Large Positive Integer Research Articles

On generalized Ramanujan primes

In this paper, we establish several results concerning the generalized Ramanujan primes. For $$n\in \mathbb {N}$$ and $$k \in \mathbb {R}_{> 1}$$ , we give estimates for the $$n$$ th $$k$$ -Ramanujan prime, which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $$k$$ -Ramanujan primes. In addition, we find explicit formulae for certain $$n$$ th $$k$$ -Ramanujan primes. As an application, we prove that a conjecture of Mitra et al. ( arXiv:0906.0104v1 , 2009) concerning the number of primes in certain intervals holds for every sufficiently large positive integer.

On Sidon sets which are asymptotic bases of order $4$

Let h ≥ 2 be an integer. We say that a set A of positive integers is an asymptotic basis of order h if every large enough positive integer can be represented as the sum of h terms from A. A set of positive integers A is called a Sidon set if all the sums a+b with a,b ∈ A, a ≤ b are distinct. In this paper we prove the existence of Sidon set A which is an asymptotic basis of order 4 by using probabilistic methods. 2000 AMS Mathematics subject classification number: primary: 11B13, secondary: 11B75.

Deligne pairing and Quillen metric

Let X → S be a smooth projective surjective morphism of relative dimension n, where X and S are integral schemes over ℂ. Let L → X be a relatively very ample line bundle. For every sufficiently large positive integer m, there is a canonical isomorphism of the Deligne pairing 〈L,…,L〉 → S with the determinant line bundle [Formula: see text] (see [D. H. Phong, J. Ross and J. Sturm, Deligne pairings and the knudsen–Mumford expansion, J. Differential Geom. 78 (2008) 475–496]). If we fix a hermitian structure on L and a relative Kähler form on X, then each of the line bundles [Formula: see text] and 〈L,…,L〉 carries a distinguished hermitian structure. We prove that the above mentioned isomorphism between 〈L,…,L〉 → S and [Formula: see text] is compatible with these hermitian structures. This holds also for the isomorphism in [Deligne pairing and determinant bundle, Electron. Res. Announc. Math. Sci. 18 (2011) 91–96] between a Deligne paring and a certain determinant line bundle.

AN -FUNCTION-FREE PROOF OF VINOGRADOV’S THREE PRIMES THEOREM

AbstractWe give a new proof of Vinogradov’s three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of $L$-functions, either explicitly or implicitly. Our proof is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2) 161 (3) (2005), 1609–1636] and used in the proof of Green and Tao’s theorem [Ann. of Math. (2) 167 (2) (2008), 481–547]. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs

We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments. As a corollary, for a convergent sequence of finite graphs, we prove that the normalized log of the chromatic polynomial converges to an analytic function outside a bounded disc. This generalizes a recent result of Borgs, Chayes, Kahn and Lovász, who proved convergence at large enough positive integers and answers a question of Borgs. Our methods also lead to explicit estimates on the number of proper colorings of graphs with large girth.

The [formula omitted]-chromatic number and [formula omitted]-chromatic vertex number of regular graphs

The b-chromatic number of a graph G, denoted by b(G), is the largest positive integer k such that there exists a proper coloring for G with k colors in which every color class contains at least one vertex adjacent to some vertex in each of the other color classes, such a vertex is called a dominant vertex. The f-chromatic vertex number of a d-regular graph G, denoted by f(G), is the maximum number of dominant vertices of distinct colors in a proper coloring with d+1 colors. El Sahili and Kouider conjectured that b(G)=d+1 for any d-regular graph G of girth 5. Blidia, Maffray and Zemir (2009) reformulated this conjecture by excluding the Petersen graph and proved it for d≤6. We study El Sahili and Kouider conjecture by giving some partial answers under supplementary conditions.

