quadratic form with coefficients in Z$\mathbb {Z}$ that is universal over the ring of integers of K$K$. In this paper, we show K=Q(ζ7+ζ7−1)$K=\mathbb {Q}(\zeta _7+\zeta _7^{-1})$ is the only such totally real cubic field. Moreover, we show that there is no such biquadratic field. [ABSTRACT FROM AUTHOR]

Most of fractional differential equations are considered on a fixed interval. In this paper, we consider a typical fractional differential equation on a symmetric interval [-α,α], where α is the order of fractional derivative. For a positive real number α we prove that the solutions are Tn,α(x) = (α+x)½ Qn,α(x) where Qn,α(x) produce a family of orthogonal polynomials with respect to the weight function wα(x) = ... on [-α,α]. For integer case α = 1, we show that these polynomials coincide with classical Chebyshev polynomials of the third kind. Orthogonal properties of the solutions lead to practical results in determining solutions of some fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper uses unit lower triangular matrices to solve the nonnegative inverse eigenvalue problem for various sets of real numbers. This problem has remained unsolved for many years for n ≥ 5: The inverse of the unit lower triangular matrices can be easily calculated and the matrix similarities are also helpful to be able to solve this important problem to a considerable extent. It is assumed that in the given set of eigenvalues, the number of positive eigenvalues is less than or equal to the number of nonpositive eigenvalues to find a nonnegative matrix such that the given set is its spectrum. [ABSTRACT FROM AUTHOR]

The Gödel–Dummett logic and Łukasiewicz one are two main many-valued logics used by the fuzzy logic community. Our goal is a quantitative comparison of these two. In this paper, we will mostly consider the 3-valued Gödel–Dummett logic as well as the 3-valued Łukasiewicz one. We shall concentrate on their implicational-negation fragments which are limited to formulas formed with a fixed finite number of variables. First, we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length built with exactly one variable. Then, we investigate such proportion for satisfiable formulas. Second, we generalise our investigation on formulas written with $ k\ge 1 $ k ≥ 1 variables. The primary goal of the paper is the research on the asymptotic behaviour of these fractions when the length n tends to infinity. If such limits exists, they are real numbers between 0 and 1, which are called the density of truth or the density of SAT. To compare the density of truth and the density of satisfiable formulas for both fragments of 3-valued Gödel–Dummett's and Łukasiewicz's logics we use the powerful theory of analytic combinatorics. This paper is a natural continuation of the previous brief conference note by Kostrzycka and Zaionc (2020) as well as enriched with some previous results from Kostrzycka and Zaionc (2003). In the conference note we computed analytically the density of truth and the density of SAT (with a determined precision) for 3-valued Łukasiewicz's logic restricted to a language with only one variable. In Kostrzycka and Zaionc (2003) we computed the same values for exactly the same fragment of the 3-valued Gödel–Dummett logic. This paper answers the more general questions of the existence of density of truth and density of SAT for both many-valued logics with an arbitrary finite number of variables. Therefore this paper gives an an interesting picture of two main families of finite-valued fuzzy logics problems treated quantitatively. This picture is taken from the perspective of classical logic. It shows that unexpectedly there is quantitatively a little distance between these two approaches. [ABSTRACT FROM AUTHOR]

In this paper, we give the form of the solutions of the following rational difference equation where an and bn are sequences of real numbers, by using Lie symmetry analysis and associated with Fibonacci numbers, respectively. We find an interesting relation between the exact solution of Equation (1) and the classical Fibonacci sequence. [ABSTRACT FROM AUTHOR]

For decades, conventional wisdom maintained that binary 0–1 Bernoulli random variables cannot contain extra-binomial variation. Taking an unorthodox stance, Hilbe actively disagreed, especially for correlated observation instances, arguing that the universally adopted diagnostic Pearson or deviance dispersion statistics are insensitive to a variance anomaly in a binary context, and hence simply fail to detect it. However, having the intuition and insight to sense the existence of this departure from standard mathematical statistical theory, but being unable to effectively isolate it, he classified this particular over-/under-dispersion phenomenon as implicit. This paper explicitly exposes his hidden quantity by demonstrating that the variance in/deflation it represents occurs in an underlying predicted beta random variable whose real number values are rounded to their nearest integers to convert to a Bernoulli random variable, with this discretization masking any materialized extra-Bernoulli variation. In doing so, asymptotics linking the beta-binomial and Bernoulli distributions show another conventional wisdom misconception, namely a mislabeling substitution involving the quasi-Bernoulli random variable; this undeniably is not a quasi-likelihood situation. A public bell pepper disease dataset exhibiting conspicuous spatial autocorrelation furnishes empirical examples illustrating various features of this advocated proposition. [ABSTRACT FROM AUTHOR]