Nonnegative Reals Research Articles

Union of Random Minkowski Sums and Network Vulnerability Analysis

Let $$\mathcal {C}=\{C_1,\ldots ,C_n\}$$C={C1,?,Cn} be a set of $$n$$n pairwise-disjoint convex sets of constant description complexity, and let $$\pi $$? be a probability density function (density for short) over the non-negative reals. For each $$i$$i, let $$K_i$$Ki be the Minkowski sum of $$C_i$$Ci with a disk of radius $$r_i$$ri, where each $$r_i$$ri is a random non-negative number drawn independently from the distribution determined by $$\pi $$?. We show that the expected complexity of the union of $$K_1, \ldots , K_n$$K1,?,Kn is $$O(n^{1+{\varepsilon }})$$O(n1+?) for any $${\varepsilon }> 0$$?>0; here the constant of proportionality depends on $${\varepsilon }$$? and the description complexity of the sets in $$\mathcal {C}$$C, but not on $$\pi $$?. If each $$C_i$$Ci is a convex polygon with at most $$s$$s vertices, then we show that the expected complexity of the union is $$O(s^2n\log n)$$O(s2nlogn). Our bounds hold in a more general model in which we are given an arbitrary multi-set $$\varTheta =\{\theta _1,\ldots ,\theta _n\}$$?={?1,?,?n} of expansion radii, each a non-negative real number. We assign them to the members of $$\mathcal {C}$$C by a random permutation, where all permutations are equally likely to be chosen; the expectations are now with respect to these permutations. We also present an application of our results to a problem that arises in analyzing the vulnerability of a network to a physical attack.

On the spectrum of the hierarchical Laplacian

Let $(X,d)$ be a locally compact separable ultrametric space. We assume that $(X,d)$ is proper, that is, any closed ball $B$ in $X$ is a compact set. Given a measure $m$ on $X$ and a function $C(B)$ defined on the set of balls (the choice function), we define the hierarchical Laplacian $L_C$ which is closely related to the concept of the hierarchical lattice of F.J. Dyson. $L_C$ is a non-negative definite, self-adjoint operator in $L^2(X,m)$. We address in this paper to the following question: How general can be the spectrum $\mathsf{Spec}(L_C)$ as a subset of the non-negative reals? When $(X,d)$ is compact, $\mathsf{Spec}(L_C)$ is an increasing sequence of eigenvalues of finite multiplicity which contains $0$. Assuming that $(X,d)$ is not compact we show that, under some natural conditions concerning the structure of the hierarchical lattice (= the tree of $d$-balls), any given closed subset $S$ of $[0,\infty)$, which contains $0$ as an accumulation point and is unbounded if $X$ is non-discrete, may appear as $\mathsf{Spec}(L_C)$ for some appropriately chosen function $C(B)$. The operator $-L_C$ extends to $L^q(X,m)$, $0 < q < \infty$, as Markov generator and its spectrum does not depend on $q$. As an example, we consider the operator $\mathfrak{D}^{\alpha}$ of fractional derivative defined on the field $\mathbb{Q}_p$ of $p$-adic numbers.

Monotone operators on Gödel logic

We consider an extension of Godel logic by a unary operator that enables the addition of non-negative reals to truth-values. Although its propositional fragment has a simple proof system, first-order validity is Π2-hard. We explain the close connection to Scarpellini’s result on Π2-hardness of Łukasiewicz’s logic.

Изотропные марковские полугруппы на ультраметрических пространствах

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure \mu on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,\mu). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

Some corollaries of the correspondence between partial metrics and multivalued equalities

Multivalued equalities originally arose in the context of the theory of sheaves, while partial metrics originally arose in the context of the theory of domains for denotational semantics. It turns out that multivalued equalities and partial metrics are closely related: one can either consider them equivalent up to the choice of dual notation, or one can formally capture the duality between logical and metric viewpoints by explicitly requiring logical values and distances to be represented by dual structures. Not only does this allow the transfer of the ideas and results between these two fields, but the most interesting situations are those of interplay between logical and metric considerations. The computational understanding of partial metrics as upper bounds and of multivalued equalities as lower bounds is discussed, together with the lower bound counterparts to partial metrics and the corresponding upper bound counterparts to multivalued equalities. It is shown that separated pre-sheaves of sets and functions over complete Heyting algebras can be understood as the pre-sheaves of ultrametrics valued in the corresponding Brouwerian algebras and non-expansive maps of those ultrametrics. It is proposed that the natural logical counterparts for partial metrics valued in non-negative reals are multivalued equalities valued in the quantale of non-positive reals. The issue of canonical partial metrics on higher-order and reflexive Scott domains is revisited. The intuition behind strong triangularity (“Vickers form”), the computational version of strong triangularity, and weighted quasi-metrics are discussed in the context of quantaloid enrichment.

