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Correction
CORRECTION TO ‘ON THE COMPLEMENT OF THE ZERO-DIVISOR GRAPH OF A PARTIALLY ORDERED SET’
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- Published online by Cambridge University Press: 18 September 2023, pp. 1-4
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Research Article
SUMSETS CONTAINING A TERM OF A SEQUENCE
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- Published online by Cambridge University Press: 18 September 2023, pp. 1-9
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Let
$S=\{s_{1}, s_{2}, \ldots \}$ be an unbounded sequence of positive integers with 
$s_{n+1}/s_{n}$ approaching 
$\alpha $ as 
$n\rightarrow \infty $ and let 
$\beta>\max (\alpha , 2)$. We show that for all sufficiently large positive integers l, if 
$A\subset [0, l]$ with 
$l\in A$, 
$\gcd A=1$ and 
$|A|\geq (2-{k}/{\lambda \beta })l/(\lambda +1)$, where 
$\lambda =\lceil {k}/{\beta }\rceil $, then 
$kA\cap S\neq \emptyset $ for 
$2<\beta \leq 3$ and 
$k\geq {2\beta }/{(\beta -2)}$ or for 
$\beta>3$ and 
$k\geq 3$.
PhD Abstract
CAYLEY GRAPHS AND GRAPHICAL REGULAR REPRESENTATIONS
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- Published online by Cambridge University Press: 15 September 2023, pp. 1-2
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GEOMETRIC AND TOPOLOGICAL SHAPE ANALYSIS: INVESTIGATING AND SUMMARISING THE SHAPE OF DATA
- Published online by Cambridge University Press: 15 September 2023, pp. 1-2
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Research Article
ON THE N-POINT CORRELATION OF VAN DER CORPUT SEQUENCES
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- Published online by Cambridge University Press: 15 September 2023, pp. 1-5
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We derive an explicit formula for the N-point correlation
$F_N(s)$ of the van der Corput sequence in base 
$2$ for all 
$N \in \mathbb {N}$ and 
$s \geq 0$. The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. This constitutes the first example of an exact closed-form expression of 
$F_N(s)$ for all 
$N \in \mathbb {N}$ and all 
$s \geq 0$ which does not require explicit knowledge about the involved sequence. Moreover, it can be immediately read off that 
$\lim _{N \to \infty } F_N(s)$ exists only for 
$0 \leq s \leq 1/2$.
THE MORDELL–LANG CONJECTURE FOR SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC
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- Published online by Cambridge University Press: 08 September 2023, pp. 1-11
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Let G be a semiabelian variety defined over an algebraically closed field K of prime characteristic. We describe the intersection of a subvariety X of G with a finitely generated subgroup of
$G(K)$.
AN IMPROVEMENT TO A THEOREM OF LEONETTI AND LUCA
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- Published online by Cambridge University Press: 01 September 2023, pp. 1-6
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Leonetti and Luca [‘On the iterates of the shifted Euler’s function’, Bull. Aust. Math. Soc., to appear] have shown that the integer sequence
$(x_n)_{n\geq 1}$ defined by 
$x_{n+2}=\phi (x_{n+1})+\phi (x_{n})+k$, where 
$x_1,x_2\geq 1$, 
$k\geq 0$ and 
$2 \mid k$, is bounded by 
$4^{X^{3^{k+1}}}$, where 
$X=(3x_1+5x_2+7k)/2$. We improve this result by showing that the sequence 
$(x_n)$ is bounded by 
$2^{2X^2+X-3}$, where 
$X=x_1+x_2+2k$.
