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A novel bivariate interval function decomposition method is proposed and applied to predict the bounds of structural response with large-range interval parameters. When the existing interval methods solve large uncertainty problems, either the calculation accuracy is poor or better accuracy is often achieved at the cost of more computational effort. To overcome this drawback, the bivariate interval function decomposition (BIFD) is first constructed for the approximation of the original response function. The univariate and the bivariate points are substituted into the second-order Taylor expansion to derive BIFD; thus, the expression of BIFD contains only the one- and two-dimensional functions. Particularly, the response function is decomposed into the sum of multiple low-dimensional functions, and solving the bounds of multi-dimensional original response can be transformed into solving those of low-dimensional interval functions. Then, the sensitivity information of structural response with respect to uncertain parameters is utilized to save computational consumption. Finally, the precision and effectiveness of the method are validated by comparing it with the other six existing interval analysis methods through several numerical examples and engineering applications.

RESUMO: No presente artigo, questionamos a função exercida por um espaço de acolhimento para crianças de até três anos e seus pais. Após uma breve descrição desse dispositivo inspirado no modelo francês da Maison Verte, criada por Françoise Dolto, discutimos o funcionamento de diferentes instituições da primeira infância e interrogamos o papel do espaço de acolhimento. Defendemos, por fim, que ele ocupa uma função denominada de intervalo, pois permite a introdução de um espaço possível no laço entre pequenas crianças e seus pais, favorecendo, assim, a separação e a emergência do sujeito do desejo.

Betweenness is a universal notion present in several disciplines of mathematics. The notion of betweenness has a profound history and many pioneers like Euclid, Pasch, Hilbert have studied betweenness axiomatically. In discrete mathematics too, betweenness is present and several authors have worked on this concept from an axiomatic view. In graph theory, betweenness is developed mainly as metric betweenness, studied using the shortest path metric in a connected graph, thus resulting in the notion of the interval function. Many interesting results are available in graph theory using the interval function. The interval function is generalized to induced path function by replacing shortest paths by induced paths. The induced path betweenness also captured attention among graph theorists with several interesting results to date. From an axiomatic point of view, two pertinent questions can be framed on these functions. Is it possible to axiomatically characterize the interval function of some special graphs using some set of first order axioms defined on an arbitrary transit function? Is it possible to characterize the graphs with the help of their interval functions? In this paper, we survey the results as answers to these questions available from the research papers. Keywords: Betweenness, Interval function, Induced path function

In this note we extend the Mulder-Nebeský characterization of the interval function of a connected graph to the disconnected case. One axiom needs to be adapted, but also a new axiom is needed in addition.

Intervals and convexity play crucial roles in the applications of graph theory such as town planning and design of graphics. In this article, the concept of geodetic interval in graphs is extended to fuzzy graphs. Intervals are useful in the study of properties of fuzzy graphs which depend on the geodetic distance between vertices. The axiomatic definition of intervals in fuzzy graphs are used to define intervals in different fuzzy graph structures like fuzzy trees and complete fuzzy graphs. Finally a set theoretic operations of intervals like union, intersection are also discussed and some results are obtained.

this article describes one of options of the main objective of external ballistics under the interval non-determination of parameters. It is supposed that initial data has been set inaccurately, i.e. their possible values belong to some interval. The algorithm of the solution of an objective within the interval analysis has been developed. The evaluation of the interval decision width has been obtained.

In 2-neighbourhood bootstrap percolation on a graph G , an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already infected neighbours become infected. Percolation occurs if eventually every vertex is infected. In this paper, we are interested to calculate the maximal time t ( G ) the process can take, in terms of the number of times the interval function is applied, to eventually infect the entire vertex set. We prove that the problem of deciding if t ( G ) ? k is NP-complete for: (a) fixed k ? 4 ; (b) bipartite graphs and fixed k ? 7 ; and (c) planar graphs. Moreover, we obtain linear and polynomial time algorithms for trees and chordal graphs, respectively.

