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The symmetric nonnegative inverse eigenvalue problem (SNIEP) is the problem of characterizing all possible spectra of entry-wise nonnegative symmetric matrices of given dimension. A list of real numbers is said to be symmetrically realizable if it is the spectrum of some nonnegative symmetric matrix. One of the most general sufficient conditions for realizability is the so-called C-realizability, which amounts to some kind of compensation between the positive and negative entries of the list of real numbers whose realizability one is trying to decide. A combinatorial characterization of C-realizable lists with zero trace was given in [11]. In this paper we make use of a recursive method for constructing simmetrically realizable lists due to Ellard and Šmigoc [3] to extend this combinatorial characterization of C-realizability to general lists with nonnegative trace. One consequence of this characterization is that the set of nonnegative C-realizable lists is a union of polyhedral cones whose faces are described by equations involving only linear combinations with coefficients 1 and −1 of the entries in the list. Another remarkable consequence is the monotonicity of C-realizability, i.e., the operation of increasing any positive entry of a C-realizable list preserves C-realizability. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

The cell-cell signaling mechanisms that are the basis for all of physiology have been used to trace evolution back to the unicellular state, and beyond, to the "First Principles of Physiology". And since our physiology derives from the Cosmos based on Symbiogenesis, it has been hypothesized that the cell behaves like a functional Mobius Strip, having no 'inside or outside' cell membrane surface - it is continuous with the Cosmos, its history being codified from Quantum Entanglement to Newtonian Mechanics, affording the cell consciousness and unconsciousness/subconsciousness as a continuum for the first time. Similarly, Klein and Maimon have concluded that their 'Soft Logic' mathematics also constitutes a Mobius Strip, using both a real number axis, combined with a zero axis, numerically representing cognition. This is congruent with the cell as 'two-tiered' consciousness, the first tier being the real-time interface between the cell membrane and its environment; the second tier constituting integrated physiology, referencing the consciousness of the Cosmos. Thus, there is coherence between physiology, consciousness and mathematics for the first time. [ABSTRACT FROM AUTHOR]

Let H be a 3-connected claw-free graph on n vertices, and define σ 2 (H) = min { d H (u) + d H (v) : u v ∉ E (H) }. Kužel et al. proved that H is Hamilton-connected if n ≥ 142 and δ (H) ≥ n + 50 8 . Let H U be an UM-closure of H. We generalize the result above with degree sum conditions by proving the following. (i) For any positive integer p and real number ϵ , there exist an integer N (p , ϵ) = 8 p 2 − (2 p + 1) | ϵ | − 2 p > 0 and a family G p (r) , which can be generated by a finite number of graphs of order r ≤ max { 8 , 6 p + 3 } , such that if n > N (p , ϵ) and σ 2 (H) ≥ n + ϵ p (or δ (H) ≥ n + ϵ 2 p ), then H is Hamilton-connected if and only if H U = L (G) for G ∉ G p (r). (i i) If n ≥ 85 and σ 2 (H) ≥ n + 4 4 (or δ (H) ≥ n + 4 8 ), then either H is Hamilton-connected or H U = L (V 8 ′) , where V 8 ′ is the graph obtained from the Wagner graph V 8 by attaching | E (V 8 ′) | − 12 8 pendant edges at each vertex of V 8. [ABSTRACT FROM AUTHOR]

A signed graph is a graph G together with a signature σ : E (G) → { 1 , − 1 }. For a real number r ≥ 1 , C r is a circle of circumference r. For two points a , b ∈ C r , the distance d (mod r) (a , b) between a and b is the length of the shorter arc of C r connecting a and b. For x ∈ C r , the antipodal x ̄ of x is the unique point in C r of distance r / 2 from x. A circular r -colouring of (G , σ) is a mapping f : V (G) → C r such that for each positive edge e = u v , d (mod r) (f (u) , f (v)) ≥ 1 , and for each negative edge e = u v , d (mod r) (f (u) , f (v) ¯) ≥ 1. The circular chromatic number of a signed graph (G , σ) is the minimum r such that (G , σ) is circular r –colourable. Let g ∗ (G , σ) be the length of the shortest cycle in (G , σ) with an odd number of positive edges. For a positive integer g , let SP g be the family of signed series–parallel graphs with g ∗ (G , σ) ≥ g. This paper determines, for any positive integer g , the supremum value of χ c (G , σ) of signed graphs in SP g. [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]

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]

Let G be a graph on n vertices. Denote by m G , θ the multiplicity of θ as an eigenvalue of the adjacency matrix of G. In a recent paper, when G is a unicyclic graph, Tian and Wang showed that m G , θ ≤ ⌊ n − 2 3 ⌋ when n ≥ 11 and θ 2 is an integer larger than 1. Motivated by it, we establish an upper bound of m G , θ for any fixed real number θ , generalizing the result of Tian and Wang. Some new techniques are developed to realize the generalization. [ABSTRACT FROM AUTHOR]