On the number of even and odd strings along the overpartitions of n

Recently, Andrews, Chan, Kim, and Osburn introduced the even strings and the odd strings in the overpartitions. We show that their conjecture $$A_k (n) \geq B_k (n)$$ holds for large enough positive integers n, where Ak (n) (resp. Bk (n)) is the number of odd (resp. even) strings along the overpartitions of n. We introduce m-strings and show how this new combinatorial object is related with another positivity conjecture of Andrews, Chan, Kim, and Osburn. Finally, we confirm that the positivity conjecture is also true for large enough integers.

A density version of the Vinogradov three primes theorem

We prove that if A is a subset of the primes, and the lower density of A in the primes is larger than 5/8, then all sufficiently large odd positive integers can be written as the sum of three primes in A. The constant 5/8 in this statement is the best possible.

Measuring and monitoring granular forecast performance

Forecasting demand is a key component of all Revenue Management (RM) processes. Measuring the performance of forecasting is not only crucial for effective inventory controls but it also provides an insight into the changing market dynamics, customer behaviour patterns and so on, which further play an important role in an airline’s strategy. During the segment-based RM era, the most detailed level of forecasting required was ‘FlightLeg-Fareclass-DepartureDate-DCP’. Forecast values at this level used to be large positive integer values. The advent of Origin–Destination (O&D) RM necessitated the detailed ‘O&D-Itinerary-PointOfSale-FareClass-DepartureDate’ level forecasts, which generally tend to be fractions less than 1, therefore are granular in nature. The actual bookings, however, arrive in integer values greater or equal to 1. Thus conventional measures such as Mean Percentage Error, Mean Absolute Percentage Error, Mean Squared Error and so on tend to become inappropriate to measure forecast performance. This article aims to briefly highlight the perceived shortcomings of existing methods in measuring and monitoring the above-mentioned granular forecasts, outline an innovative method and discuss its suitability.

The general Goldbach problem with Beatty primes

In this paper, we are concerned with the representation of large positive integers as the sum of a fixed number of prime numbers taken from Beatty sequences. We are able to avoid any conditions on the Beatty sequences that they be of finite type which have been required hitherto. We use a form of the Hardy–Littlewood–Vinogradov method, but the proofs are quite delicate.

ON PAIRS OF ONE PRIME, TWO PRIME SQUARES AND POWERS OF 2

In this short paper, it is proved that every pair of large positive odd integers satisfying some necessary conditions can be represented in the form of a pair of one prime, two prime squares and k powers of 2. In particular, we have k = 332.

ON WARING'S PROBLEM: TWO SQUARES, TWO CUBES AND TWO SIXTH POWERS

We investigate the number of representations of a large positive integer as the sum of two squares, two positive integral cubes and two sixth powers, showing that the anticipated asymptotic formula fails for at most O((log X)3+ε) positive integers not exceeding X.

Square root algorithm in 𝔽 q for q ≡ 2 s + 1 (mod 2 s +1 )

Presented is a square root algorithm in F q which generalises Atkins's square root algorithm [see reference 6] for q ≡ 5 (mod 8) and Muller's algorithm [see reference 7] for q ≡ 9 (mod 16). The presented algorithm precomputes a primitive 2 s -th root of unity ξ where s is the largest positive integer satisfying 2 s | q - 1, and is applicable for the cases when s is small. The proposed algorithm requires one exponentiation for square root computation and is favourably compared with the algorithms of Atkin, Muuller and Kong et al.

Fourier expansions of GL(2) newforms at various cusps

This paper studies the Fourier expansion of Hecke–Maass eigenforms for GL(2,ℚ) of arbitrary weight, level, and character at various cusps. Translating well known results in the theory of adelic automorphic representations into classical language, a multiplicative expression for the Fourier coefficients at any cusp is derived. In general, this expression involves Fourier coefficients at several different cusps. A sufficient condition for the existence of multiplicative relations among Fourier coefficients at a single cusp is given. It is shown that if the level is 4 times (or in some cases 8 times) an odd squarefree number then there are multiplicative relations at every cusp. We also show that a local representation of GL(2,ℚ p ) which is isomorphic to a local factor of a global cuspidal automorphic representation generated by the adelic lift of a newform of arbitrary weight, level N, and character χ (mod N) cannot be supercuspidal if χ is primitive. Furthermore, it is supercuspidal if and only if at every cusp (of width m and cusp parameter=0) the mp l Fourier coefficient, at that cusp, vanishes for all sufficiently large positive integers l. In the last part of this paper, a three term identity involving the Fourier expansion at three different cusps is derived.

