1. Introduction. Let d(n) denote the divisor function, **ν**(n) the **number** of distinct prime factors, and Ω(n) the total **number** of prime factors of n, respectively. In 1984 Heath-Brown [4] proved the well-known Erd˝os–Mirsky conjecture [1] (which seemed at one time as hard as the twin prime conjec- ture, cf. [4, p. 141]):

one considers Ꮽ (s) multiplied by a suitable constant. The Laurent coeﬃcients of such **functions**, which frequently occur in analytic **number** theory, were investigated by the author in [10]. The function u(x) is to be considered as the error term in the asymp- totic formula for A(x). For this to hold, it is enough to assume that the mean-square estimate

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The importance of the Theorem-1 lies in the fact that whenever one knows a bilateral gener- ating relation (1.2), the corresponding mixed trilateral generating relation can at once be written down from (1.3). Thus one can get a large **number** of mixed trilateral generating relations from (1.3) by attributing diﬀerent suitable values to a n in (1.2).

In this paper, we investigate the infinitary analogues of such familiar number theoretic functions as the divisor sum function, Euler’s phi function and the Mbbius function... KEY WORDS [r]

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Introduction: One of the most important **functions** in **number** theory is the Euler function ∅(𝒏) first introduced by Euler. Euler (1707 – 1783), is universally considered as one of the greatest mathematicians in history and is certainly the most prolific among them. His researches covered almost all the fields of mathematics of his time: algebra astronomy, calculus, calculus of variations, finite differences, mechanics, **number** theory and several other topics. An idea of the huge output of mathematical results he discovered will be gained if one considers the fact that it required 60 to 80 large volumes to publish them. His mathematical genius was so penetrating that he discovered great principles in trivial problems. As an example of this he solved the Konigsberg riddle and thereby laid the foundation of that great branch of mathematics called Topology. During the last seventeen years of his life totally blind, but this calamity did not prevent him from continuing his researches with greater vigor and producing results of first importance {[2.2.1] S. G. Telang}.

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In what follows, we suppose that Y is a Delone set in R , with packing radius r and covering radius R. For x ∈ R and ρ > 0, we use B(x, ρ) to denote the closed ball of radius ρ centered at x. The symbol λ denotes Lebesgue measure on R , and card(S) denotes the cardinality of a set S. We also use the standard Vinogradov ≪ notation, so that if f, g : D → R are two **functions** on some common domain D, then f ≪ g

existing graph **theoretic** parameters such as independence **number**, outer connected independence **number**, vertex covering **number**, total domination **number** and connected domination **number**. However there is a wide scope for further research on this concept. Some problems are listed below.

We show that the coverability problem in **ν**-Petri nets is complete for ‘double Ackermann’ time, thus closing an open complexity gap between an Ackermann lower bound and a hyper-Ackermann upper bound. The coverability problem captures the verification of safety properties in this nominal extension of Petri nets with name management and fresh name creation. Our completeness result establishes **ν**-Petri nets as a model of intermediate power among the formalisms of nets enriched with data, and relies on new algorithmic insights brought by the use of well-quasi-order ideals.

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Basic to the game theoretic approach is the assumption that each bargainer possesses a yon Neumann-Morgenstern utility function.2 Given these utility functions it is usual to treat the b[r]

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where, for types A, B, A ↔ B def = (A → B)×(B → A). Provided that T does not have any additional rules for forming the types of ML the converse implication to part 1 holds. For part 2, the rule (PaT) can be used to prove (AC) as in Martin-L¨ of’s original proof of the type **theoretic** axiom of choice in his type theory and the rule ( 0 ⊥) is derived using the instance of (PaT) when φ is ⊥. For the other direction of part 2, (PaT) is proved by induction on the formation of the formula φ. The rule ( 0 ⊥) is needed to deal with ⊥ and (AC) is needed to deal with the implication and universal quantification cases.

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The DFT becomes very a ttra c tiv e to use as i t can be implemented e f f ic ie n t ly using the Fast Fourier Transform (FFT) type algorithm [1 5 ]. The two main disadvantages associated with the FFT are the m u ltip lic atio n by irra tio n a l co efficients and the inherent **number** growth. Both o f the above introduce truncation and/or round-off errors when implemented on a f in it e wordlength machine.

