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## Real Numbers (real + number)
## Selected Abstracts## Which truth values in fuzzy logics are definable? INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 10 2003Hung T. NguyenIn fuzzy logic, every word or phrase describing uncertainty is represented by a real number from the interval [0, 1]. There are only denumerable many words and phrases and continuum many real numbers; thus, not every real number corresponds to some common sense degree of uncertainty. In this article, for several fuzzy logics, we describe which numbers are describing such degrees, i.e., in mathematical terms, which real numbers are definable in the corresponding fuzzy logic. © 2003 Wiley Periodicals, Inc. [source] ## On the sample-complexity of ,, identification INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 7 2001S. R. VenkateshAbstract In this paper we derive the sample complexity for discrete time linear time-invariant stable systems described in the ,, topology. The problem set-up is as follows: the ,, norm distance between the unknown real system and a known finitely parameterized family of systems is bounded by a known real number. We can associate, for every feasible real system, a model in the finitely parameterized family that minimizes the ,, distance. The question now arises as to how long a data record is required to identify such a model from noisy input,output data. This question has been addressed in the context of l1, ,2 and several other topologies, and it has been shown that the sample-complexity is polynomial. Nevertheless, it turns out that for the ,, topology the sample-complexity in the worst case can be infinite. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Phylogenetic analysis indicates transmission of hepatitis C virus from an infected orthopedic surgeon to a patient JOURNAL OF MEDICAL VIROLOGY, Issue 4 2002R. Stefan RossAbstract During recent years, a controversial discussion has emerged in the medical community on the real number and possible public health implications of hepatitis C virus (HCV) transmissions from infected medical staff to susceptible patients. We report here on molecular virological and epidemiological analyses involving 229 patients who underwent exposure-prone operations by an HCV-infected orthopedic surgeon. Of the 229 individuals affected, 207 could be tested. Three were positive for HCV antibodies. Molecular and epidemiological investigation revealed that two of them were not infected by the surgeon. The third patient, a 50-year-old man, underwent complicated total hip arthroplasty with trochanteric osteotomy. He harbored an HCV 2b isolate that in phylogenetic analysis of the hypervariable region 1 (HVR 1) was closely related to the HCV strain recovered from the infected surgeon, indicating that HCV-provider-to-patient transmission occurred intraoperatively. To our knowledge, this is the first documented case of HCV transmission by an orthopedic surgeon. The recorded transmission rate of 0.48% (95% confidence interval: 0.09,2.68%) was within the same range reported previously for the spread of hepatitis B virus during orthopedic procedures. Since the result of our investigation sustains the notion that patients may contract HCV from infected health-care workers during exposure-prone procedures, a series of further retrospective exercises is needed to assess more precisely the risk of HCV provider-to-patient transmission and to delineate from these studies recommendations for the guidance and management of HCV-infected medical personnel. J. Med. Virol. 66:461,467, 2002. © 2002 Wiley-Liss, Inc. [source] ## Semilinear wave equation with time dependent potential MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 18 2004Nicola ViscigliaAbstract We consider the following semilinear wave equation: (1) for (t,x) , ,t × ,. We prove that if the potential V(t,x) is a measurable function that satisfies the following decay assumption: ,V(t,x),,C(1+t)(1+,x,) for a.e. (t,x) , ,t × , where C, ,0>0 are real constants, then for any real number , that satisfies there exists a real number ,(f,g,,)>0 such that the equation has a global solution provided that 0<,,,(f,g,,). Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Detecting and creating oscillations using multifractal methods MATHEMATISCHE NACHRICHTEN, Issue 11 2006Stéphane SeuretAbstract By comparing the Hausdorff multifractal spectrum with the large deviations spectrum of a given continuous function f, we find sufficient conditions ensuring that f possesses oscillating singularities. Using a similar approach, we study the nonlinear wavelet threshold operator which associates with any function f = ,j ,kdj,k,j,k , L2(,) the function series ft whose wavelet coefficients are dtj,k = dj,k1, for some fixed real number , > 0. This operator creates a context propitious to have oscillating singularities. As a consequence, we prove that the series ft may have a multifractal spectrum with a support larger than the one of f . We exhibit an example of function f , L2(,) such that the associated thresholded function series ft effectively possesses oscillating singularities which were not present in the initial function f . This series ft is a typical example of function with homogeneous non-concave multifractal spectrum and which does not satisfy the classical multifractal formalisms. