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Journal articles on the topic 'Least integer function'

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Published: 4 June 2021

Last updated: 5 February 2022

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1

Hong, Shaofang, and Guoyou Qian. "The least common multiple of consecutive arithmetic progression terms." Proceedings of the Edinburgh Mathematical Society 54, no. 2 (2011): 431–41. http://dx.doi.org/10.1017/s0013091509000431.

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AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to

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2

QIAN, GUOYOU, QIANRONG TAN, and SHAOFANG HONG. "THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN A QUADRATIC PROGRESSION." Bulletin of the Australian Mathematical Society 86, no. 3 (2012): 389–404. http://dx.doi.org/10.1017/s0004972712000202.

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AbstractLet k be any given positive integer. We define the arithmetic function gk for any positive integer n by We first show that gk is periodic. Subsequently, we provide a detailed local analysis of the periodic function gk, and determine its smallest period. We also obtain an asymptotic formula for log lcm0≤i≤k {(n+i)2+1}.

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3

Ma, Wu-Xia, Yong-Gao Chen, and Bing-Ling Wu. "Distribution of the primes involving the ceiling function." International Journal of Number Theory 15, no. 03 (2019): 597–611. http://dx.doi.org/10.1142/s1793042119500313.

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The distribution of the primes of the forms [Formula: see text] and [Formula: see text] are studied extensively, where [Formula: see text] denotes the largest integer not exceeding [Formula: see text]. In this paper, we will consider several new type problems on the distribution of the primes involving the ceiling (floor) function. For any real number [Formula: see text] with [Formula: see text], let [Formula: see text] be the number of integers [Formula: see text] with [Formula: see text] such that [Formula: see text] is prime and let [Formula: see text] be the number of primes [Formula: see

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4

Leonetti, Paolo, and Carlo Sanna. "A note on primes in certain residue classes." International Journal of Number Theory 14, no. 08 (2018): 2219–23. http://dx.doi.org/10.1142/s1793042118501336.

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Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

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5

Ballantine, Cristina, and Mircea Merca. "Combinatorial proof of the minimal excludant theorem." International Journal of Number Theory 17, no. 08 (2021): 1765–79. http://dx.doi.org/10.1142/s1793042121500615.

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The minimal excludant of a partition [Formula: see text], [Formula: see text], is the smallest positive integer that is not a part of [Formula: see text]. For a positive integer [Formula: see text], [Formula: see text] denotes the sum of the minimal excludants of all partitions of [Formula: see text]. Recently, Andrews and Newman obtained a new combinatorial interpretation for [Formula: see text]. They showed, using generating functions, that [Formula: see text] equals the number of partitions of [Formula: see text] into distinct parts using two colors. In this paper, we provide a purely combi

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Du, Julia Q. D., Edward Y. S. Liu, and Jack C. D. Zhao. "Congruence properties of pk(n)." International Journal of Number Theory 15, no. 06 (2019): 1267–90. http://dx.doi.org/10.1142/s1793042119500714.

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We present a unified approach to establish infinite families of congruences for [Formula: see text] for arbitrary positive integer [Formula: see text], where [Formula: see text] is given by the [Formula: see text]th power of the Euler product [Formula: see text]. For [Formula: see text], define [Formula: see text] to be the least positive integer such that [Formula: see text] and [Formula: see text] the least non-negative integer satisfying [Formula: see text]. Using the Atkin [Formula: see text]-operator, we find that the generating function of [Formula: see text] (respectively, [Formula: see

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7

Popkov, Kirill Andreevich. "On self-correcting logic circuits of unreliable gates." Keldysh Institute Preprints, no. 49 (2021): 1–18. http://dx.doi.org/10.20948/prepr-2021-49.

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The following statements are proved: 1) for any integer m ≥ 3 there is a basis consisting of Boolean functions of no more than m variables, in which any Boolean function can be implemented by a logic circuit of unreliable gates that self-corrects relative to certain faults in an arbitrary number of gates; 2) for any positive integer k there are bases consisting of Boolean functions of no more than two variables, in each of which any Boolean function can be implemented by a logic circuit of unreliable gates that self-correct relative to certain faults in no more than k gates; 3) there is a func

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8

Lu, Qian, and Qilong Liao. "Normal criterion and shared values by derivatives of meromorphic functions." Tamkang Journal of Mathematics 45, no. 2 (2014): 109–17. http://dx.doi.org/10.5556/j.tkjm.45.2014.1014.

