Ramanujan journal

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The Ramanujan Journal. Journal updates. The Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan. Publishing model: Hybrid.

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2015Panaitopol, L.: A formula for \(\pi (x)\) applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. 5(1), 55–56 (2000)MathSciNet Google Scholar Ramanujan, S.: A proof of Bertrand’s postulate. J. Indian Math. Soc. 11, 181–182 (1919) Google Scholar Rosser, J.B.: The \(n\)-th prime is greater than \(n \log n\). Proc. Lond. Math. Soc. 45, 21–44 (1939)Article Google Scholar Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)MATH MathSciNet Google Scholar Shevelev, V.: Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012), Article 12.1.1Sondow, J.: Ramanujan primes and Bertrand’s Postulate. Am. Math. Monthly 116, 630–635 (2009)Article MATH MathSciNet Google Scholar Sondow, J.: Sequence A104272. The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015Sondow, J.: Sequence A233739. The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015Sondow, J., Nicholson, J.W., Noe, T.D.: Ramanujan primes: Bounds, Runs, Twins, and Gaps. J. Integer Seq. 14 (2011), Article 11.6.2Srinivasan, A.: An upper bound for Ramanujan primes. Integers 19 (2014), #A19Tchebychev, P.: Mémoire sur les nombres premiers. Mémoires des savants étrangers de l’Acad. Sci. St.Pétersbourg 7 (1850), 17–33 [Also, Journal de mathématiques pures et appliques 17 (1852), 366–390]Trost, E.: Primzahlen. Birkhäuser, Basel/Stuttgart (1953)MATH Google Scholar Download referencesAcknowledgmentsI would like to thank Benjamin Klopsch for the helpful conversations. Also I would like to thank Elena Klimenko and Anitha Thillaisundaram for their careful reading of the paper.Author informationAuthors and AffiliationsMathematisches Institut, Heinrich-Heine-Universität Düsseldorf, 40225, Düsseldorf, GermanyChristian AxlerAuthorsChristian AxlerYou can also search for this author in PubMed Google ScholarCorresponding authorCorrespondence to Christian Axler.Rights and permissionsAbout this articleCite this articleAxler, C. On generalized Ramanujan primes. Ramanujan J 39, 1–30 (2016). citationReceived: 07 March 2014Accepted: 01 April 2015Published: 30 April 2015Issue Date: January 2016DOI: Subject Classification

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Few explicit methods for constructing Ramanujan graphs. Many of the currently known techniques are reprised in Sect. 4.3 Two new families of unbalanced Ramanujan bigraphsThe existence of Ramanujan bigraphs of specified degrees and sizes has been well studied in recent years. A “road map” for constructing Ramanujan bigraphs is given in [4], and some abstract constructions of Ramanujan bigraphs are given in [5, 14]. These bigraphs have degrees \((p+1,p^3+1)\) for various values of p, such as \(p \equiv 5 ~\mathrm{mod}~12\), \(p \equiv 11 ~\mathrm{mod}~12\) [5], and \(p \equiv 3 ~\mathrm{mod}~4\) [14].Footnote 3 For each suitable choice of p, these papers lead to an infinite family of Ramanujan bigraphs. At present, these constructions are not explicit in terms of resulting in a biadjacency matrix of 0 s and 1 s. There is a paper under preparation by the authors of these papers to make these constructions explicit.The celebrated results of [23,24,25] show that there exist bipartite Ramanujan graphs of all degrees and all sizes. However, these results do not imply the existence of Ramanujan graphs of all sizes and degrees. To understand the approach of [24], we recall the notion of a Ramanujan covering as follows: A covering of a graph G refers to a graph \(G'\) and a surjection \(f:\,\mathcal {V}(G') \rightarrow \mathcal {V}(G)\) such that for any vertex \(v \in \mathcal {V}(G')\), the neighborhood of any vertex in \(G'\) [that is, the set of all vertices in \(G'\) which are joined to v by an edge] is mapped bijectively to the neighborhood of f(v) in G. Moreover, for a positive integer s, a covering \(G'\) of G is called an s-covering if every vertex in G has exactly s preimages in \(G'\). Finally, if an s-covering \(G'\) of a Ramanujan bigraph G satisfies the Ramanujan property in (3), it is said to be a Ramanujan s-covering of G.A special case of one of the key results in [24] implies the existence of a Ramanujan 2-covering of any Ramanujan bigraph G. Subsequently, this was generalized to the existence of a Ramanujan s-covering of a Ramanujan bigraph G for \(s=3,\,4\) in