Given a graph G , for a real number α ∈ [ 0 , 1 ] , Nikiforov (2017) proposed the A α -matrix of G as A α (G) = α D (G) + (1 − α) A (G) , where A (G) and D (G) are the adjacency matrix and the degree diagonal matrix of G , respectively. The largest eigenvalue of A α (G) is called the A α -index of G. For n > k , let S n , k be the join of a clique on k vertices with an independent set of n − k vertices. Then S n , k + denotes the graph obtained from S n , k by adding an edge to connect two vertices of degree k. Very recently, Cioabǎ, Desai and Tait resolved the Nikiforov's conjecture: For fixed k ≥ 2 , and sufficiently large n , the { C 2 k + 1 , C 2 k + 2 } -free graph of order n with maximum adjacency spectral radius is S n , k and the C 2 k + 2 -free graph of order n with maximum adjacency spectral radius is S n , k +. In this paper, we confirm the A α -spectral version of this conjecture: For fixed k ≥ 2 , 0 < α < 1 and n ≥ 324 k 6 (k + 1) 2 α 6 (1 − α) 2 , the { C 2 k + 1 , C 2 k + 2 } -free graph of order n with maximum A α -index is S n , k and the C 2 k + 2 -free graph of order n with maximum A α -index is S n , k +. [ABSTRACT FROM AUTHOR]

Binary relations, coverings and neighborhood systems/operators are useful tools to study fuzzy rough set theory. In this paper, we use the notion of real-valued hemimetric, a weak version of the standard metric, as the basic structure to define and study fuzzy rough sets by using the usual addition and subtraction of real numbers. We define a pair of fuzzy upper and lower rough approximation operators and investigate their properties and interrelations. These two operators have nice logical descriptions by using the Lukasiewicz logical system. It is shown that upper definable fuzzy subsets, lower definable fuzzy subsets and Lipschitz fuzzy subsets are the same thing in this model. Definable fuzzy subsets are exactly the upper sets with respect to the induced fuzzy preorder, which form a stratified Alexandrov fuzzy topology. A comparison between hemimetric-based fuzzy rough sets and fuzzy preorder-based fuzzy rough sets has been made. Results show that the former can be considered as a real-valued extension of the latter. At the end, an application of the hemimetric-based rough set model to contour extraction of digital surfaces is proposed. [ABSTRACT FROM AUTHOR]

We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that c 2 ∈ [ 1 / 2 , 2 + 5 ] , where c 2 is the minimum real number c for which every real number greater than c is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem. [ABSTRACT FROM AUTHOR]

A multigraph is a graph with possible multiple edges, but no loops. Let t be a positive integer. Let G t be the set of simple graphs (or multigraphs) such that for each G ∈ G t there exist at least t + 1 non-empty disjoint proper subsets V 1 , V 2 , ... , V t + 1 ⊆ V (G) satisfying V (G) ∖ (V 1 ∪ V 2 ∪ ⋯ ∪ V t + 1) ≠ ϕ and edge connectivity κ ′ (G) = e (V i , V (G) ∖ V i) for i = 1 , 2 , ... , t + 1. Let D (G) and A (G) denote the degree diagonal matrix and adjacency matrix of a simple graph (or a multigraph) G , and let μ i (G) be the i th largest eigenvalue of the Laplacian matrix L (G) = D (G) + A (G). In this paper, we investigate the relationship between μ n − 2 (G) and edge connectivity or spanning tree packing number of a graph G ∈ G 1 , respectively. We also give the relationship between μ n − 3 (G) and edge connectivity or spanning tree packing number of a graph G ∈ G 2 , respectively. Moreover, we generalize all the results about L (G) to a more general matrix a D (G) + A (G) , where a is a real number with a ≥ − 1. [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) denote respectively the adjacency matrix and the diagonal matrix of vertex degrees of G. The largest eigenvalue of A α (G) is called the A α -index of G. Chang and Tam (2011) have proved that for every pair of integers n , k with − 1 ≤ k ≤ n − 3 , H n , k , the graph obtained from the star K 1 , n − 1 by joining a vertex of degree 1 to k + 1 other vertices of degree 1, is the unique connected graph that maximizes the Q -index (i.e., the signless Laplacian spectral radius or, equivalently, the A 1 2 -index) over all connected graphs with n vertices and n + k edges. In this paper it is proved that for every pair of integers n , k with − 1 ≤ k ≤ n − 3 , when 1 2 < α < 1 or α = 1 2 and k ≠ 2 , the graph H n , k is the unique connected graph that maximizes the A α -index over all connected graphs with n vertices and n + k edges. This work extends (and also provides an alternative proof for) the above-mentioned result of Chang and Tam. A complete overview of the history of the maximal index problems is also given. [ABSTRACT FROM AUTHOR]

In this paper, we study the global stability of delayed genetic regulatory networks (DGRNs) with Hill-type activation (or inhibition) functions based on the mixing monotone semiflows approach, where Hill coefficients can be arbitrary positive real number. A new result on the global stability of DGRN is given in the case that all Hill coefficients are less than or equal to 1. In addition, for all Hill coefficients greater than 1, a new sufficient condition for global convergence of DGRN is given, which is less conservative than the condition of the existing correlation results. Finally, two numerical examples and simulations are given to explain the effect of obtained results. [ABSTRACT FROM AUTHOR]

