To conclude, we have provided **exact** **results** for strongly repulsive dirty bosons in 1D, which can be ob- tained from a Bose-Fermi mapping to non-interacting dis- ordered fermions. A Bose-glass phase is thereby mapped to an Anderson-localized fermionic phase. A similar map- ping is also available for arbitrary interaction strength, but involves interacting fermions with a non-standard contact interaction [36]. For strong (but finite) repulsive bosonic interactions, the weak fermionic interactions can safely be treated on a perturbative level [13] and cause no substantial differences to our predictions. Finally, other quantities not discussed here can also be inferred, e.g., the crossover from short-time diffusive wave-packet ex- pansion to localized behavior at long times [35].

Let us stress that the many-body Hamiltonian of [1] is the restriction to the N -particle Fock space of the well-known nonlinear Schr¨odinger (NLS) Hamiltonian (see e.g. [13] for a review). The NLS model is one of most studied examples of integrable field theory for which a huge amount of **exact** **results** is known. In the same way, the Hamiltonian of [8] is the counterpart of the NLS model on the half-line whose symmetry is given by the reflection algebra [14], showing the consistency of the approach of [9]. In [14], the concept of boundary algebra [15] was crucial to establish all the properties of NLS on the half-line as an integrable system.

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Some models have sampling or measurement error built into the computer code so that the model output includes a realization of this noise. Rather than coding the noise process into the model, it will sometimes be possible to rewrite the model so that it outputs the latent underlying signal. If the likelihood of the data given the latent signal is computable (as it often is), then it may be possible to analytically account for the noise with the acceptance probability π ε (·). ABC methods have proven most popular in fields such as genetics, epidemiology, and population biology, where a common occurrence is to have data generated by sampling a hidden underlying tree structure. In many cases, it is the partially observed tree structure which causes the likelihood to be intractable, and given the underlying tree the sampling process will have a known distribution. If this is the case (and if computational constraints allow), we can use the probabilistic ABC algorithm to do the sampling to give **exact** inference without any assumption of model error. Note that if the sampling process gives continuous data, then **exact** inference using the rejection algorithm would not be possible, and so this approach has the potential to give a significant improvement over current methods.

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It is said that, above the threshold, the network/graph ‘percolates’ such that a single connected component appears that is of the order of N in size, this being the ‘giant component’ (Newman et al., 2001). Unlike the branching process, we therefore have two interpretations of the probability 1 − S. It is the probability that a randomly selected vertex be connected to a positive fraction of the network, and it is also the relative size of the giant component. Indeed, there is a branch of mathematics named ‘percolation theory’ which addresses the issue of the existence and size of giant components for different random graphs (and the size distribution for non-giant connected components). This kind of analysis was especially simple for configuration networks because of the locally tree like structure. In contrast, starting with an infinite square lattice and then removing every edge independently with probability p **results** in a giant component if and only if p < 1/2 - this took decades of work to prove (see, for example, Grimmett (2010)).

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probability N −n . Thus, each conceivable numerical characteristic of a random allocation becomes a random variable with a speciﬁed probability distribu- tion. Various **exact** and asymptotic distributional **results** on this subject are included in classical textbooks and monographs (see, e.g., Feller [2], Kolchin et al. [4]). If in this model, we let η j denote the number of balls of the j th cell,

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Initially only free bending vibration is considered for validation purposes us- ing different boundary conditions and different thickness and Young’s mod- ulus ratios. The **results** are compared with those in the literature and with finite element **results** obtained by NASTRAN. Subsequently, in-plane free vibration is carried out and the **results** are compared with finite element re- sults rather than **exact** **results** due to their unavailability in the literature for composites. Next, a thicker composite plate which exhibits both in-plane and out of plane vibrational modes between the first twenty is investigated. The **results** obtained by DySAP have been validated against those in the literature and also against those obtained by using Carrera’s Unified Formu- lation. In all cases, excellent agreement was found. Comparison of DySAP **results** against NASTRAN shows that if a sufficiently fine mesh in the latter is used, generally the error made by the finite element is small but it increases for higher frequencies, as expected. It has also been noted that FEM may produce **results** where modes are interchanged.