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## The admission of students to UK Dental Schools , Recent trends (1983,1998) EUROPEAN JOURNAL OF DENTAL EDUCATION, Issue 2 2000R. DuguidDetails of student applications and admissions to UK Dental Schools from 1983 to 1998 have been recorded and analyzed. Trends observed include a rise in the proportion of female dental students, a drop in real numbers of male dental students, a recent drop in the proportion of mature students and an increase in the number of EU and overseas entrant dental students. Some implications of these and other factors on workforce planning in the UK are discussed. [source] ## Fuzzy harmonic mean operators INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 2 2009Zeshui XuHarmonic mean is a conservative average, which is widely used to aggregate central tendency data. In the existing literature, the harmonic mean is generally considered as a fusion technique of numerical data information. In this paper, we investigate the situations in which the input data are expressed in fuzzy values and develop some fuzzy harmonic mean operators, such as fuzzy weighted harmonic mean operator, fuzzy ordered weighted harmonic mean operator, fuzzy hybrid harmonic mean operator, and so on. Especially, all these operators can be reduced to aggregate interval or real numbers. Then based on the developed operators, we present an approach to multiple attribute group decision making and illustrate it with a practical example. © 2008 Wiley Periodicals, Inc. [source] ## Which truth values in fuzzy logics are definable? INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 10 2003Hung T. NguyenIn fuzzy logic, every word or phrase describing uncertainty is represented by a real number from the interval [0, 1]. There are only denumerable many words and phrases and continuum many real numbers; thus, not every real number corresponds to some common sense degree of uncertainty. In this article, for several fuzzy logics, we describe which numbers are describing such degrees, i.e., in mathematical terms, which real numbers are definable in the corresponding fuzzy logic. © 2003 Wiley Periodicals, Inc. [source] ## Computational verb systems: Verb numbers INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 5 2001Tao YangIn this paper the concepts of (computational) verb number and arithmetical operations of verb numbers are presented. A verb number is a kind of special computational verb, which is derived from structures of (host) verb+(real, interval, fuzzy) number. From the linguistic point of view, a verb number is a computational verb with contexts defined by (real, interval, fuzzy) numbers. Verb numbers can be classified into three basic types based on the outer systems of host verbs. The arithmetic for verb numbers and its rules are presented and proved. If host computational verbs degrade to BE, then verb numbers collapse to real numbers, interval numbers, fuzzy numbers (sets), or other numbers. The set of all verb numbers can be a metric space. The distance between verb numbers can be defined based on the collapses of verb numbers. The cases when verb numbers collapse to triangular fuzzy numbers, trapezoidal fuzzy numbers, and interval numbers are presented and proved. © 2001 John Wiley & Sons, Inc. [source] ## Between ends and fibers JOURNAL OF GRAPH THEORY, Issue 2 2007C. Paul BonningtonAbstract Let , be an infinite, locally finite, connected graph with distance function ,. Given a ray P in , and a constant C , 1, a vertex-sequence is said to be regulated by C if, for all n,,, never precedes xn on P, each vertex of P appears at most C times in the sequence, and . R. Halin (Math. Ann., 157, 1964, 125,137) defined two rays to be end-equivalent if they are joined by infinitely many pairwise-disjoint paths; the resulting equivalence classes are called ends. More recently H. A. Jung (Graph Structure Theory, Contemporary Mathematics, 147, 1993, 477,484) defined rays P and Q to be b-equivalent if there exist sequences and VQ regulated by some constant C , 1 such that for all n,,; he named the resulting equivalence classes b-fibers. Let denote the set of nondecreasing functions from into the set of positive real numbers. The relation (called f-equivalence) generalizes Jung's condition to . As f runs through , uncountably many equivalence relations are produced on the set of rays that are no finer than b -equivalence while, under specified conditions, are no coarser than end-equivalence. Indeed, for every , there exists an "end-defining function" that is unbounded and sublinear and such that implies that P and Q are end-equivalent. Say if there exists a sublinear function such that . The equivalence classes with respect to are called bundles. We pursue the notion of "initially metric" rays in relation to bundles, and show that in any bundle either all or none of its rays are initially metric. Furthermore, initially metric rays in the same bundle are end-equivalent. In the case that , contains translatable rays we give some sufficient conditions for every f -equivalence class to contain uncountably many g -equivalence classes (where ). We conclude with a variety of applications to infinite planar graphs. Among these, it is shown that two rays whose union is the boundary of an infinite face of an almost-transitive planar map are never bundle- equivalent. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 125,153, 2007 [source] ## Estimating red deer Cervus elaphus populations: an analysis of variation and cost-effectiveness of counting methods MAMMAL REVIEW, Issue 3 2006MIKE J. DANIELSABSTRACT 1Different counting methods are currently used to estimate red deer populations in the open range in Scotland, but there are few data available to compare variation in estimates, or relative cost-effectiveness. 2While it is impossible to determine the accuracy of counts (as real numbers are unknown), variation within and between different methods can be measured by repeat counts of the same area within as short a period as possible. 3This study aimed to quantify the variation observed from repeat counts using each of four methods (ground, helicopter, infrared helicopter and dung-counting methods) at one of three study sites in late winters 2003, 2004 and 2005. Additional data from digital camera images of groups from counts in other areas of Scotland were also used to assess the accuracy of visual counts. 