• Four errors in the traditional forecast model were identified and mitigated. • Grey forecast prediction model was developed for interval sequences. • Weight parameter comprehensively characterized the development trend of the interval. • Proposed model demonstrated superior performance in two real-world examples. With the increasing complexity of systems, it is difficult to describe a huge amount of information accurately with real numbers. To make predictions using uncertain information, this study uses interval numbers as descriptors and proposes a weight-adaptive fusion grey prediction model based on interval sequences (WAFGM(1,N) model). Unlike most interval prediction models; the proposed model uses the adaptive weight and compensation coefficient to directly predict interval grey numbers without converting interval sequences into real number sequences. Moreover, it improves the four errors of the traditional GM(1,N) model, thereby further improving model accuracy. First, by fully considering the integrity and interaction of the upper and lower boundaries of the interval, this study introduced a weight parameter to comprehensively characterize the development trend of the interval, and by optimizing the background value and adjusting the compensation coefficient, the univariate and multivariate weight-adaptive grey prediction models for interval prediction were constructed, respectively. Second, the interval development and driving coefficients, compensation coefficient, and weight parameter were estimated and optimized using the least squares method, accumulation method, and particle swarm algorithm, respectively. Finally, the WAFGM(1,N) model was constructed by combining the two models. The validity and applicability of the WAFGM(1,N) model were proved using two examples: the gross agricultural output value of Northeast China and the production of dairy products in the golden milk source areas. [ABSTRACT FROM AUTHOR]

We study the family of all fuzzy sets of the n-dimensional Euclidean space, which are upper-semicontinuous, fuzzy convex and normal with compact supports contained in a non-degenerate convex subset Y, and prove that the following statements are equivalent: (i) The family of fuzzy sets with the topology induced by the supremum metric is homeomorphic to a non-separable Hilbert space whose weight is the cardinality of the set of all real numbers; (ii) the non-degenerate convex subset Y is topologically complete, or equivalently, it is a countable intersection of open sets in the n-dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

Program extraction from proofs can be used to obtain verified algorithms in exact real arithmetic for e.g., the signed-digit code representation. In Minlog this has been done in the past with the use of certain coinductive predicates. In a next step we want to analyze the lookahead of these extracted programs. Doing it by hand is quite cumbersome, so instead we change our definition. Instead of a coinductive predicate we use an inductive predicate for the representation of reals that already incorporates the lookahead. In this way the lookahead becomes part of the specification which is carried through all the proofs. In the end we extract programs for computations on a signed-digit representation and we can just read off the lookahead from the specification. • We use the theory TCF for extracting verified algorithms for exact real number arithmetic based on stream representations. • The lookahead of the stream-transformers is explicitly part of the specification. • We use an explicit representation of real numbers rather than an axiomatization of abstract real numbers. [ABSTRACT FROM AUTHOR]

Emerging wireless and mobile applications, such as border intrusion detection with station-based drones, brought a new barrier coverage problem of using sink-based mobile sensors to cover a given line barrier with minimum energy consumption. In this paper, we focus on the uniform sink-based line barrier coverage (SLBC) problem, in which we are given a line barrier and k sink stations distributed on the plane which can emit an infinite number of sensors with an identical sensing radius. The problem aims to find their final positions on the barrier for the sensors emitted by the stations, such that the total moving distance of the sensors is minimized and each point of the barrier is within the sensing area of at least one sensor. We first observe the geometric structure of an optimal solution that any optimal solution can be considered as a set of intersecting tangent (disk) segments, where a tangent (disk) segment is a sequence of tangent disks. Then, we devise an algorithm to calculate all possible tangent (disk) segments and another algorithm to calculate the near-optimal positions for each of such segments. After computing all tangent (disk) segments and their near-optimal positions, an algorithm is proposed to transform uniform SLBC into an instance of the shortest path problem. It is shown the whole algorithm deserves a runtime O (k 2 log ⁡ k r ε) and consumes at most ε more movement than an optimal solution, where ε is any given positive real number, and r and k are the sensor radius and the number of sink stations, respectively. • The algorithm consumes a runtime O (k 2 log ⁡ k r ε) and produces a solution with at most ε more movement than that of an optimum solution. • With the runtime and the performance guarantee, the proposed algorithm behaves in a manner similar to a polynomial-time exact algorithm. • We proved several geometric properties of an optimum solution, which might have possible applications in other variants of the problem. [ABSTRACT FROM AUTHOR]

This paper enriches a pre-existing decision algorithm, which in turn augmented a fragment of Tarski's elementary algebra with one-argument real functions endowed with a continuous first derivative. In its present (still quantifier-free) version, our decidable language embodies the addition of functions and multiplication of functions by scalars; the issue we address is the one of satisfiability. As regards real numbers, individual variables and constructs designating the basic arithmetic operations are available, along with comparison relators. As regards functions, we have variables of another sort, out of which compound terms are formed by means of constructs designating addition and differentiation. An array of predicates designates various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between the derivative of a function and a real-valued term. Our decision method consists in preprocessing the given formula into an equi-satisfiable quantifier-free formula of the elementary algebra of real numbers, whose satisfiability can then be checked by means of Tarski's decision method. No direct reference to functions will appear in the target formula, each function variable having been superseded by a collection of stub real variables; hence, in order to prove that the proposed translation is satisfiability-preserving, we must figure out a flexible-enough family of interpolating C 1 functions that can accommodate a model for the source formula whenever the target formula turns out to be satisfiable. With respect to the results announced in earlier papers of the same stream, a significant effort went into designing the family of interpolating functions so that it could meet the new constraints stemming from the presence of function addition (along with differentiation) among the constructs of our fragment of mathematical analysis. • A formal language RDF* concerned with differentiable real functions is proposed. • The class of differentiable functions treated is closed under addition. • The expressive power of RDF* is illustrated through a gallery of examples. • A satisfiability-preserving algorithm reducing RDF* formulas into an existential sentence of Tarskian algebra is presented. • The correctness of the reduction algorithm is reported. [ABSTRACT FROM AUTHOR]