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We shall consider three data sets that we had formally introduced in Chapter 1. These data sets are from three different applications. The first which we called the Animal Toxicology data is from a study to compare the toxicity of two different chemicals. The second data set, which we are calling the Childhood Nephroblastoma data is from a clinical trial to compare two different kinds of treatment for childhood nephroblastoma. And the third data, which we are calling the Influenza Vaccine data set is from a vaccine trial where patients were randomized to receive either the influenza vaccine, or placebo. We will construct the confidence intervals for the two **exact** intervals and an asymptotic interval. The asymptotic intervals are constructed by assuming that the test statistic Z (see 2.6) follows a standard normal distribution under the null hypothesis. Then one can obtain the large sample confidence intervals for R as follows. We know that

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Where the data recipient is applying newer analytical approaches to shared clinical trial data, resulting publi- cations should compare and contrast back to the **results** obtained from the original clinical trial(s). Possible rea- sons for differences in **results** should be described and differences in assumptions used by the different methods clarified. An important aspect to consider is the reso- lution of conflicting **results** between the original analysis and any new analyses. During drug licencing, differing analyses are taken into consideration by the regulatory authorities in light of which analyses were primary, were pre-specified as secondary or sensitivity analyses, and which were done post-hoc. The final interpretation of benefit-risk is then taken by the drug licensing authority. If a new analysis potentially alters the benefit-risk profile of a drug, perhaps in a newly-investigated subgroup, it is imperative that the methods, assumptions, and possible reasons for important discrepancies are clearly de- scribed. The data holder should be informed of the re- sults, and evaluation of the implications by a licensing authority might sometimes be required.

This study introduces that, by applying Galerkin method to linear second order ordinary differential equations with mixed and Neumann boundary conditions, it is possible to find their approximate solutions. The numerical **results** that are obtained using this method converges to the **exact** solution as the number of Chebyshev polynomial increases, that will be used as a trial function; and also using small step size h , while converting the given linear second order ordinary differential equation from mixed type to Neumann boundary condition, increases the accuracy of the approximate solution. So that using this method better **results** will be obtained as the number of Chebyshev polynomial increases and using small step size while using Rung-Kutta method.

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The aim of the article is to propound a simplest and **exact** definition of mathematics in a single sentence. It is observed that all mathematical and non-mathematical subjects whether science, arts, language or commerce, follow the same steps and roots to develop, they all consist of three parts: assumptions, properties and applications. These three terms make the **exact** definition of Mathematics, which can be applied to all subjects also. Therefore all subjects can be brought under the same umbrella of definition consisting of these three terms. Following this mathematics has been defined as the study of assumptions, its properties and applications. Then different branches of mathematics have been discussed. A short paragraph has been devoted to technical teachers and students on engineering mathematics. In last how should we teach mathematics has been emphasized? A special focus on the type of assignment has been mentioned. This article will be useful for mathematics teachers and its learners, if it is discussed in the first few lectures of undergraduate and post graduate level as well as will be more fruitful for technical students as they can understand and apply it better than non-technical students. Key Words: Axiom, Theorem, Properties, Conjecture.

Abstract. An **exact** truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as **exact** truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for **exact** truthmaking and characterise the resulting notion of entailment, showing that it is compact and decidable. We then investigate the effect of various restrictions on the semantics. We also formulate a sequent-style proof system for **exact** entailment and give soundness and completeness **results**.

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In this paper some aspects of hypermodules are studied. We will show that the cate- gory Hmod is **exact** in the sense that it is normal and conormal with kernels and cok- ernels in which every arrow f has a factorization f = ν q, with ν being a monomor- phism and q an epimorphism (see [6]). Two of the most used **results** of the paper are those which state that the monomorphisms of Hmod (in the categorical sense) are the one- to one-homomorphisms and the epimorphisms of Hmod are the onto homomor- phisms.

Space frame roof supported by 22 columns each placed at 3m c/c on the two opposite perimeter edge. The Finite element linear elastic analysis for two opposite edge supported condition was made using ANSYS and the **results** of central deflection were tabulated as in Table 2.8 and the graph **results** shown in Figure 2.9. Figure 2.8 represent the deflection diagram of 30m x 30m opposite edge supported space frame.

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The optimal control of a system is one of the most practical subjects in science and engineering [5, 6, 17, 21, 25, 33]. The optimal control problems involve minimization of a performance index subject to a dynamical system. As generalizations of the classical optimal control, FOCPs are problems in which fractional derivatives or integrals are used in the performance index or constraints. Similar to the other types of fractional functional equations, most of FOCPs do not have **exact** and analytic solutions [7, 34, 35, 37]. Fur- thermore, a time-delay fractional optimal control problem (TDFOCP) is a fractional optimal control problem in which the dynamic of system contains some time-delay equations. The control of time-delay systems frequently in electronic, age structure, biological, chemical, electronic, and transportation systems [14, 20, 23, 41]. Due to their variety of applications in the realistic models of phenomena, the control of time-delay systems has been investi- gated by many engineers and scientists. Also, considerable attention has been focused on the approximate and numerical solution of them [7,34,35,37]. Generally, there are two main approaches to the approximate solution of TD- FOCPs. The first one is based on constructing a Hamiltonian system and then solve the arising two-point boundary value problem, which is the indirect approach, and the other involves facing directly the problems by discretizing or approximating the functions without constructing the Hamiltonian equa- tions. Recently, more attention has been done to direct approaches [7,22,35]. The main purpose of this work is to introduce a new kind of discrete orthonormal polynomials. Some properties of these discrete polynomials are studied and a general formulation of their operational matrix of fractional integration, in the Riemann–Liouville sense, is derived. Make use of these orthogonal polynomials and their operational matrix, an eﬃcient direct nu- merical method is proposed to approximate the solution of the following TDFOCP:

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1. Introduction. The relative homotopy theory of modules, including the (module) homotopy **exact** sequence, was developed by Peter Hilton and stated in [1, Chapter 13]. The approach in this paper produces an alternative proof of the existence of the injective homotopy **exact** sequence without involving any reference to elements of sets in the arguments, so that one can deﬁne the necessary homotopy concepts in arbitrary abelian categories with enough injectives and projectives, and obtain, automatically, the projective relative homotopy theory as the dual.

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DOI: 10.4236/am.2019.106030 420 Applied Mathematics In the consulted bibliography we have not found any **results** of the application of the HAM to differential problems with discontinuities. For this reason, this paper systematically analyzes its application to IVPs of ODEs of second order with independent non-continuous term. We have treated functions with a dis- continuous derivative, with some of Heaviside step function and with Dirac del- ta function.

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As early as in 1983 the present author introduced the so called “chemical Hamiltonian approach” [10] devoted to connect the Hilbert space description of molecules (the use of atom-centered basis sets) with the genuine chemical concept of atoms exhibiting pairwise interatomic interactions. Already that paper contained the formulae of an energy decomposition scheme; later on it was improved somewhat but these **results** were left unpublished for about a decade. A discussion with Pierre Valiron, however, changed the situation and he convinced the author that these formulae are worth to be programmed. That was done first by using Pierre’s version of the HONDO ab initio program system [11]. The scheme thus obtained [12, 13] was named “Chemical Energy Component Analysis” (CECA). In this approach the energy decomposition is **exact** for diatomic molecules, while the three- and four-center e ﬀ ects are compressed into one- and two-center ones as much as it is possible by performing appro- priate projections. Thus one gets an approximate version of the decomposition Eq. (1): the energy of a molecule calculated at the SCF level is expressed approximately but to a good accuracy as a sum of atomic and diatomic contributions, the computation of which requires the use of one- and two-center integrals only.

The overall bad quality of the self-consistent KS po- tential also compromises the evaluation of post-self- consistency corrections based on more sophisticated methods, like Many-Body Perturbation Theory or time- dependent DFT [3], if the wrong effective potential is not corrected as well. On the other hand, an accurate description of the XC potential is required, for instance, when studying neutral excitations in finite systems us- ing time-dependent DFT [15]. As shown recently by Della Sala and G¨orling [16], for those systems having an HOMO orbital with nodal surfaces, the **exact** exchange potential tends to a constant if going to infinity over a set of zero measure directions. This leads to the appear- ance of potential barriers that, although could be of mi- nor importance when obtaining the self-consistent static **results**, might be essential if a proper description of all the unoccupied KS orbitals was required. Here, although in a very different context, potential barriers induced by the non-locality of the many-body effects in an electron system have been observed as well.

These findings clearly contradict corresponding running time experiments [Wild, 2012, Chap- ter 8], where Yaroslavskiy’s algorithm was significantly faster across implementations and pro- gramming environments. One might object that the poor performance of Yaroslavskiy’s algorithm is a peculiarity of counting Bytecode instructions. Wild [2012, Section 7.1] also gives implementations and analyses thereof in MMIX, the new version of Knuth’s imaginary processor architecture. Every MMIX instruction has well-defined costs, chosen to closely resemble actual execution time on a simple processor. The **results** show the same trend: Classic Quicksort is more efficient. Together with the Bytecode **results** of this paper, we see strong evidence for the following conjecture: Conjecture 5.1. The efficiency of Yaroslavskiy’s algorithm in practice is caused by advanced features of modern processors. In models that assign constant cost contributions to single instruc- tions — i. e., locality of memory accesses and instruction pipelining are ignored — classic Quicksort is more efficient.

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Abstract: This paper presents a centre and edge crack analysis using meshless methods which is based on moving least squares (MLS) approximation. The unknown displacement function u(x) is approximated by moving least square approximation u h (x). These approximation are constructed by using a weight function which is based a monomial basis function and a set of non-constant coefficients. A subdivision that is similar to finite element method is used to provide a background mesh for numerical integration. An enriched EFG formulation with fracture problems is proposed to improve the solution accuracy for linear elastic fracture problem. The essential boundary conditions are enforced by Lagrange multipliers method. A code has been written in Matlab for the analysis of a crack tip. The obtained **results** of the developed EFG-code were compared to available experimental data and other numerical (**exact** methods and finite element method) methods.