# Top PDF On the number-theoretic functions ν(n) and Ω(n)

### On the number-theoretic functions ν(n) and Ω(n)

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]):

### Estimates of convolutions of certain number theoretic error terms

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

### On Some Mixed Trilateral Generating Functions of Modified Jacobi Polynomials by Group Theoretic Method

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).

### Arithmetic functions associated with infinitary divisors of an integer

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]

### A REMARKABLE OBSERVATION ON NUMBER THEORETIC FUNCTION ∅(𝒏)

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}.

### A measure theoretic result for approximation by Delone sets

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

### Restrained Independence In Graphs

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.

### The complexity of coverability in ν Petri nets

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.

### The Theory of Bargaining: A Selective Survey with Particular Reference to Union Employer Negotiations and the Occurence of Strikes

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]

### Collection Principles in Dependent Type Theory

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.

### A read-only-memory oriented implementation of the number theoretic transform butterfly unit.

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.

### 3. Generalized q-Bessel operator

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]

### Geometric Construction of Quantum Hall Clustering Hamiltonians

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

### Numerical Investigation on the Performance of Various Wall Functions on Heat Transfer Enhancement

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.

### The Number of Boolean Functions with Multiplicative Complexity 2

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

### A SURVEY ON BESSEL TYPE FUNCTIONS AS MULTI-INDEX MITTAG-LEFFLER FUNCTIONS: DIFFERENTIAL AND INTEGRAL RELATIONS

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

### On Henstock Stieltjes Integrals of Interval Valued Functions and Fuzzy Number Valued Functions

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.

### On AP Henstock Integrals of Interval Valued Functions and Fuzzy Number Valued Functions

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.

### Inequalities for the number of monotonic functions of partial orders

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]

### An Empirical Evaluation of the Indicators for Performance Regression Test Selection

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.