In this work, we study a range of constrained versions of the k-supplier and k-center problems. In the classical (unconstrained) k -supplier problem, we are given a set of clients C in a metric space X , with distance function d (. ,.). We are also given a set of feasible facility locations L ⊆ X. The goal is to open a set F of k facilities in L to minimize the maximum distance of any client to the closest open facility, i.e., minimize, cost (F , C) ≡ max j ∈ C ⁡ { d (F , j) } , where d (F , j) is the distance of client j to the closest facility in F. The k -center problem is a special case of the k -supplier problem where L = C. We study various constrained versions of the k -supplier problem such as: capacitated , fault-tolerant , ℓ -diversity, etc. These problems fall under a broad framework of constrained clustering. A unified framework for constrained clustering was proposed by Ding and Xu [Algorithmica 2020] in the context of the k -median and k -means objectives. We extend this framework to the k -supplier and k -center objectives in this work. This unified framework allows us to obtain results simultaneously for the following constrained versions of the k -supplier problem: r-gather, r-capacity, balanced, chromatic, fault-tolerant, strongly private, ℓ-diversity, and fair k-supplier problems, with and without outliers. We design Fixed-Parameter Tractable (FPT) algorithms for these problems. FPT algorithms have polynomial running time if the parameter under consideration is a constant. This may be relevant even to a practitioner since the parameter k is a small number in many real clustering scenarios. We obtain the following results: • We give 3 and 2 approximation algorithms for the constrained k -supplier and k -center problems, respectively, with FPT running time k O (k) ⋅ n O (1) , where n = | C ∪ L |. Moreover, these approximation guarantees are tight; that is, for any constant ε > 0 , no algorithm can achieve (3 − ε) and (2 − ε) approximation guarantees for the constrained k -supplier and k -center problems in FPT time, assuming FPT ≠ W [ 2 ]. • We study the constrained clustering problem with outliers. Our algorithm gives 3 and 2 approximation guarantees for the constrained outlier k -supplier and k -center problems, respectively, with FPT running time (k + m) O (k) ⋅ n O (1) , where n = | C ∪ L | and m is the number of outliers. • Our techniques generalise for distance function d (. ,.) z. That is, for any positive real number z , if the cost of a client is defined by d (. ,.) z instead of d (. ,.) , then our algorithm gives 3 z and 2 z approximation guarantees for the constrained k -supplier and k -center problems, respectively. • We design 3 approximation algorithm for a wide range of constrained k -supplier problems with FPT running time f (k). p o l y (n). • We design 2 approximation algorithm for a range of constrained k -center problems with FPT running time f (k). p o l y (n). • We extend the algorithm to outlier setting with FPT running time f (k , m). p o l y (n) , where m = number of outliers. • We show matching lower bounds for the constrained k -supplier and k -center problems in the outlier and non-outlier settings. [ABSTRACT FROM AUTHOR]

According to the Jordan Decomposition Theorem, for any function f : [ 0 , 1 ] → R of bounded variation there exists a pair of nondecreasing functions f 1 , f 2 : [ 0 , 1 ] → R with f = f 1 − f 2. Such a pair is called a Jordan decomposition of f. Ko (1991) and Zheng and Rettinger (2005) have analyzed computability-theoretic and complexity-theoretic aspects of this theorem. We complement their observations by three new results. One of our results says that there exists a polynomial time computable, absolutely continuous function f : [ 0 , 1 ] → R with polynomial modulus of absolute continuity such that, on the one hand, the so-called standard Jordan decomposition of f is computable but, on the other hand, neither the standard Jordan decomposition of f nor any other Jordan decomposition of f can be computed easily, say, in polynomial time, or in exponential time. • A function with a non-computable standard Jordan decomposition can have a polytime computable Jordan decomposition. • A computable but not polytime computable standard Jordan decomposition allows a polytime computable Jordan decomposition. • There is a PAC function without polytime computable Jordan decomposition but with computable standard Jordan decomposition. [ABSTRACT FROM AUTHOR]

The independent set polynomial is important in many areas of combinatorics, computer science, and statistical physics. For every integer Δ ≥ 2 , the Shearer threshold is the value λ ⁎ (Δ) = (Δ − 1) Δ − 1 / Δ Δ. It is known that for λ < − λ ⁎ (Δ) , there are graphs G with maximum degree Δ whose independent set polynomial, evaluated at λ , is at most 0. Also, there are no such graphs for any λ > − λ ⁎ (Δ). This paper is motivated by the computational problem of approximating the independent set polynomial when λ < − λ ⁎ (Δ). The key issue in complexity bounds for this problem is "implementation". Informally, an implementation of a real number λ ′ is a graph whose hard-core partition function, evaluated at λ , simulates a vertex-weight of λ ′ in the sense that λ ′ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any λ < − λ ⁎ (Δ) , it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any λ > λ ⁎ (Δ). Our result has already been used in a paper with Bezáková (STOC 2018) to show that it is #P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most Δ at any value λ < − λ ⁎ (Δ). In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness). [ABSTRACT FROM AUTHOR]