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Let $\mathscr{F}$ be a family of meromorphic functions in a plane domain $D$. If for every function $f\in\mathscr{F}$, all of whose zeros have,at least,multiplicity $l$ and poles have, at least,multiplicity $p$, and for each pair functions $f$ and $g$ in $\mathscr{F}$, $f^{(k)}$ and $g^{(k)}$ share 1 in $D$, where $k,l,$ and $p$ are three positive integer satisfying $\frac{k+1}{l}+\frac{1}{p}\leq 1$, then $\mathscr{F}$ is normal.

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9

Cusick, T. W. "Units in real cyclic quartic fields." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 1 (1990): 5–17. http://dx.doi.org/10.1017/s0305004100068328.

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Let F be a totally real quartic field. For any α in F, let α, α′, α″, α‴ or α(0) = α(1), α(2), α(3) denote the conjugates of α. Define the function T(α) byWe define a triple of units ε1, ε2, ε3 in F as follows. Let ε1 be a unit which gives the least value of T(ε) for any unit ε ≠ = ± 1 in F. Let ε2 be a unit which gives the least value of T(ε) for any unit ε ≠ = ± ε1m with m a rational integer. Let ε3 be a unit which gives the least value of T(ε) for ε ≠ = ± ε1m ε2n with m and n rational integers. We call ε1, ε2, ε3 the successive unit minima for T(ε).

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10

Chen, Jun-Fan. "Exceptional functions and normal families of holomorphic functions with multiple zeros." gmj 18, no. 1 (2011): 31–38. http://dx.doi.org/10.1515/gmj.2011.0005.

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Abstract Let k be a positive integer, and let ℱ be a family of functions holomorphic on a domain D in C, all of whose zeros are of multiplicity at least k + 1. Let h be a function meromorphic on D, h ≢ 0, ∞. Suppose that for each ƒ ∈ ℱ, ƒ(k)(z) ≠ h(z) for z ∈ D. Then ℱ is a normal family on D. The condition that the zeros of functions in ℱ are of multiplicity at least k + 1 cannot be weakened, and the corresponding result for families of meromorphic functions is no longer true.

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MORIYA, B. K., and C. J. SMYTH. "INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS." International Journal of Number Theory 09, no. 07 (2013): 1841–53. http://dx.doi.org/10.1142/s1793042113500607.

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We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is pr

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12

Huang, Xiaojun, and Yongxing Gu. "On the value distribution of f2f(k)." Journal of the Australian Mathematical Society 78, no. 1 (2005): 17–26. http://dx.doi.org/10.1017/s1446788700015536.

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AbstractIn this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

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13

Kalita, Manashee, Themrichon Tuithung, and Swanirbhar Majumder. "A New Steganography Method Using Integer Wavelet Transform and Least Significant Bit Substitution." Computer Journal 62, no. 11 (2019): 1639–55. http://dx.doi.org/10.1093/comjnl/bxz014.

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Abstract Steganography is a data hiding technique, which is used for securing data. Both spatial and transform domains are used to implement a steganography method. In this paper, a novel transform domain method is proposed to provide a better data hiding method. The method uses a multi-resolution transform function, integer wavelet transform (IWT) that decomposes an image into four subbands: low-low, low-high, high-low and high-high subband. The proposed method utilizes only the three subbands keeping the low-low subband untouched which helps to improve the quality of the stego image. The met

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Fang, Mingliang, and Lawrence Zalcman. "A note on normality and shared values." Journal of the Australian Mathematical Society 76, no. 1 (2004): 141–50. http://dx.doi.org/10.1017/s1446788700008752.

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AbstractLet k be a positive integer and b a nonzero constant. Suppose that F is a family of meromorphic functions in a domain D. If each function f ∈ F has only zeros of multiplicity at least k + 2 and for any two functions f, g ∈ F, f and g share 0 in D and f(k) and g(k) share b in D, then F is normal in D. The case f ≠ 0, f(k) ≠ b is a celebrated result of Gu.

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15

OBERSNEL, FRANCO, and PIERPAOLO OMARI. "MULTIPLE BOUNDED VARIATION SOLUTIONS OF A PERIODICALLY PERTURBED SINE-CURVATURE EQUATION." Communications in Contemporary Mathematics 13, no. 05 (2011): 863–83. http://dx.doi.org/10.1142/s0219199711004488.