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We analyze the computational effort required in actually implementing the various construction methods reviewed in Sect. 4. A pertinent issue that is addressed here is whether one can give a polynomial-time algorithm for implementing the known constructions.In Sect. 7 (Sects. 7.1 and 7.2), we accomplish the third goal of this article, namely to construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be disjoint from a given set of “prohibited” edges. In Sect. 7.3, we analyze our new construction of Ramanujan bigraphs from Sect. 3 as well as the previously known constructions of Ramanujan graphs and bigraphs, in terms of how many edges can be relocated while retaining the Ramanujan property.2 Review of Ramanujan graphs and bigraphsIn this subsection, we review the basics of Ramanujan graphs and Ramanujan bigraphs. Further details about Ramanujan graphs can be found in [12, 28].Recall that a graph consists of a vertex set \(\mathcal {V}\) and an edge set \(\mathcal {E}\subseteq \mathcal {V}\times \mathcal {V}\). If \((v_i,v_j) \in \mathcal {E}\) implies that \((v_j,v_i) \in \mathcal {E}\), then the graph is said to be undirected. A graph is said to be bipartite if \(\mathcal {V}\) can be partitioned into two sets \(\mathcal {V}_r, \mathcal {V}_c\) such that \(\mathcal {E}\cap (\mathcal {V}_r \times \mathcal {V}_r) = \emptyset \), \(\mathcal {E}\cap (\mathcal {V}_c \times \mathcal {V}_c) = \emptyset \). Thus, in a bipartite graph, all edges connect one vertex in \(\mathcal {V}_r\) with another vertex in \(\mathcal {V}_c\). A bipartite graph is said to be balanced if \(|\mathcal {V}_r| = |\mathcal {V}_c|\), and unbalanced otherwise.A graph is said to be d-regular if every vertex has the same degree d. A bipartite graph is said to be \((d_r,d_c)\)-biregular if every vertex in \(\mathcal {V}_r\) has degree \(d_r\), and every vertex in \(\mathcal {V}_c\) has degree \(d_c\). Clearly this implies that \(d_c |\mathcal {V}_c| = d_r |\mathcal {V}_r|\).Suppose \((\mathcal {V},\mathcal {E})\) is a graph. Then its adjacency matrix \(A \in \{0,1\}^{|\mathcal {V}| \times |\mathcal {V}|}\) is defined by setting \(A_{ij} = 1\) if there is an edge \((v_i,v_j) \in. The Ramanujan Journal. Journal updates. The Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan. Publishing model: Hybrid.

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Their adjacency or biadjacency matrices. We also focus on the two earlier-known constructions of Ramanujan bipartite graphs due to Lubotzky–Phillips–Sarnak [22] and Gunnells [19] and show that each can be converted into a nonbipartite graph. Note that every graph can be associated with a bipartite graph, but the converse is not true in general. Also, the research community prefers nonbipartite graphs over bipartite graphs. Thus, our proof that the LPS and Gunnells constructions can be converted to nonbipartite graphs is of some interest. All of this is addressed in Sect. 4. 3. [Construction of Ramanujan graphs with prohibited edges]: The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? The approach that we follow to answer this question is to start with an existing Ramanujan bigraph, and then to perturb its edge set so as to eliminate the prohibited edges and replace them by other edges that are nonprohibited. This procedure retains the biregularity of the graph. We then show that our replacement procedure also retains the Ramanujan nature of the bigraph, provided the gap between the second largest singular value of the biadjacency matrix and the “Ramanujan bound” is larger than twice the maximum number of prohibited edges at each vertex. These questions are studied in Section 7. 1.1 Organization of paperThis article is organized as follows. In Sect. 2, we present a brief review of Ramanujan graphs and bigraphs.In Sect. 3, we address the first goal of this article, and present the first explicit construction of an infinite family of unbalanced Ramanujan bigraphs.In Sect. 4 (Sects 4.1–4.4), we address the second goal of this article. We review many of the known methods for constructing Ramanujan graphs, based on the original publications.In Sect. 5, we shed further light on the two constructions of Ramanujan bipartite graphs due to Lubotzky–Phillips–Sarnak [22] and Gunnells [19] and show how each can be converted into a nonbipartite graph.In Sect. 6,