Statistical Inferences and Applications of the Half Exponential Power Distribution

We investigate the statistical inferences and applications of the half exponential power distribution for the first time. The proposed model defined on the nonnegative reals extends the half normal distribution and is more flexible. The characterizations and properties involving moments and some measures based on moments of this distribution are derived. The inference aspects using methods of moment and maximum likelihood are presented. We also study the performance of the estimators using the Monte Carlo simulation. Finally, we illustrate it with two real applications.

Infinitely divisible distributions arising from first crossing times and related results

Results of Jain and Khan (1979) and Khan and Jain (1978) on the time to first emptiness of a reservoir are generalized to include the case of defective random variables where the mass at ∞ can be positive. The assumption of an underlying exponential family is not needed — the general condition is infinite divisibility and closure under convolutions. The support of the distributions can be nonnegative reals or nonnegative integers. Examples are given to illustrate the general theory and show the bivariate extension.

Kalman Filtering With Intermittent Observations: Weak Convergence to a Stationary Distribution

The paper studies the asymptotic behavior of discrete time Random Riccati Equations (RRE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli independent and identically distributed (i.i.d.) process. We model the RRE as an order-preserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limit-set dichotomy result for order-preserving, strongly sublinear RDS, we establish the asymptotic properties of the RRE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. For stabilizable and detectable systems, stochastic boundedness (and hence weak convergence) holds for any nonzero observation packet arrival probability and, in particular, we can establish weak convergence at operating arrival rates well below the critical probability for mean stability (the resulting invariant measure in that situation does not possess a first moment). We apply the weak-Feller property of the Markov process governing the RRE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semi-definite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure (a.s.) ergodicity of the sample paths to easily compute statistics of the invariant distribution. A one-dimensional example illustrates that the support is a fractured subset of the non-negative reals with self-similarity properties.

Transversals and domination in uniform hypergraphs

Let H = ( V , E ) be a hypergraph with vertex set V and edge set E of order n H = | V | and size m H = | E | . A transversal in H is a subset of vertices in H that has a nonempty intersection with every edge of H . The transversal number τ ( H ) of H is the minimum size of a transversal in H . A dominating set in H is a subset of vertices D ⊆ V such that for every vertex v ∈ V ∖ D there exists an edge e ∈ E for which v ∈ e and e ∩ D ≠ 0̸ . The domination number γ ( H ) is the minimum cardinality of a dominating set in H . A hypergraph H is k -uniform if every edge of H has size k . We establish the following relationship between the transversal number and the domination number of uniform hypergraphs. For any two nonnegative reals a and b and for every integer k ≥ 3 the equality sup H ∈ H k γ ( H ) / ( a n H + b m H ) = sup H ∈ H k − 1 τ ( H ) / ( a n H + ( a + b ) m H ) holds, where H k denotes the class of all k -uniform hypergraphs containing no isolated vertices. As a consequence of this result, we establish upper bounds on the domination number of a k -uniform hypergraph with minimum degree at least 1. In particular, we show that if k ≥ 3 , then γ ( H ) ≤ ( n H + ⌊ k − 3 2 ⌋ m H ) / ⌊ 3 ( k − 1 ) 2 ⌋ for all H ∈ H k , and this bound is sharp. More generally, for k = 2 and k = 3 we prove that all the essential upper bounds can be written in the unified form γ ( H ) ≤ ( a n H + b m H ) / ( a k + b ) for constants b ≥ 0 and a > − b / k .

Graded bundles and homogeneity structures

We introduce the concept of a graded bundle which is a natural generalization of the concept of a vector bundle and whose standard examples are higher tangent bundles T n Q playing a fundamental role in higher order Lagrangian formalisms. Graded bundles are graded manifolds in the sense that we can choose an atlas whose local coordinates are homogeneous functions of degrees 0 , 1 , … , n . We prove that graded bundles have a convenient equivalent description as homogeneity structures, i.e. manifolds with a smooth action of the multiplicative monoid ( R ≥ 0 , ⋅ ) of non-negative reals. The main result states that each homogeneity structure admits an atlas whose local coordinates are homogeneous. Considering a natural compatibility condition of homogeneity structures we formulate, in turn, the concept of a double ( r - tuple, in general) graded bundle–a broad generalization of the concept of a double ( r -tuple) vector bundle. Double graded bundles are proven to be locally trivial in the sense that we can find local coordinates which are simultaneously homogeneous with respect to both homogeneity structures.