в данной статье рассматривается один из вариантов основной задачи внешней баллистики в условиях интервальной недетерминированности параметров. Предполагается, что начальные данные заданы неточно, т.е. их возможные значения принадлежат к некоторому интервалу. Разработан алгоритм решения поставленной задачи в рамках интервального анализа и получены оценки ширины интервального решения., this article describes one of options of the main objective of external ballistics under the interval non-determination of parameters. It is supposed that initial data has been set inaccurately, i.e. their possible values belong to some interval. The algorithm of the solution of an objective within the interval analysis has been developed. The evaluation of the interval decision width has been obtained.

Context. In this paper we propose the dedetermination as the new method designed to solving a problem of calculation of deterministicfunctions with the so-called singular points where the function does not take a certain value.Objective. The approach is developed that allows for division by zero and thus exclude singular points of functions.Method. The method proposed in this article is to move from the problematic (from point of view of calculating) determined function to the corresponding not determined (interval) function by replacing determined function parameters by corresponding interval parameters.Due to this change values of the function at the singular points will be well-defined interval and values.Results. The latter allows you to solve the problem of calculating the function. For the simplified by cutting out interval function theeffective formulas are derived based on main provisions of interval mathematics and make it easy to calculate value of this function. Theproposed in the article approach to the problem of calculating functions with singular points is important for all those classes of systems inwhich the problem really exists. It is about the systems which functions have any number of specific points. Such systems are found mostly intelemetry, reliability theory and practice, humanitarian and others areas. The features of these areas is that they do not always apply theclassical methods of deterministic mathematics. This leads us to search for new approaches to solving problems that arise here.Conclusions. The solution to this problem is achieved by legalization division by zero by intervalization of calculations. It uses theprinciple of cutting out a neighborhood of zero in the interval being the denominator of the fraction representing studied function., Актуальность. При моделировании организационно-технических систем в ряде случаев возникают сложности в исследованиифункционирования таких систем, если они формализованы на основе аналитико-детерминированных функций. В работе предложенновый метод – раздетерминизация, созданный для решения проблемы вычисления детерминированных функций, имеющих так называ-емые особые точки, в которых у функции не существует определенного значения.Цель. Целью является работка подхода, позволяющего осуществлять деление на нуль и тем самым исключать особые точки функций.Метод. Предложенный в статье метод заключается в переходе от проблематичной, с точки зрения вычисления, детерминированнойфункции к соответствующей недетерминированной, а именно, интервальной функции, путем замены детерминированных параметров функции соответствующими интервальными параметрами. Благодаря этой замене значения функции в особых точках становятсяинтервальными и вполне определенными значениями. Последнее и позволяет решить проблему вычисления функции.Результат. Путем вырезания интервальной функции выведены рабочие формулы, основанные на основных положенияхинтервальной математики и позволяющие легко вычислять значения этой функции. Предложенный в статье подход к решениюпроблемы вычисления функций с особыми точками имеет важное значение для всех классов прикладных систем, в которых этапроблема реально существует. Речь здесь идет о тех системах, функции-характеристики которых имеют некоторое число особыхточек. Такие системы встречаются чаще всего в телеметрии, теории и практике надежности, гуманитарной сфере и ряде другихобластей. Особенности этих областей в том, что в них не всегда применимы классические методы детерминистской математики, чтопобуждает разрабатывать новые подходы к решению возникающих здесь задач.Выводы. Решение проблемы вычисления функции достигается легализацией деления на нуль путем интервализации вычислений.При этом используется принцип вырезания окрестности нуля из интервала, являющегося делителем интервальной дроби,представляющей исследуемую функцию.