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We prove the existence of at least two T-periodic solutions, not differing from each other by an integer multiple of 2π, of the sine-curvature equation [Formula: see text] We assume that A ∈ ℝ and [Formula: see text] is a T-periodic function such that [Formula: see text] and, e.g. ‖h‖L∞ < 4/T. Our approach is variational and makes use of basic results of non-smooth critical point theory in the space of bounded variation functions.

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16

LIN, ZONGBING, and SIAO HONG. "MORE ON A CERTAIN ARITHMETICAL DETERMINANT." Bulletin of the Australian Mathematical Society 97, no. 1 (2017): 15–25. http://dx.doi.org/10.1017/s0004972717000788.

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Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212

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17

Landman, Bruce M., and Beata Wysocka. "Collections of sequences having the Ramsey property only for few colours." Bulletin of the Australian Mathematical Society 55, no. 1 (1997): 19–28. http://dx.doi.org/10.1017/s0004972700030501.

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A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g(r)(k) such that for every r-colouring of {1, 2, …, g(r)(k)} there is a monochromatic k-term member of 𝑐. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x1, …, xk} such that xi − xi−1 ≡ a (mod m) for i = 2, …, k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let be the collection of sequences that are either a(mod m)-sequences or m-a.p.

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18

Brown, Tom C., and Bruce M. Landman. "Monochromatic arithmetic progressions with large differences." Bulletin of the Australian Mathematical Society 60, no. 1 (1999): 21–35. http://dx.doi.org/10.1017/s0004972700033293.

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A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

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WEI, ERLING, YE CHEN, PING LI, and HONG-JIAN LAI. "EVERY N2-LOCALLY CONNECTED CLAW-FREE GRAPH WITH MINIMUM DEGREE AT LEAST 7 IS Z3-CONNECTED." Discrete Mathematics, Algorithms and Applications 03, no. 02 (2011): 193–201. http://dx.doi.org/10.1142/s1793830911001140.

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Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A-{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b : V(G) ↦ A satisfying Συ∈V(G)b(υ) = 0, there is a function f : E(G) ↦ A* such that for each vertex υ ∈ V(G), the total amount of f values on the edges directed out from υ minus the total amount of f values on the edges directed into υ equals b(υ). Let Z3denote the group of order 3. Jaeger et al. conjectured that there exists an integer k such that every k-edge-connected graph is Z3-connected. In this paper, we prove

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20

Olanrewaju, Rasaki Olawale. "Integer-valued Time Series Model via Generalized Linear Models Technique of Estimation." International Annals of Science 4, no. 1 (2018): 35–43. http://dx.doi.org/10.21467/ias.4.1.35-43.

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The paper authenticated the need for separate positive integer time series model(s). This was done from the standpoint of a proposal for both mixtures of continuous and discrete time series models. Positive integer time series data are time series data subjected to a number of events per constant interval of time that relatedly fits into the analogy of conditional mean and variance which depends on immediate past observations. This includes dependency among observations that can be best described by Generalized Autoregressive Conditional Heteroscedasticity (GARCH) model with Poisson distribute

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Cenzer, D., V. W. Marek, and J. B. Remmel. "On the complexity of index sets for finite predicate logic programs which allow function symbols." Journal of Logic and Computation 30, no. 1 (2020): 107–56. http://dx.doi.org/10.1093/logcom/exaa005.

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Abstract We study the recognition problem in the metaprogramming of finite normal predicate logic programs. That is, let $\mathcal{L}$ be a computable first-order predicate language with infinitely many constant symbols and infinitely many $n$-ary predicate symbols and $n$-ary functions symbols for all $n \geq 1$. Then we can effectively list all the finite normal predicate logic programs $Q_0,Q_1,\ldots $ over $\mathcal{L}$. Given some property $\mathcal{P}$ of finite normal predicate logic programs over $\mathcal{L}$, we define the index set $I_{\mathcal{P}}$ to be the set of indices $e$ suc

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Sun, Cheng Xiong. "Normal Families and Shared Functions." Fasciculi Mathematici 60, no. 1 (2018): 173–80. http://dx.doi.org/10.1515/fascmath-2018-0011.

