• OCR based feature extraction and template matching algorithms for Qatari number plate

      Farhat, Ali; Al-Zawqari, Ali; Al-Qahtani, Abdulhadi; Hommos, Omar; Bensaali, Faycal; Amira, Abbes; Zhai, Xiaojun; University of Derby (IEEE, 2016-03-13)
      There are several algorithms and methods that could be applied to perform the character recognition stage of an automatic number plate recognition system; however, the constraints of having a high recognition rate and real-time processing should be taken into consideration. In this paper four algorithms applied to Qatari number plates are presented and compared. The proposed algorithms are based on feature extraction (vector crossing, zoning, combined zoning and vector crossing) and template matching techniques. All four proposed algorithms have been implemented and tested using MATLAB. A total of 2790 Qatari binary character images were used to test the algorithms. Template matching based algorithm showed the highest recognition rate of 99.5% with an average time of 1.95 ms per character.
    • On a result of Bunder involving horadam sequences: A new proof

      Larcombe, Peter J.; Bagdasar, Ovidiu; Fennessey, Eric J.; University of Derby (Fibonacci Association, 2014-05-02)
      This note offers a new proof of a 1975 result due to M. W. Bunder which has recently been proven (inductively), extended empirically and generalized in this journal. The proof methodology, while interesting, cannot be applied realistically beyond the original order two case of Bunder dealt with here.
    • On a result of Bunder involving Horadam sequences: A proof and generalization

      Larcombe, Peter J.; Bagdasar, Ovidiu; University of Derby (Fibonacci Association, 2013-05-03)
      This note introduces, proves, extends empirically and generalizes a short 1975 offering of M.W. Bunder who, in this journal, gave an isolated observation involving Horadam sequences on which work has been conducted for nearly half a century.
    • On a scaled balanced-power product recurrence

      Larcombe, Peter J.; Fennessey, Eric J.; University of Derby (The Fibonacci Association, 2016-08)
      A power product recurrence (due to M. W. Bunder) is extended here by the introduction of a scaling factor, and delivers a sequence whose general term closed form is derived for both degenerate and non-degenerate characteristic root cases. It is shown how recurrence parameter conditions dictate the nature of each solution type, and a fundamental link between them is highlighted together with some other observations and results.
    • On an arithmetic triangle of numbers arising from inverses of analytic functions.

      Bagdasaryan, Armen G.; Bagdasar, Ovidiu; American University of the Middle East; University of Derby (Elsevier, 2018-12-06)
      The Lagrange inversion formula is a fundamental tool in combinatorics. In this work, we investigate an inversion formula for analytic functions, which does not require taking limits. By applying this formula to certain functions we have found an interesting arithmetic triangle for which we give a recurrence formula. We then explore the links between these numbers, Pascal’s triangle, and Bernoulli’s numbers, for which we obtain a new explicit formula. Furthermore, we present power series and asymptotic expansions of some elementary and special functions, and some links to the Online Encyclopedia of Integer Sequences (OEIS).
    • On certain series expansions of the sine function: Catalan numbers and convergence

      Larcombe, Peter J.; O'Neill, Sam T.; Fennessey, Eric J.; University of Derby (The Fibonacci Association, 2014-08)
      The appearance of Catalan numbers in certain infinite series expansions of the sine function was first reported well over a decade ago. A combination of computation and analysis is employed as we return to this topic and examine the outstanding issue of convergence for this suite of results and also for the general case expansion.
    • On cyclicity and density of some Catalan polynomial sequences

      Larcombe, Peter J.; Fennessey, Eric J.; University of Derby (The Institute of Combinatorics and its Applications, 2014-05)
      We give proofs of cyclic and density properties of some sequences generated by Catalan polynomials, and other observations.
    • On generalised multi-index non-linear recursion identities for terms of the Horadam sequence.

      Larcombe, Peter J.; Fennessey, Eric J.; University of Derby (Palestine Polytechnic University, 2019)
      We state and prove a non-linear recurrence identity for terms of the so called Horadamsequence,andthenofferitsgeneralisationwhichisavailablefromthesamemethodology. We illustrate how the overarching idea may be used to sequentially produce extended versions, each in turn with an extra level of non-linearity and term index complexity. These identities can all be captured in matrix determinant form.
    • On horadam sequence periodicity: A new approach

      Larcombe, Peter J.; Fennessey, Eric J.; University of Derby (The Institute of Combinatorics and its Applications (ICA), 2015-01)
      A so called Horadam sequence is one delivered by a general second order recurrence formula with arbitrary initial conditions. We examine aspects of self-repeating Horadam sequences by applying matrix based methods in new ways, and derive some conditions governing their cyclic behaviour. The analysis allows for both real and complex sequence periodicity.
    • On Locky ransomware, Al Capone and Brexit

      MacRae, John; Franqueira, Virginia N. L.; Ulster University; University of Derby (Springer, 2018-01-06)
      The highly crafted lines of code which constitute the Locky cryptolocker ransomware are there to see in plain text in an infected machine. Yet, this forensic evidence does not lead investigators to the identity of the extortionists nor to the destination of the ransom payments. Perpetrators of this ransomware remain unknown and unchallenged and so the ransomware cyber crimewave gathers pace. This paper examines what Locky is, how it works, and the mechanics of this malware to understand how ransom payments are made. The financial impact of Locky is found to be substantial. The paper describes methods for “following the money” to assess how effectively such a digital forensic trail can assist ransomware investigators. The legal instruments that are being established by the authorities as they attempt to shut down ransomware attacks and secure prosecutions are evaluated. The technical difficulty of following the money coupled with a lack of registration and disclosure legislation mean that investigators of this cybercrime are struggling to secure prosecutions and halt Locky.
    • On sequence-based closed form entries for an exponentiated general $2 \times 2$ matrix: A re-formulation and an application.