Let I G ( x , y ) and J G ( x , y ) be the geodesic and induced path intervals between x and y in a connected graph G , respectively. The following three betweenness axioms are considered for a set V and R : V × V → 2 V : (i) x ∈ R ( u , y ) , y ∈ R ( x , v ) , x ≠ y , | R ( u , v ) | > 2 ⇒ x ∈ R ( u , v ) ; (ii) x ∈ R ( u , v ) ⇒ R ( u , x ) ∩ R ( x , v ) = { x } ; (iii) x ∈ R ( u , y ) , y ∈ R ( x , v ) , x ≠ y ⇒ x ∈ R ( u , v ) . We characterize the class of graphs for which I G satisfies (i), the class for which J G satisfies (ii) and the class for which I G or J G satisfies (iii). The characterization is given in terms of forbidden induced subgraphs. It turns out that the class of graphs for which I G satisfies (i) is a proper subclass of distance hereditary graphs and the class for which J G satisfies (ii) is a proper superclass of distance hereditary graphs. We also give an axiomatic characterization of chordal and Ptolemaic graphs.

SUMMARY First, this paper derives the prefix sum-of-products expression (PreSOP) and the number of products in a PreSOP for an interval function. Second, it derives Ψ(n ,τ p), the number of n-variable interval functions that can be represented with τp products. Finally, it shows that more than 99.9% of the n-variable interval functions can be represented with � 3n − 1� products, when n is sufficiently large. These results are use

We can consider the Riemann-Stieltjes integral $$ \int\limits_a^b f $$ dg as an integral of a point function f with respect to an interval function g. We could extend it to the Henstock-Stieltjes integral. In this paper, we extend it to a generalized Stieltjes integral $$ \int\limits_a^b f $$ dg of a point function f with respect to a function g of divisions of an interval. Then we prove for this integral the standard results in the theory of integration, including the controlled convergence theorem.

This paper introduces the theory of ϕ-Jensen variance. Our main motivation is to extend the connotation of the analysis of variance and facilitate its applications in probability, statistics and higher education. To this end, we first introduce the relevant concepts and properties of the interval function. Next, we study several characteristics of the log-concave function and prove an interesting quasi-log concavity conjecture. Next, we introduce the theory of ϕ-Jensen variance and study the monotonicity of the interval function $\operatorname{JVar}_{\phi}\varphi ( {{X _{ [ {a,b} ] }}} )$ by means of the log concavity. Finally, we demonstrate the applications of our results in higher education, show that the hierarchical teaching model is ‘normally’ better than the traditional teaching model under the appropriate hypotheses, and study the monotonicity of the interval function $\operatorname{Var} \mathscr{A} (X _{ [ {a,b} ] } )$ .

Let G be a (finite) connected graph. Intervals and steps in G are objects that depend on the distance function d of G. If u,v∈V(G), then by the u–v interval in G we mean the set{x∈V(G);d(u,x)+d(x,v)=d(u,v)}.By the interval function of G we mean the mapping I of V(G)×V(G) into the power set of V(G) such that I(u,v) is the u–v interval of G. By a step in G we mean an ordered triple (u,v,w) where u,v,w∈V(G), d(u,v)=1 and d(v,w)=d(u,w)−1. A characterization of the interval function of G and a characterization of the set of all steps in G were published by this author in 1994 and 1997, respectively.This paper is a review of author's results on intervals and steps in a connected graph. Some small results or short proofs are new.

For the interval function [f]: I ( R n )→ I ( R n ) defined by [f]([x])=[A][x]+[b],|[A]| irreducible with ρ(|[A]|)⩾1, we derive necessary and sufficient criteria for the existence and uniqueness of fixed points [x]*. In addition, we show how [x]* can be represented by means of the input data [A] and [b].

1.1.1 Example 1 shows aspects of an imaginary passage for strings (bass clef) and winds (treble clef). The strings (at a piano dynamic, making frequent bow changes) sustain an "Ode to Napoleon" hexachord, labeled "OTN" on the example. The winds (playing louder) make figuration based on forms of a hexachord H, bracketed on the example, namely H itself, an inverted form h, the complement H* of H, the Tl 1-transpose of h, and so forth. When we explore the pc-intervals spanned between notes of OTN and notes of H, an interesting feature is manifest: There are three ways to span interval 0 between OTN and H, namely F-to-F, G#-to-Ab, and C#-to-Db. There are three ways to span interval 1 between OTN and H, namely C-to-Db, C#-to-D, and E-to-F. There are three ways to span interval 2 between OTN and H, namely C-to-D, F-toG, and C#-to-E. There are three ways to span interval 3 between OTN and H, namely C-to-Eb, F-to-Ab, and E-to-G. There are three ways to span interval 4 between OTN and H, namely A-to-Db, C#-to-F, and E-toAb. There are three ways to span interval 5 between OTN and H, namely A-to-D, C-to-F, and G#-to-Db. There are three ways to span interval 6 between OTN and H, namely A-to-Eb, G#-to-D, and C#-to-G. There are