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As application, we give the Heisenberg uncertainty inequality for functions in Lq,2,ν space and the Hardy’s inequality which give an information about how a function and its generalized [r]

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The main sequence of steps is stated as follows: (i) Assuming that the ground-state wave function is known in the first-quantized form, find its thin-torus pattern (this step has been frequently discussed in the literature; for completeness, we provide a brief summary in Appendix C). (ii) Using the formalism developed in the previous sections, write down the parent Hamiltonian corresponding to the root pattern in the second-quantized form. (iii) Use numerics (exact diagonalization) to verify that the ground state of the proposed Hamiltonian is indeed given by the initial wave function. The verification criteria include testing for the unique zero-energy ground state at the given filling factor, the correct thin-cylinder root patterns as the circumference of the cylinder is taken to zero, and the level counting of entanglement spectra. For the purpose of numerical calculations, we place a finite total **number** of particles N on the surface of a torus or an open cylinder. The two linear dimensions of the Hall surface ( L and H ) satisfy the relation LH ¼ 2πl 2 B N orb , where N orb is the

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The Parabolic Sub-Layer approach (PSL) developed by Iacovides & Launder For buoyant flows with simple flow geometry may be used. The PSL treatment is similar to the low-Reynolds-**number** approach where a fine grid resolution is used in the near wall region. The main difference is that the static pressure distribution is assumed to be uniform across near-wall region; therefore, the pressure-correction algorithm is not solved in the near-wall cells. Instead, the wall normal velocity is calculated from continuity. This method gives substantial savings and has the same level accuracy as the full low-Reynolds-**number** formulation, but it is still having some instability for flows in complex geometries and consequently cannot be measured as general.

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Abstract. Multiplicative complexity is a complexity measure defined as the minimum **number** of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementa- tions of ciphers with a small **number** of AND gates are preferred in pro- tocols for fully homomorphic encryption, multi-party computation and zero-knowledge proofs. In 2002, Fischer and Peralta [12] showed that the **number** of n-variable Boolean **functions** with multiplicative complexity one equals 2 2 3 n . In this paper, we study Boolean **functions** with multi- plicative complexity 2. By characterizing the structure of these **functions** in terms of affine equivalence relations, we provide a closed form formula for the **number** of Boolean **functions** with multiplicative complexity 2. Keywords:Affine equivalence; Boolean **functions**; Cryptography; Mul- tiplicative complexity; Self-mappings

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one). For the applications of this class of special **functions** in the solutions of fractional order differential equations and models, see e.g. in Kiryakova and Luchko [12]. The survey by Kilbas et al. [5] describes the historical development of the theory of these multi-index (2m-parametric) Mittag-Leffler **functions** as a subclass of the Wright generalized hypergeometric **functions** p Ψ q (z). The

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As it is well known, the Henstock (H) integral for a real function was first defined by Henstock [1] in 1963. The Henstock (H) integral is a lot powerful and easier than the Lebesgue, Wiener and Richard Phillips Feynman integrals. Furthermore, it is also equal to the Denjoy and the Perron integrals [1] [2]. In 2000, Congxin Wu and Zengtai Gong [3] introduced the notion of the Henstock (H) integrals of interval-valued **functions** and fuzzy- **number**-valued **functions** and obtained a **number** of their properties. In 2016, Yoon [4] introduced the interval- valued Henstock-Stieltjes integral on time scales and investigated some properties of these integrals. In 1998, Lim et al. [5] introduced the notion of the Henstock-Stieltjes (HS) integral of real-valued function which was a generalization of the Henstock (H) integral and obtained its properties.

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The paper is organized as follows. In Section 2, we have a tendency to provide the preliminary terminology used in this paper. Section 3 is dedicated to discussing the AP-Henstock integral of interval-valued **functions**. In Section 4, we introduce the AP- Henstock integral of fuzzy-**number**-valued **functions**. The last section provides conclu- sions.

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Unpublished CS-RR-065 Permanent WRAP url: http://wrap.warwick.ac.uk/60765 Copyright and reuse: The Warwick Research Archive Portal WRAP makes this work by researchers of the University o[r]

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Note that all the data we gathered here is independent of the order of the commits. At this point we did not calculate the **number** of **functions** deleted between two commits, but rather the list of **functions** that existed in each commit. Because of this, and because of how git source control works, we could technically compare commits which were not actually developed one after the other and see whether or not the tool could predict a regression, and validate whether the performance was in fact different from commit to commit. We did not take advantage of this for the most part, but we discuss why this property is useful in the Threats and Limitations section of the paper.

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