PhD Abstract
BAYESIAN SPATIOTEMPORAL MODELS FOR INFECTIOUS DISEASES IN RELATION TO HEALTH INEQUALITY
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- Published online by Cambridge University Press: 31 August 2023, pp. 1-2
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MONTE CARLO VARIANCE REDUCTION METHODS WITH APPLICATIONS IN STRUCTURAL RELIABILITY ANALYSIS
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- Published online by Cambridge University Press: 31 August 2023, pp. 1-4
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Research Article
A NOTE ON BRØNDSTED’S FIXED POINT THEOREM
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- Published online by Cambridge University Press: 31 August 2023, pp. 1-6
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ON SUMS INVOLVING THE EULER TOTIENT FUNCTION
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- Published online by Cambridge University Press: 24 August 2023, pp. 1-12
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Let
$\gcd (n_{1},\ldots ,n_{k})$ denote the greatest common divisor of positive integers 
$n_{1},\ldots ,n_{k}$ and let 
$\phi $ be the Euler totient function. For any real number 
$x>3$ and any integer 
$k\geq 2$, we investigate the asymptotic behaviour of 
$\sum _{n_{1}\ldots n_{k}\leq x}\phi (\gcd (n_{1},\ldots ,n_{k})). $
ON THE EXPECTED UNIFORM ERROR OF BROWNIAN MOTION APPROXIMATED BY THE LÉVY–CIESIELSKI CONSTRUCTION
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- Published online by Cambridge University Press: 24 August 2023, pp. 1-13
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The Brownian bridge or Lévy–Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. We focus on the uniform error. In particular, we show constructively that at level N, at which there are
$d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order 
$\mathcal {O}(\sqrt {\ln d/d})$, matching the known orders. We apply the results to an option pricing example.
PhD Abstract
EXTRACTING FEATURES FROM EIGENFUNCTIONS: HIGHER CHEEGER CONSTANTS AND SPARSE EIGENBASIS APPROXIMATION
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- Published online by Cambridge University Press: 22 August 2023, pp. 1-2
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Research Article
A COUNTEREXAMPLE TO A RESULT OF JABERI AND MAHMOODI
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- Published online by Cambridge University Press: 10 August 2023, pp. 1-3
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We show that
$\ell ^1(\mathbb {N}_\wedge )$ is 
$\varphi $-amenable for each multiplicative linear functional 
$\varphi :\ell ^1(\mathbb {N}_\wedge )\rightarrow \mathbb {C}.$ This is a counterexample to the final corollary of Jaberi and Mahmoodi [‘On 
$\varphi $-amenability of dual Banach algebras’, Bull. Aust. Math. Soc. 105 (2022), 303–313] and shows that the final theorem in that paper is not valid.
GROUPS WITH FEW NONPOWER SUBGROUPS
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- Published online by Cambridge University Press: 10 August 2023, pp. 1-12
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For a group G and
$m\ge 1$, let 
$G^m$ denote the subgroup generated by the elements 
$g^m$, where g runs through G. The subgroups not of the form 
$G^m$ are the nonpower subgroups of G. We classify the groups with at most nine nonpower subgroups.
PhD Abstract
MECHANISTIC MARKOV MODELS FOR THE EVOLUTION OF GENE FAMILIES
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- Published online by Cambridge University Press: 10 August 2023, pp. 1-3
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Correction
CORRECTION TO ‘EXISTENCE AND BOX DIMENSION OF GENERAL RECURRENT FRACTAL INTERPOLATION FUNCTIONS’
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- Published online by Cambridge University Press: 07 August 2023, pp. 1-3
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PhD Abstract
USING MACHINE LEARNING APPROACHES TO DEVELOP PRICE OPTIMISATION AND DEMAND PREDICTION MODELS FOR MULTIPLE PRODUCTS WITH DEMAND CORRELATION
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- Published online by Cambridge University Press: 01 August 2023, pp. 1-3
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Research Article
EVERY ARITHMETIC PROGRESSION CONTAINS INFINITELY MANY b-NIVEN NUMBERS
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- Published online by Cambridge University Press: 31 July 2023, pp. 1-5
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For an integer
$b\geq 2$, a positive integer is called a b-Niven number if it is a multiple of the sum of the digits in its base-b representation. In this article, we show that every arithmetic progression contains infinitely many b-Niven numbers.
SUM OF VALUES OF THE IDEAL CLASS ZETA-FUNCTION OVER NONTRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION
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- Published online by Cambridge University Press: 31 July 2023, pp. 1-13
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