It is shown that the concept of bounded variation in the mean is not a meaningful generalization of ordinary bounded variation. In fact, it is a characterization of functions which differ from functions of bounded variation on a zero set. Let f be a real-valued function in L1 on the circle group T. We define the corresponding interval function by f (I) = f (b) -f (a), where I denotes the interval [a, b]. Let 0 = to < tj < ... < t4 = 27r be a partition of [0, 27r], and Ikx = [x + tkl-, X+tk]. If ,n Vm(f) =sSUP{j E If(Ikx) I dx} < oo, Tk=1 where the supremum is taken over all partitions, then f is said to be of bounded variation in the mean (or of bounded variation in the L1 norm). We denote the class of all functions which are of bounded variation in the mean by BVM. This concept was introduced by Moricz and Siddiqi [MS], who investigated the convergence in the mean of the partial sums of S[f], the Fourier series of f. If f is of bounded variation (f E BV) with variation V(f, T), then

One of the main sources of destruction during earthquake is resonance. Therefore, the following idea has been proposed. We design special control linkages between floors that are normally unattached to the building but can be attached if necessary. They are so designed that adding them changes the building's characteristic frequency. We continuously monitor displacements within the structure, and when they exceed specified limits, the linkages are engaged in a way to control structural motion. This idea can also be applied to avoid vibrational destruction of large aerospace structures.

We provide a proof of Sholander's claim (Trees, lattices, order, and betweenness, Proc. Amer. Math. Soc. 3, 369-381 (1952)) concerning the representability of collections of so-called segments by trees, which yields a characterization of the interval function of a tree. Furthermore, we streamline Burigana's characterization (Tree representations of betweenness relations defined by intersection and inclusion, Mathematics and Social Sciences 185, 5-36 (2009)) of tree betweenness and provide a relatively short proof., 8 pages

The aim of this note is to show that for a given continuous function $F$ on a set $E \subset \R,$ the associated interval function need not be continuous. We also give an example to show that the associated interval function can be continuous even if $F$ is not continuous.