      Larcombe, Peter J.; Fennessey, Eric J.; University of Derby (The Institute of Combinatorics and its Applications, 2017-01)
      Closed form entries for an exponentiated (and arbitrary) 2 ⇥ 2 matrix are established here, and expressed in terms of a specialized Horadam sequence; two proofs of the result are given accordingly, along with examples and observations derived therefrom. The result o↵ers a new formulation of a general class of polynomial families associated with sequences whose ordinary generating functions are governed by quadratic equations.
    • On some functions involving the lcm and gcd of integer tuples

      Bagdasar, Ovidiu; University of Derby (2014-06)
      In this paper an explicit formula for the number of tuples of positive integers having the same lowest common multiple n is derived, and some of the properties of the resulting arithmetic function are analyzed. The tuples having also the same greatest common divisor are investigated, while some novel or existing integer sequences are recovered as particular cases. A formula linking the gcd and lcm for tuples of integers is also presented.
    • On some new arithmetic functions involving prime divisors and perfect powers.

      Bagdasar, Ovidiu; Tatt, Ralph-Joseph; University of Derby (Elsevier, 2018-12-06)
      Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers i j obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences.
    • On some new arithmetic properties of the generalized Lucas sequences

      Andrica, Dorin; Bagdasar, Ovidiu; Babeş-Bolyai University of Cluj-Napoca, Cluj-Napoca, Romania; University of Derby (Springer Science and Business Media LLC, 2021-01-21)
      Some arithmetic properties of the generalized Lucas sequences are studied, extending a number of recent results obtained for Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These properties are then applied to investigate certain notions of Fibonacci, Lucas, Pell, and Pell–Lucas pseudoprimality, for which we formulate some conjectures.
    • On some results concerning generalized arithmetic triangles.

      Bagdasaryan, Armen G.; Bagdasar, Ovidiu; University of Derby; University of the Middle East (Elsevier, 2018-06-14)
      In this paper we present theoretical and computational results regarding generalized arithmetic m-triangles. The numerical values recover well-known number sequences, indexed in the OEIS including binomial coefficients and their extensions. Some combinatorial interpretations, generating functions and also asymptotic formulae for these triangles are provided.
    • On some results concerning the polygonal polynomials.

      Andrica, Dorin; Bagdasar, Ovidiu; Babeș-Bolyai University; University of Derby (Technical University of Cluj-Napoca., 2019-02-13)
      In this paper we define the $n$th polygonal polynomial $P_n(z) = (z-1)(z^2-1)\cdots(z^n-1)$ and we investigate recurrence relations and exact integral formulae for the coefficients of $P_n(z)$ and for those of the Mahonian polynomials $Q_n(z)=(z+1)(z^2+z+1)\cdots(z^{n-1}+\cdots+z+1)$. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.
    • On the characterization of periodic complex Horadam sequences

      Bagdasar, Ovidiu; Larcombe, Peter J.; University of Derby (Fibonacci Association, 2013-02-01)
      Horadam sequences are second-order linear recurrence sequences which depend on a family of four parameters (two in the defining recursion itself, and two initial values). In this article we find necessary and sufficient conditions for the periodicity of complex Horadam sequences, under general initial values, characterizing sequence behavior for degenerate and non-degenerate characteristic solution types. Inner and outer boundaries for regions containing periodic orbits are also determined.
    • On the characterization of periodic generalized Horadam sequences

      Bagdasar, Ovidiu; Larcombe, Peter J.; University of Derby (Taylor and Francis, 2014-03-13)
      The Horadam sequence is a direct generalization of the Fibonacci numbers in the complex plane, which depends on a family of four complex parameters: two recurrence coefficients and two initial conditions. In this article a computational matrix-based method is developed to formulate necessary and sufficient conditions for the periodicity of generalized complex Horadam sequences, which are generated by higher order recurrences for arbitrary initial conditions. The asymptotic behaviour of generalized Horadam sequences generated by roots of unity is also examined, along with upper boundaries for the disc containing periodic orbits. Some applications are suggested, along with a number of future research directions.
    • On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach

      Larcombe, Peter J.; University of Derby (The Fibonacci Association, 2016-08)
      Infinite series comprising exponentiated multiples of p-term linear combinations of Catalan numbers arise naturally from a related power series expansion for sin(2pα) (in odd powers of sin(α)) which itself has an interesting history. In this article some explicit results generated previously by the author (for p = 1, 2, 3) are discussed in the context of this general problem of series summation, and new evaluations made for the cases p = 4, 5 by way of further examples. A powerful hypergeometric approach is adopted which offers, from the analytical formulation developed, a means to achieve these particular evaluations and in principle many others for even greater values of p.
    • On the geometry of certain periodic non-homogeneous Horadam sequences.

      Bagdasar, Ovidiu; Popa, Ioan-Lucian; University of Derby; University of Alba Iulia (Elsevier, 2016-12)
      Horadam sequences are second-order recurrences depending on a family of four complex parameters: two initial conditions and two recurrence coefficients. The periodicity conditions, as well as the number and geometric structure of self-repeating Horadam sequences of fixed length have been recently investigated. In this paper various geometric properties of non-homogeneous Horadam sequences are explored, including periodicity conditions and the structure of certain periodic orbits.