4Coefficients of variation (CVs) within any method of between 5% and 16% were recorded, consistent with previous comparisons for red deer open range counts in Scotland. CVs were lowest for ground and helicopter counts. The infrequency of optimal conditions was likely to limit the applicability of infrared counts in Scotland. 5In terms of cost-effectiveness, helicopter counting was the least labour-intensive, with costs of other techniques depending on the availability of existing manpower as an overhead cost. 6It is concluded that helicopter counts are most likely to minimize errors while maximizing cost-efficiency. Accuracy can be improved by the use of digital photography for counting larger deer groups. Estimates are likely to be improved further by increasing the frequency of counts and using the same methods, counters and routes for repeat counts. [source] ## Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2002Yang ZhijianWe consider the blowup of solutions of the initial boundary value problem for a class of non-linear evolution equations with non-linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, ,}, where m, , and p are non-negative real numbers and m+1, ,+1, p+1 are, respectively, the growth orders of the non-linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above-mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non-linear terms, the states of the initial energy and the existence and non-existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd. [source] ## On the point spectrum of ,,2 -singular perturbations MATHEMATISCHE NACHRICHTEN, Issue 1-2 2007Sergio AlbeverioAbstract We prove that for any self-adjoint operator A in a separable Hilbert space , and a given countable set , = {,i}i ,, of real numbers, there exist ,,2 -singular perturbations Ã of A such that , , ,p(Ã). In particular, if , = {,1,,, ,n} is finite, then the operator Ã solving the eigenvalues problem, Ã,k = ,k,k, k = 1,,, n, is uniquely defined by a given set of orthonormal vectors {,k}nk =1 satisfying the condition span {,k}nk =1 , dom (|A |1/2) = {0}. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## A note on the axiomatisation of real numbers, MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 3 2008Thierry CoquandAbstract Is it possible to give an abstract characterisation of constructive real numbers? A condition should be that all axioms are valid for Dedekind reals in any topos, or for constructive reals in Bishop mathematics. We present here a possible first-order axiomatisation of real numbers, which becomes complete if one adds the law of excluded middle. As an application of the forcing relation defined in [3, 2], we give a proof that the formula which specifies the maximum function is not provable in this theory. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Primitive recursive real numbers MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2007Qingliang ChenAbstract In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure , Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if "computable" is replaced by "primitive recursive" (p. r., for short), these definitions lead to a number of different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4-5 2004Andrej BauerAbstract We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to , extends to a sequentially continuous function from X to ,. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between "continuity principles" asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle "all functions from X to , are continuous", when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on "domain realizability", in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On a wave map equation arising in general relativity COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2004Hans RingströmWe consider a class of space-times for which the essential part of Einstein's equations can be written as a wave map equation. The domain is not the standard one, but the target is hyperbolic space. One ends up with a 1 + 1 nonlinear wave equation, where the space variable belongs to the circle and the time variable belongs to the positive real numbers. The main objective of this paper is to analyze the asymptotics of solutions to these equations as t , ,. For each point in time, the solution defines a loop in hyperbolic space, and the first result is that the length of this loop tends to 0 as t,1/2 as t , ,. In other words, the solution in some sense becomes spatially homogeneous. However, the asymptotic behavior need not be similar to that of spatially homogeneous solutions to the equations. The orbits of such solutions are either a point or a geodesic in the hyperbolic plane. In the nonhomogeneous case, one gets the following asymptotic behavior in the upper half-plane (after applying an isometry of hyperbolic space if necessary): 1The solution converges to a point. 2The solution converges to the origin on the boundary along a straight line (which need not be perpendicular to the boundary). 3The solution goes to infinity along a curve y = const. 4The solution oscillates around a circle inside the upper half-plane. Thus, even though the solutions become spatially homogeneous in the sense that the spatial variations die out, the asymptotic behavior may be radically different from anything observed for spatially homogeneous solutions of the equations. This analysis can then be applied to draw conclusions concerning the associated class of space-times. For instance, one obtains the leading-order behavior of the functions appearing in the metric, and one can conclude future causal geodesic completeness. © 2004 Wiley Periodicals, Inc. [source] |