Definitions and integrals are of the subdivision-refinement type, and functions are from R × R R \times R to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm | ⋅ | | \cdot | with respect to which N is complete and | 1 | = 1 |1| = 1 . If G is a function from R × R R \times R to N, then G ∈ O M ∗ G \in O{M^\ast } on [a, b] only if (i) x Π y ( 1 + G ) _x{\Pi ^y}(1 + G) exists for a ≤ x > y ≤ b a \leq x > y \leq b and (ii) if ε > 0 \varepsilon > 0 , then there exists a subdivision D of [a, b] such that, if { x i } i = 0 n \{ {x_i}\} _{i = 0}^n is a refinement of D and 0 ≤ p > q ≤ n 0 \leq p > q \leq n , then \[ | x p ∏ x q ( 1 + G ) − ∏ i = p + 1 q ( 1 + G i ) | > ε ; \left |{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod \limits _{i = p + 1}^q {(1 + {G_i})} \right | > \varepsilon ; \] and G ∈ O M ∘ G \in O{M^ \circ } on [a, b] only if (i) x Π y ( 1 + G ) _x{\Pi ^y}(1 + G) exists for a ≤ x > y ≤ b a \leq x > y \leq b and (ii) the integral ∫ a b | 1 + G − Π ( 1 + G ) | \smallint _a^b|1 + G - \Pi (1 + G)| exists and is zero. Further, G ∈ O P ∘ G \in O{P^ \circ } on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if { x i } i = 0 n \{ {x_i}\} _{i = 0}^n is a refinement of D and 0 > p ≤ q ≤ n 0 > p \leq q \leq n , then | Π i = p q ( 1 + G i ) | > B |\Pi _{i = p}^q(1 + {G_i})| > B . If F and G are functions from R × R R \times R to N, F ∈ O P ∘ F \in O{P^ \circ } on [a, b], each of lim x , y → p + F ( x , y ) {\lim _{x,y \to {p^ + }}}F(x,y) and lim x , y → p − F ( x , y ) {\lim _{x,y \to {p^ - }}}F(x,y) exists and is zero for p ∈ [ a , b ] p \in [a,b] , each of lim x → p + F ( p , x ) , lim x → p − F ( x , p ) , lim x → p + G ( p , x ) {\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x) and lim x → p − G ( x , p ) {\lim _{x \to {p^ - }}}G(x,p) exists for p ∈ [ a , b ] p \in [a,b] , and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) F + G ∈ O M ∗ F + G \in OM^\ast on [a, b], (2) F ∈ O M ∗ F \in OM^\ast on [a, b], and (3) G ∈ O M ∗ G \in OM^\ast on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1) F + G ∈ O M ∘ F + G \in OM^\circ on [a, b], (2) F ∈ O M ∘ F \in OM^\circ on [a, b], and (3) G ∈ O M ∘ G \in OM^\circ on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].

A formal interval system (FIS) is an ordered triple (THINGS, IVLS, int), where THINGS is a set, IVLS is a mathematical group, and int is a function from THINGS × THINGS to IVLS satisfying three conditions: (1) from all r,s, and t in THINGS, int (r,t) = int (r,s)int(s,f) [group product in IVLS]; (2) for all s and t in THINGS, int(t,s) = int(s,t) [group inverse in IVLS]; (3) for every s in THINGS and every i in IVLS, there exists a unique t in THINGS satisfying the equation int(s,t) = i. The FIS is a useful general model for our intuitions about "intervals" between "things" in many specific musical contexts. A "time-span" is an ordered pair (a,x), where a is a real number and x is a positive real. This pair is meant to model a temporal event that begins a units of time after (or – a before) some referential moment and extends x units of time therefrom. A change of referential time-unit and a change of referential moment relabel the events (a, x) and (b, y) as events (au + m,xu) and (bu + m,yu). We seek a FIS with time-spans for its THINGS whose interval function is invariant under such transformations: int((au + m,xu), (bu + m,yu)) = int((a,x),(b,y)). There is in fact exactly one such FIS, up to isomorphism in the pertinent sense. This FIS is discussed and explored.

The effect of varying trials per day conditions on the CS-US interval or interstimulus interval (ISI) function in rabbit nictitating membrane response conditioning was studied in two experiments. Experiment 1 showed that a 1250-msec ISI was more effective than a 250-msec ISI when there was 1 trial/day. Experiment 2 showed that as the number of trials per day decreased from 20 to 1, the superiority of the 250-msec ISI group over the 1250-msec ISI group declined, with a reversal at 1 trial/day. Results are interpreted in terms of the role of a hypothesized CS-elicited short-duration orienting response in CR performance.

We extend, to a certain class of differentiation bases, some results on the variational measure and the δ-variation obtained earlier for the full interval basis. In particular the theorem stating that the variational measure generated by an interval function is σ-finite whenever it is absolutely continuous with respect to the Lebesgue measure is extended to any Busemann–Feller basis.

Let V be a finite nonempty set. In this paper, a road system on V (as a generalization of the set of all geodesics in a connected graph G with V(G)=V) and an intervaloid function on V (as a generalization of the interval function (in the sense of Mulder) of a connected graph G with V(G)=V) are introduced. A natural bijection of the set of all intervaloid functions on V onto the set of all road systems on V is constructed. This bijection enables to deduce an axiomatic characterization of the interval function of a connected graph G from a characterization of the set of all geodesics in G.