A characterization of horizontal visibility graphs and combinatorics on words

A Horizontal Visibility Graph (HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos Luque et al. [B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: exact results for random time series, Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if it is outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics Flajolet and Noy [P. Flajolet, M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204 (1999) 203–229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique, we determine asymptotically the average number of edges of HVGs.

NONBIJECTIVE IDEMPOTENTS PRESERVERS OVER SEMIRINGS

We classify linear maps which preserve idempotents on n◊n matrices over some classes of semirings. Our results include many known semirings like the semiring of all nonnegative integers, the semiring of all nonnegative reals, any unital commutative ring, which is zero divisor free and of characteristic not two (not necessarily a principal ideal domain), and the ring of integers modulo m, where m is a product of distinct odd primes.

First-Fit coloring of bounded tolerance graphs

Let G = ( V , E ) be a graph. A tolerance representation of G is a set I = { I v : v ∈ V } of intervals and a set t = { t v : v ∈ V } of nonnegative reals such that x y ∈ E iff I x ∩ I y ≠ 0̸ and ‖ I x ∩ I y ‖ ≥ min { t x , t y } ; in this case G is a tolerance graph. We refine this definition by saying that G is a p -tolerance graph if t v / | I v | ≤ p for all v ∈ V . A Grundy coloring g of G is a proper coloring of V with positive integers such that for every positive integer i , if i < g ( v ) then v has a neighbor u with g ( u ) = i . The Grundy number Γ ( G ) of G is the maximum integer k such that G has a Grundy coloring using k colors. It is also called the First-Fit chromatic number. For fixed 0 ≤ p < 1 we prove that if G is a p -tolerance graph then, Γ ( G ) = Θ ( ω ( G ) 1 − p ) , and in particular, Γ ( G ) ≤ 8 ⌈ 1 1 − p ⌉ ω ( G ) . Also, we show how restricting p forbids induced copies of K s , s . Finally, we observe that there exist 1-tolerance graphs G with ω ( G ) = 2 and arbitrarily large Grundy number.

Greedy approximation for the source location problem with vertex-connectivity requirements in undirected graphs

Let G=(V,E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v∈V has a demand d(v)∈Z+, and a cost c(v)∈R+, where Z+ and R+ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing ∑v∈Sc(v) such that there are at least d(v) pairwise vertex-disjoint paths from S to v for each vertex v∈V−S. It is known that the problem is not approximable within a ratio of O(ln∑v∈Vd(v)), unless NP has an O(NloglogN)-time deterministic algorithm. Also, it is known that even if every vertex has a uniform cost and d∗=4 holds, then the problem is NP-hard, where d∗=max{d(v)|v∈V}.In this paper, we consider the problem in the case where every vertex has uniform cost. We propose a simple greedy algorithm for providing a max{d∗,2d∗−6}-approximate solution to the problem in O(min{d∗,|V|}d∗|V|2) time, while we also show that there exists an instance for which it provides no better than a (d∗−1)-approximate solution. Especially, in the case of d∗⩽4, we give a tight analysis to show that it achieves an approximation ratio of 3. We also show the APX-hardness of the problem even restricted to d∗⩽4.

Higher vector bundles and multi-graded symplectic manifolds

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F , or, equivalently, from an Euler vector field. This is used in showing that double (or higher) vector bundles present in the literature can be equivalently defined as manifolds with a family of commuting Euler vector fields. Higher vector bundles can be therefore defined as manifolds admitting certain N n -grading in the structure sheaf. Consequently, multi-graded (super)manifolds are canonically associated with higher vector bundles that is an equivalence of categories. Of particular interest are symplectic multi-graded manifolds which are proven to be associated with cotangent bundles. Duality for higher vector bundles is then explained by means of the cotangent bundles as they contain the collection of all possible duals. This gives, moreover, higher generalizations of the known “universal Legendre transformation” T ∗ E ≃ T ∗ E ∗ , identifying the cotangent bundles of all higher vector bundles in duality. The symplectic multi-graded manifolds, equipped with certain homological Hamiltonian vector fields, lead to an alternative to Roytenberg’s picture generalization of Lie bialgebroids, Courant brackets, Drinfeld doubles and can be viewed as geometrical base for higher BRST and Batalin–Vilkovisky formalisms. This is also a natural framework for studying n -fold Lie algebroids and related structures.