For a bounded quaternionic operator T on a right quaternionic Hilbert space H and ε > 0 , the pseudo S -spectrum of T is defined as Λ ε S (T) : = σ S (T) ⋃ { q ∈ H ∖ σ S (T) : ‖ Q q (T) − 1 ‖ ≥ 1 ε } , where H denotes the division ring of quaternions, σ S (T) is the S -spectrum of T and Q q (T) = T 2 − 2 Re (q) T + | q | 2 I. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo S -spectrum and explicitly compute the pseudo S -spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of S -functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number r if and only if for every ε > 0 the pseudo S -spectrum of the operator is the circularization of a closed disk in the complex plane centred at r with the radius ε. Further, we propose a G 1 -condition for quaternionic operators and prove some results in this setting. [ABSTRACT FROM AUTHOR]

Ando, Li, Mathias [1] defined the geometric mean M # (A , B , C) for positive definite matrices A , B , C by recursive procedure. Petz and Temesi [10] gave a simple proof of it and conjectured that any symmetric mean σ for two matrices would be extended to one for three matrices in the same way. Although it was proved by Palfia [12] , we give, nevertheless, a simple and natural proof. In [7] it was shown that if A , B , C are 2 × 2 positive definite matrices with determinant one such that t r a c e (A − 1 B) = t r a c e (B − 1 C) = t r a c e (C − 1 A) , then M # (A , B , C) is a scalar multiple of A + B + C. We extend this to N × N matrices. If A , B , C are linearly independent and A − 1 B + B − 1 A = 2 γ I , B − 1 C + C − 1 B = 2 α I , C − 1 A + A − 1 C = 2 β I for α , β , γ > 1 , then there are non-negative real numbers a , b , c such that M # (A , B , C) = a A + b B + c C. Moreover, if α = β = γ , then a = b = c = 1 3 (2 α + 1) . [ABSTRACT FROM AUTHOR]

The graph grabbing game is a two-player game on a connected graph with a vertex-weight function. In the game, they alternately remove a non-cut vertex from the graph (i.e., the resulting graph remains connected) and get the weight assigned to the vertex. Each player's aim is to maximize his or her outcome, when all vertices have been taken. In this paper, we consider the graph grabbing game on totally-weighted graphs that are graphs with weight functions from a set of elements in the vertex set and the edge set to non-negative real numbers. In this version, when a player removes a non-cut vertex v , that player gets the weight of v plus the total weight assigned to the edges incident to v. In particular, we give some results of interest for the graph grabbing game on edge-weighted trees, i.e., every vertex has weight zero. Moreover, we consider the game on edge-weighed graphs in the altered rule that each player must keep the connectedness of graphs induced by edges. [ABSTRACT FROM AUTHOR]

Let T be a tree on n vertices. Denote by m T , θ the multiplicity of θ as an eigenvalue of the adjacency matrix of T. From two previous publications of Rowlinson and Wong et al., it is shown that m T , 0 ≤ n − 2 , m T , θ ≤ n − 3 2 if θ ≠ 0 , and m T , θ ≤ n − 4 3 if θ ≠ 0 , ± 1 and n ≥ 7. In this note, we exploit the Parter-Wiener Theorem (different from the star complement technique used in previous publications) to obtain an upper bound for m T , θ for any fixed real number θ , thereby generalizing the results of Rowlinson and Wong et al., as well as two new papers from Tian and Wang, and Zhang et al. [ABSTRACT FROM AUTHOR]

We propose a novel mapping from a space of interval-valued functions defined on an arbitrary set to the family of closed subintervals of nonnegative extended real numbers, called the interval-valued Choquet-Sugeno-like operator. It is based on the notion of the Choquet-Sugeno-like operator and the interval-valued seminormed fuzzy operator introduced by Boczek et al. (2021) and generalizes many interval-valued operators known in the literature, such as n -ary interval-valued aggregation operators, interval-valued ordered weighting average operators, interval-valued seminormed fuzzy operators, and discrete interval-valued Choquet integral. We study its properties and show that it can be used to aggregate interval-valued data. [ABSTRACT FROM AUTHOR]