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Of Ramanujan graphs and balanced bigraphs, and no explicit constructions of an unbalanced Ramanujan bigraph. In this paper, we presented for the first time an infinite family of unbalanced Ramanujan bigraphs with explicitly constructed biadjacency matrices. In addition, we have also shown how to construct the adjacency matrices for the currently available families of Ramanujan graphs. These explicit constructions, as well as forthcoming ones based on [5, 14], are available for only a few combinations of degree and size. In contrast, it is known from [24, 25] that Ramanujan graphs are known to exist for all degrees and all sizes. The main limiting factor is that these are only existence proofs and do not lead to explicit constructions. A supposedly polynomial-time algorithm for constructing Ramanujan graphs of all degrees and sizes is proposed in [11]. But it is still a conceptual algorithm and no code has been made available. Therefore it is imperative to develop efficient implementations of the ideas proposed in [25], and/or to develop other methods to construct Ramanujan graphs of most degrees and sizes. We should also note that the work of [24, 25] shows the existence of balanced, bipartite graphs of all degrees and sizes. Therefore, in Sect. 5 of this article, we have also looked at how the constructions of [22] and Gunnells [19] can be further analyzed to derive nonbipartite Ramanujan graphs.It is worth pointing out that efficient solutions of the matrix completion problem do not really require the existence of Ramanujan graphs of all sizes and degrees. It is enough if the “gaps” in the permissible values for the degrees and the sizes are very small. If this extra freedom leads to substantial simplification in the construction procedures, then it would be a worthwhile tradeoff. However, research on this problem is still at a nascent stage.Finally, in Sect. 7, we address another issue in the matrix completion problem, namely the “missing measurements” problem. This leads to the problem of the construction of a Ramanujan bigraph (not necessarily balanced) in which a certain set of edges is prohibited. In a typical real-life application, the

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1 IntroductionA fundamental theme in graph theory is the study of the spectral gap of a regular (undirected) graph, that is, the difference between the two largest eigenvalues of the adjacency matrix of such a graph. Analogously, the spectral gap of a bipartite, biregular graph is the difference between the two largest singular values of its biadjacency matrix (please see Sect. 2 for detailed definitions). Ramanujan graphs (respectively Ramanujan bigraphs) are graphs with an optimal spectral gap. Explicit constructions of such graphs have multifaceted applications in areas such as computer science, coding theory, and compressed sensing. In particular, in [8] the first and third authors show that Ramanujan graphs can provide the first deterministic solution to the so-called matrix completion problem. Prior to the publication of [8], the matrix completion problem had only a probabilistic solution. The explicit construction of Ramanujan graphs has been classically well studied. Some explicit methods are known, for example, by the work of Lubotzky, Phillips, and Sarnak [22], Margulis [26], Li [21], Morgenstern [27], Gunnells [19], Bibak et al. [6]. These methods are drawn from concepts in linear algebra, number theory, representation theory and the theory of automorphic forms. In contrast, however, no explicit methods for constructing unbalanced Ramanujan bigraphs are known. There are a couple of abstract constructions in [5, 14], but these are not explicit.Footnote 1This article has three goals. 1. [Explicit construction of unbalanced Ramanujan bigraphs]: First and foremost, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. Our construction, presented in Sect. 3, is based on “array code” matrices from LDPC (low-density parity check) coding theory. Apart from being the first explicit constructions, they also have an important computational feature: the biadjacency matrices are obtained immediately upon specifying two parameters, a prime number q and an integer \(l \ge 2\). 2. [Comparison of computational aspects of known constructions]: The second goal of this article is to revisit some of the earlier-known constructions of Ramanujan graphs and (balanced) bigraphs and compare the amount of work involved in constructing the various classes of graphs and obtaining. The Ramanujan Journal. Journal updates. The Ramanujan Journal. An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan. Publishing model: Hybrid.