Labeling the [formula omitted]-path with a condition at distance two

For integer r ≥ 2 , the infinite r -path P ∞ ( r ) is the graph on vertices … v − 3 , v − 2 , v − 1 , v 0 , v 1 , v 2 , v 3 … such that v s is adjacent to v t if and only if | s − t | ≤ r − 1 . The r -path on n vertices is the subgraph of P ∞ ( r ) induced by vertices v 0 , v 1 , v 2 , … , v n − 1 . For non-negative reals x 1 and x 2 , a λ x 1 , x 2 -labeling of a simple graph G is an assignment of non-negative reals to the vertices of G such that adjacent vertices receive reals that differ by at least x 1 , vertices at distance two receive reals that differ by at least x 2 , and the absolute difference between the largest and smallest assigned reals is minimized. With λ x 1 , x 2 ( G ) denoting that minimum difference, we derive λ x 1 , x 2 ( P n ( r ) ) for r ≥ 3 , 1 ≤ n ≤ ∞ , and x 1 x 2 ∈ [ 2 , ∞ ] . For x 1 x 2 ∈ [ 1 , 2 ] , we obtain upper bounds on λ x 1 , x 2 ( P ∞ ( r ) ) and use them to give λ x 1 , x 2 ( P ∞ ( r ) ) for r ≥ 5 and x 1 x 2 ∈ [ 1 , 2 r − 2 2 r − 3 ] ⋃ [ 4 3 , 2 ] . We also determine λ x 1 , x 2 ( P ∞ ( 3 ) ) and λ x 1 , x 2 ( P ∞ ( 4 ) ) for all x 1 x 2 ∈ [ 1 , 2 ] .

Predicate transformers for extended probability and non-determinism

We investigate laws for predicate transformers for the combination of non-deterministic choice and (extended) probabilistic choice, where predicates are taken to be functions to the extended non-negative reals, or to closed intervals of such reals. These predicate transformers correspond to state transformers, which are functions to conical powerdomains, which are the appropriate powerdomains for the combined forms of non-determinism. As with standard powerdomains for non-deterministic choice, these come in three flavours – lower, upper and (order-)convex – so there are also three kinds of predicate transformers. In order to make the connection, the powerdomains are first characterised in terms of relevant classes of functionals.Much of the development is carried out at an abstract level, a kind of domain-theoretic functional analysis: one considers d-cones, which are dcpos equipped with a module structure over the non-negative extended reals, in place of topological vector spaces. Such a development still needs to be carried out for probabilistic choice per se; it would presumably be necessary to work with a notion of convex space rather than a cone.

Linear Preservers of Perimeters of Nonnegative Real Matrices

Abstract. For a nonnegative real matrix A of rank 1, A can be factored as ab t for somevectors a and b. The perimeter of A is the number of nonzero entries in both a and b. If Bis a matrix of rank k, then B is the sum of k matrices of rank 1. The perimeter of B is theminimum of the sums of perimeters of k matrices of rank 1, where the minimum is takenover all possible rank-1 decompositions of B. In this paper, we obtain characterizations ofthe linear operators which preserve perimeters 2 and k for some k ≥ 4. That is, a linearoperator T preserves perimeters 2 and k(≥ 4) if and only if it has the form T(A) = UAV ,or T(A) = UA t V with some invertible matrices U and V . 1. IntroductionThere is much literature on the study of linear operators that preserve theranks of matrices over several semirings([1]-[8]). Nonnegative matrices also havebeen the subject of research by many authors([2], [5], [6], [8]). Beasley, Gregoryand Pullman [2] obtained characterizations of linear operators which preserve therank of nonnegative real matrices. In [8], Song and Hwang characterized spanningcolumn ranks and their preservers of nonnegative matrices. Beasley, Song, Kangand Sarma [5] treated column ranks of nonnegative real matrices and characterizedtheir preservers.But there are few papers on the characterizations of linear operators preserv-ing the perimeter of matrices. Beasley et al. characterized those linear operatorspreserving the rank and perimeter of Boolean rank-1 matrices([1]).In this paper, we consider the set of linear operators that preserve the perimeterof matrices of rank k(≥ 2) over the nonnegative reals.

Regular matrices and their strong preservers over semirings

An m × n matrix A over a semiring S is called regular if there is an n × m matrix G over S such that AGA = A . We study the problem of characterizing those linear operators T on the matrices over a semiring such that T ( X ) is regular if and only if X is. Complete characterizations are obtained for many semirings including the Boolean algebra, the nonnegative reals, the nonnegative integers and the fuzzy scalars.

Enhanced synchronizability via age-based coupling

In this Brief Report, we study the synchronization of growing scale-free networks. An asymmetrical age-based coupling method is proposed with only one free parameter alpha . Although the coupling matrix is asymmetric, our coupling method could guarantee that all the eigenvalues are non-negative reals. The eigenratio R will approach 1 in the large limit of alpha .