For integers s ≥ 0 and t ≥ 0 , a graph G is (s , t) -supereulerian if for any disjoint edge sets X , Y ⊆ E (G) with | X | ≤ s and | Y | ≤ t , G has a spanning closed trail that contains X and avoids Y. Pulleyblank (1979) showed that determining whether a graph is (0 , 0) -supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in (Catlin, 1988) showed that every simple graph G on n vertices with δ (G) ≥ n 5 − 1 , when n is sufficiently large, is (0 , 0) -supereulerian or is contractible to K 2 , 3. A function j 0 (s , t) has been found that every (s , t) -supereulerian graph must have edge connectivity at least j 0 (s , t). For any nonnegative integers s and t , we obtain best possible Ore conditions to assure a simple graph on n vertices to be (s , t) -supereulerian as stated in the following. (i) For any real numbers α and β with 0 < α < 1 , there exists a family of finitely many graphs F (α , β ; s , t) such that if κ ′ (G) ≥ j 0 (s , t) and if for any nonadjacent vertices u , v ∈ V (G) , d G (u) + d G (v) ≥ α n + β , then either G is (s , t) -supereulerian, or G is contractible to a member in F (α , β ; s , t). (ii) If κ ′ (G) ≥ j 0 (s , t) and if for any nonadjacent vertices u , v ∈ V (G) , d G (u) + d G (v) ≥ n − 1 , then when n is sufficiently large, either G is (s , t) -supereulerian, or G is contractible to one of the six well specified graphs. (iii) Suppose that δ (G) ≥ 5. If (1) for any vertices u , v , w ∈ V (G) with E (G [ { u , v , w } ]) = 0̸ , d G (u) + d G (v) + d G (w) > n − 3. then G is (s , t) -supereulerian if and only if κ ′ (G) ≥ j 0 (s , t). [ABSTRACT FROM AUTHOR]

• The prevention of sudden cardiac death in athletes (SDA) is a subject of ongoing debate. • Two milestones are pre-participation screening and the use of automated external defibrillators. • Conditions potentially associated with SDA can be identified in 0.4% of screened athletes although overall sensitivity is low. • On the other hand, widespread use of automated external defibrillators increased the survival chain. • The effectiveness of the reactive strategy suggests pushing for homogeneous implementation worldwide. Sudden death in young athletes is a rare but always dramatic condition. Unlike all other rare diseases, the estimate of its real incidence is made complex not only because of the uncertain number of correctly identified cases (numerator) but also because of the uncertain estimation of the real number of the reference population, the athletes (denominator). New elements of complexity are also emerging with regard to prevention. The current two pillars of prevention are pre-participation screening (proactive strategy) and promoting use / access to the automated external defibrillator (reactive strategy). The standardization of procedures implemented over the past two decades for pre-participation screening can now allow us to assess the impact of this approach. The result is complex to evaluate. While screening may allow the identification of conditions potentially associated with SDA, and therefore the adoption of specific treatments, in about 0.4% of screened subjects, a single study investigated the yield in terms of mortality showing a positive predictive value of 4.7% with 25% sensitivity. Conversely, the reactive strategy appears considerably effective, due to the widespread use of the automated external defibrillators in sports facilities, calling for a homogeneous implementation worldwide. On a broader perspective, the vast attention devoted to SDA prevention in the world of sports represents a major driver for transfer of a reactive prevention strategy to the general population. [ABSTRACT FROM AUTHOR]

Let G be a graph and let ω (G) denote the number of components of G. The graph G is said to be t-tough if t ⋅ ω (G − X) ≤ | X | for all X ⊆ V (G) with ω (G − X) > 1 , where t is a nonnegative real number. The toughness τ (G) of the graph G is the maximum value of t such that G is t -tough (taking τ (K n) = ∞ for all n ≥ 1). A graph G is said to be minimally t-tough if τ (G) = t and τ (G − e) < t for every e ∈ E (G). Kriesell (2003) conjectured that every minimally 1-tough graph has a vertex of degree 2. Katona and Varga (2018) proposed a generalized version that every minimally t -tough graph has a vertex of degree ⌈ 2 t ⌉ for any positive real number t. In this note, we disprove the Generalized Kriesell's Conjecture by constructing an infinite family of counterexamples having toughness close to 1, and thus also show that the toughness condition in Kriesell's Conjecture would be sharp. [ABSTRACT FROM AUTHOR]

A graph is a k-threshold graph with thresholds θ 1 , θ 2 , ... , θ k if we can assign a real number r v to each vertex v such that for any two distinct vertices u and v , uv is an edge if and only if the number of thresholds not exceeding r u + r v is odd. The threshold number of a graph is the smallest k for which it is a k -threshold graph. Multithreshold graphs were introduced by Jamison and Sprague as a generalization of classical threshold graphs. They asked for the exact threshold numbers of complete multipartite graphs. Recently, Chen and Hao solved the problem for complete multipartite graphs where each part is not too small, and they asked for the case when each part has size 3. We determine the exact threshold numbers of K 3 , 3 , ... , 3 , K 4 , 4 , ... , 4 and their complements n K 3 , n K 4. This improves a result of Puleo. [ABSTRACT FROM AUTHOR]

In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L -fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces. [ABSTRACT FROM AUTHOR]

For a stationary random function ξ , sampled on a subset D of R d , we examine the equivalence and orthogonality of two zero-mean Gaussian measures P 1 and P 2 associated with ξ. We give the isotropic analog to the result that the equivalence of P 1 and P 2 is linked with the existence of a square-integrable extension of the difference between the covariance functions of P 1 and P 2 from D to R d. We show that the orthogonality of P 1 and P 2 can be recovered when the set of distances from points of D to the origin is dense in the set of non-negative real numbers. • We characterize equivalent isotropic Gaussian measures using the Hankel transform • We show how sufficiently dense sampling domains lead to orthogonal Gaussian measures • We derive orthogonality when the random function is sampled along a continuous path [ABSTRACT FROM AUTHOR]