Interpolation theorem for partial Grundy coloring

An ordered partition Π=(V1,V2,…,Vk) of G is a partial Grundy coloring if for each i, 2≤i≤k, there exists a vertex xi in Vi such that xi is adjacent to at least one vertex in Vj for each j<i. The partial Grundy number of G is the largest positive integer k for which G has a partial Grundy coloring using k colors and it is denoted by ∂Γ(G). For any graph G, we have χ(G)≤∂Γ(G). P. Erdős, S.T. Hedetniemi, R.C. Laskar and G.C.E. Prins (2003) [2] raised the following question: For any graph G and any positive integer k with χ(G)≤k≤∂Γ(G), does G have a partial Grundy coloring using k colors? In this note, we provide an affirmative answer to this question.

B-Chromatic Number of Cartesian Product of Some Families of Graphs

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It is known that for any two graphs G and H, $${b(G \square H) \geq {\rm {max}} \{b(G), b(H)\}}$$ , where $${\square}$$ stands for the Cartesian product. In this paper, we determine some families of graphs G and H for which strict inequality holds. More precisely, we show that for these graphs G and H, $${b(G \square H) \geq b(G) + b(H) - 1}$$ . This generalizes one of the results due to Kouider and Maheo.

Grimm's conjecture and smooth numbers

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm. We establish upper and lower bounds for $g(n)$ by relating its study to the distribution of smooth numbers. Standard conjectures concerning smooth numbers in short intervals imply $g(n) =O(n^\epsilon)$ for any $\epsilon >0$. We also prove unconditionally that $g(n) =O(n^\al)$ with $0.45<\al <0.46$. The study of $g(n)$ and cognate functions has some interesting implications for gaps between consecutive primes.

Unstable Adams operations onp–local compact groups

A p-local compact group is an algebraic object modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. In the study of these objects unstable Adams operations, are of fundamental importance. In this paper we define unstable Adams operations within the theory of p-local compact groups, and show that such operations exist under rather mild conditions. More precisely, we prove that for a given p-local compact group G and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of p-adic units which are congruent to 1 modulo p^m to the group of unstable Adams operations on G

Generalized Frobenius numbers: bounds and average behavior

Let n ≥ 2 and s ≥ 1 be integers and a = (a1, . . . , an) be a relatively prime integer n-tuple. The s-Frobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as ∑n i=1 aixi in at least s different ways, where x1, ..., xn are non-negative integers. This natural generalization of the classical Frobenius number, F1(a), has been studied recently by a number of authors. We produce new upper and lower bounds for the s-Frobenius number by relating it to the so called s-covering radius of a certain convex body with respect to a certain lattice; this generalizes a well-known theorem of R. Kannan for the classical Frobenius number. Using these bounds, we obtain results on the average behavior of the s-Frobenius number, extending analogous recent investigations for the classical Frobenius number by a variety of authors. We also derive bounds on the s-covering radius, an interesting geometric quantity in its own right.

Formula omitted]-coloring of Kneser graphs

A b -coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b -coloring using k colors is the b -chromatic number b ( G ) of G . The b -spectrum S b ( G ) of a graph G is the set of positive integers k , χ ( G ) ≤ k ≤ b ( G ) , for which G has a b -coloring using k colors. A graph G is b -continuous if S b ( G ) = the closed interval [ χ ( G ) , b ( G ) ] . In this paper, we obtain an upper bound for the b -chromatic number of some families of Kneser graphs. In addition we establish that [ χ ( G ) , n + k + 1 ] ⊂ S b ( G ) for the Kneser graph G = K ( 2 n + k , n ) whenever 3 ≤ n ≤ k + 1 . We also establish the b -continuity of some families of regular graphs which include the family of odd graphs.