For 0 < p < ∞ , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space A α 2 on the unit ball of C m. The main results of this paper are twofold: (1) For α , β ∈ R , we show that the necessary and sufficient condition for the embedding operator I : A α 2 → A β 2 is in the Schatten p -class is that p (β − α) − 2 m > 0. (2) For a positive Borel measure μ on B m and a real number τ , we prove that the fractional embedding operator I α + τ τ : A α 2 → L 2 (B m , d μ) is in the Schatten p -class if and only if the Berezin transform of μ , denoted by μ ˜ α α + τ , is in L p 2 (B m , d λ) , where d λ (z) = (1 − | z | 2) − m − 1 d v (z) is the Möbius invariant volume measure and d v is the Lebesgue measure on B m , or equivalently, the averaging function of μ , denoted by μ ˆ r α , is in L p 2 (B m , d λ). • For p > 0 , we completely characterize the Schatten p -class embedding and fractional embedding of the Bergman-type space on the unit ball. • The range 0 < p < 2 is an open problem left by H. T. Kaptanoğlu (see [5]). • We prove that the fractional embedding operator is in Schatten p -class iff the Berezin transform of the positive Borel measure is in L − space. [ABSTRACT FROM AUTHOR]

In the famous paper in which he introduced what is now known as the Turing machine, Alan Turing gave a definition of computable real numbers under which it turns out that multiplication by 3 is uncomputable. This shortcoming vanished in a Correction to his paper that Turing himself published shortly afterwards, but it clearly illustrates the subtlety of defining computability issues correctly. In this paper, we give the name "printable" to real numbers that Turing originally called "computable", we recall what is now the generally accepted definition of computable real numbers (which is not quite Turing's amended definition, but is equivalent to it), and we contrast the two notions. Despite the fact that the multiplication by 3 of printable numbers is uncomputable, as opposed to the same operation on computable numbers, a real number is computable if and only if it is printable. The resolution of this apparent paradox is that no machine can transform the "computable" description of a real number to its "printable" description, as Turing proved in his Correction. Finally, we address the subtle issue of allowing or not the printable description of a real number to end with an infinite sequence of 9s (or of 1s in binary), which was left open by Turing in his Correction. Several of these results were already known, as they appear in scattered places, some in non-refereed publications, but we give a unified treatment with some different proofs and a historical perspective. • Alan Turing's original definition of computable real numbers was inadequate. • Turing's use of the word Entscheidungsproblem led us to invent the Druckproblem. • The correct definition of computable real numbers is very subtle. • All computable real numbers are printable, but this is inherently non-constructive. • Discontinuous functions of the real numbers on closed intervals are uncomputable. [ABSTRACT FROM AUTHOR]

This paper abstracts the complex heterogeneous multi-attribute group decision-making (CHMAGDM) characterized by two-layer decision-makers (DMs), individual attribute sets, complex relationships among attributes and heterogeneous evaluation information. DMs are composed by internal decision-makers (IDMs) and external decision-makers (EDMs). The evaluation information is represented by linguistic variables (LVs), intervals and real numbers. To handle such CHMAGDM, this paper puts forward an integrated method of prospect theory, DEMATEL (decision-making trial and evaluation laboratory) and QUALIFLEX (qualitative flexible). Firstly, a novel reference point determining approach is proposed based on three special indicators. A linguistic variable inverse prospect value function is designed and introduced to the transformation process from LVs to clouds to reduce the impacts of DMs' psychological behaviors on decision-making. Secondly, a cloud-based DEMATEL approach is proposed to determine the attribute weights, where LVs are employed to judge the relationships among attributes. A new linguistic scale function is initiated and introduced to another transformation process from clouds to LVs. A representative indicator of cloud is designed for facilitating the normalization of direct relation matrix. Moreover, a linear programming model is built to calculate the attribute weights. Thirdly, an extended QUALIFLEX approach is developed to obtain the optimal ranking and EDM weights, where adoption coefficient is provided by IDMs. Another linear programming model is constructed to calculate the optimal collective concordance/discordance index of each permutation. The optimal ranking is generated by the maximum collective concordance/discordance index. Finally, a photovoltaic power station site selection example validates the proposed method. The advantages are demonstrated with sensitivity and comparison analyses. [ABSTRACT FROM AUTHOR]

Given a graph G and a real number 0 ≤ p ≤ 1 , we define the random set B p (G) ⊆ V (G) by including each vertex independently and with probability p. We investigate the probability that the random set B p (G) is a zero forcing set of G. In particular, we prove that for large n , this probability for trees is upper bounded by the corresponding probability for a path graph. Given a minimum degree condition, we also prove a conjecture of Boyer et al. regarding the number of zero forcing sets of a given size that a graph can have. [ABSTRACT FROM AUTHOR]

Let G be a family of subsets of an n -element set. The family G is called 3-wise t -intersecting if the intersection of any three subsets in G is of size at least t. 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. For example, if G consists of all subsets containing a fixed t -element set, then it is a 3-wise t -intersecting family with the measure p t. Let 0 < p ≤ 2 / (4 t + 9 − 1) , δ > 0 , and let G be a 3-wise t -intersecting family. It is known that the measure of G is at most p t. Suppose, moreover, that G has the measure at least (1 2 + δ) p t. We show that, by choosing t sufficiently large depending on δ , the structure of G is one of (i) and (ii): (i) every subset in G contains a fixed t -element set, (ii) every subset in G contains at least t + 2 elements from a fixed (t + 3) -element set. [ABSTRACT FROM AUTHOR]

Williamson-Hall, Stocks-Wilson, Scherrer, Halder-Wagner, and Size-Strain Plot (SSP) methods are used essentially to ensure the material particle size falls at the nano-level. They treat the broadening in the XRD peak as a sum of Gauss and Lorentz diffraction probability functions. In this work, an approach to the microstrain distribution is presented as a strain distribution (SD) model, assuming a nanostructure as a liquid drop where surface tension controls the particle positions while strain controls the geometry and spacing of the lattice parameters. The number of diffraction planes is considered in the model, treated as a Gaussian-like (or Lorentzian-like) function, and estimated with numerical analysis. The SD model writes an equation about the broadening, peak position, and lattice parameters to estimate the crystalline size and strain exponent. Williamson-Hall, Stocks-Wilson, and Scherrer can be explained as approximations for this model, and the presence of negative strain is explained. Possible approximations can show Halder-Wagner and SSP as another face of the SD model equation. The strain exponent, which is estimated here, is more useful than the average micro-strain, which is obtained from previous models. The strain exponent role in the nanoparticle reactions with materials can be discussed and explained. The change in crystal system as bulk material is reduced to nanostructure can be negated according to the SD model. • Introducing a novel XRD peak profile based on the real number of diffraction planes. • Showing the distribution of micro-strain inside the nanoparticle. • Explanation the change in crystal system as reduce bulk material to nanostructure. • Illustrate the reason for negative micro-strain due to the Williamson-Hall method. • Proving Williamson-Hall, and Scherrer are an approximation to the SD model. [ABSTRACT FROM AUTHOR]

Lanthanide metal-organic frameworks (Ln-MOFs) have exhibited remarkable capabilities in color change, luminescence, proton conduction through the strategic modification of metal nodes, organic ligands and channels or pores. By modifying the electron-deficient bipyridinium to contain carboxylic acid moieties and coordinating them with the luninescent Eu3+ ion, a unique and versatile Eu-MOF has been synthesized. Under UV light conditions, the complex shows an obvious color change from colorless to blue, which is due to the formation of viologen radicals after photoinduced electron transfer (PIET) process. As we all know, the changed color is caused by the alteration of absorption intensity, ultimately impacting its luminescence as well. Thus, the bright and strong red fluorescence of this complex is quenched upon continuous UV light irradiation, which may be attributed to intramolecular energy transfer from the excited state of the europium ion to the colored state of the CV ligand. Based on the control of multiple colors of the complex, the designed encryption model can realize multiple anti-counterfeiting of real numbers. In addition, the conductivity of this compound increases significantly after UV illumination, showing excellent photo-modulated proton conductivity. In short, our work has proposed a multifunctional bipyridinium-based Ln-MOF with photochromism, photo-switchable photoluminescence and proton conduction, which offers valuable insights for the creation of versatile switchable materials based on MOFs. [Display omitted] • A novel viologen-based Eu-MOF exhibits reversible photochromism and photo-modulated photoluminescence. • The colored Eu-MOF presents higher proton conductivity after UV irradiation. • The Eu-MOF can be used for multiple anti-counterfeiting encryption of information. [ABSTRACT FROM AUTHOR]

For any real number α ∈ [ 0 , 1 ] and a graph G , Nikiforov [20] defined matrix A α (G) as A α (G) = α D (G) + (1 − α) A (G). The A α - spectral radius of G is the largest eigenvalue of A α (G). In this paper, we determine the first three largest extremal graphs in Halin graphs by their A α - spectral radii when 1 2 ≤ α < 1. For 0 ≤ α ≤ 1 , the 3-regular Halin graph alone minimizes the A α - spectral radius for Halin graphs. [ABSTRACT FROM AUTHOR]

We describe a model for m vertex reinforced interacting random walks on complete graphs with d ≥ 2 vertices. The transition probability of a random walk to a given vertex depends exponentially on the proportion of visits made by all walks to that vertex. The individual proportion of visits is modulated by a strength parameter that can be set equal to any real number. This model covers a large variety of interactions including different vertex repulsion and attraction strengths between any two random walks as well as self-reinforced interactions. We show that the process of empirical vertex occupation measures defined by the interacting random walks converges (a.s.) to the limit set of the flow induced by a smooth vector field. Further, if the set of equilibria of the field is formed by isolated points, then the vertex occupation measures converge (a.s.) to an equilibrium of the field. These facts are shown by means of the construction of a strict Lyapunov function. We show that if the absolute value of the interaction strength parameters are smaller than a certain upper bound, then, for any number of random walks (m ≥ 2) on any graph (d ≥ 2), the vertex occupation measures converge towards a unique equilibrium. We provide two additional examples of repelling random walks for the cases m = d = 2 and m ∈ { 3 , 4 , 5 } with d = 2. The latter is used to study some properties of three, four and five exponentially repelling random walks on Z. [ABSTRACT FROM AUTHOR]

This paper shows that the Hukuhara difference and generalized gH-difference, as well as the gH-division, are not inclusion-isotonic interval extensions of the difference and division of real numbers, respectively, and discusses how this fact may affect the reliability of some results in specific epistemic mathematical models with respect to the semantics being considered. [ABSTRACT FROM AUTHOR]