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Abstract Let k ∈ ℕ, m ∈ℕ ∪{0}, and let a(z)(≢ 0) be a holomorphic function, all zeros of a(z) have multiplicities at most m. Let ℱ be a family of meromorphic functions in D. If for each f ∈ℱ, the zeros of f have multiplicities at least k + m + 1 and all poles of f are of multiplicity at least m + 1, and for f,g ∈ℱ, ff(k)−a(z) and gg(k)−a(z) share 0, then ℱ is normal in D. Some examples are given to show that the conditions are best, and the result removes the condition “m is an even integer” in the result due to Sun [Kragujevac Journal of Math 38(2), 173-282, 2014].

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Hong, Siao, Shuangnian Hu, and Shaofang Hong. "Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions." Open Mathematics 14, no. 1 (2016): 146–55. http://dx.doi.org/10.1515/math-2016-0014.

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AbstractLet f be an arithmetic function and S= {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S1, …, Sk with k ≥ 1 being an integer and

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Baldwin, Stewart. "An extension of Šarkovskiư's Theorem to the n-od." Ergodic Theory and Dynamical Systems 11, no. 2 (1991): 249–71. http://dx.doi.org/10.1017/s0143385700006131.

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AbstractThe n-od is defined to be the set of all complex numbers z such that zn is a real number in the interval [0,1], i.e., a central point with n copies of the unit interval attached at their endpoints. Given a space X and a function f:X → X, Per (f) is defined to be the set {k: f has for a point of (least) period k, k a positive integer}. The main result of this paper is to give, for each n, a complete characterization of all possible sets Per (f), where f ranges over all continuous functions on the n-od.

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MADRITSCH, MANFRED G. "q-ADDITIVE FUNCTIONS ON POLYNOMIAL SEQUENCES." International Journal of Number Theory 08, no. 02 (2012): 377–93. http://dx.doi.org/10.1142/s1793042112500224.

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The present paper deals with functions acting only on the digits of an q-ary expansion. In particular, let n be a positive integer, then we denote by [Formula: see text] its q-ary expansion. We call a function f strictly q-additive if it acts only on the digits of a representation, i.e. [Formula: see text] The goal is to prove that if p is a polynomial having at least one coefficient with bounded continued fraction expansion, then [Formula: see text] This result is motivated by the asymptotic distribution result of Bassily and Kátai and a similar result of Peter.

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Fagin, Barry. "Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers." Information 12, no. 8 (2021): 305. http://dx.doi.org/10.3390/info12080305.

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Previous work established the set of square-free integers n with at least one factorization n=p¯q¯ for which p¯ and q¯ are valid RSA keys, whether they are prime or composite. These integers are exactly those with the property λ(n)∣(p¯−1)(q¯−1), where λ is the Carmichael totient function. We refer to these integers as idempotent, because ∀a∈Zn,ak(p¯−1)(q¯−1)+1≡na for any positive integer k. This set was initially known to contain only the semiprimes, and later expanded to include some of the Carmichael numbers. Recent work by the author gave the explicit formulation for the set, showing that t

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Ali Larnene, Yamina, Samir LADACI, and Aissa Belemeguenai. "RLS-based Identification of fractional order H n1,n2 system using the Singularity Function approximation." Algerian Journal of Signals and Systems 5, no. 4 (2020): 197–202. http://dx.doi.org/10.51485/ajss.v5i4.117.

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This paper presents a study of fractional order systems modeling and identification by recursive least squares (RLS) with forgetting factor estimation technique. The fractional order integrators are implemented using the Singularity Function approximation method. Parametric Identification of fractional order differential equations (FDE) is investigated when estimating system parameters by a linear model with respect to parameters, as well as non-integer orders from temporal data (H n1,n2 )-type model. A numerical simulation example illustrates the effectiveness of the proposed identification a

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LLIBRE, JAUME, ANA CRISTINA MEREU, and MARCO ANTONIO TEIXEIRA. "Limit cycles of the generalized polynomial Liénard differential equations." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 2 (2009): 363–83. http://dx.doi.org/10.1017/s0305004109990193.

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AbstractWe apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.

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Cichacz, Sylwia, and Karolina Szopa. "Modular Edge-Gracefulness of Graphs without Stars." Symmetry 12, no. 12 (2020): 2013. http://dx.doi.org/10.3390/sym12122013.

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We investigate the modular edge-gracefulness k(G) of a graph, i.e., the least integer k such that taking a cyclic group Zk of order k, there exists a function f:E(G)→Zk so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on k(G) for a general graph G is 2n, where n is the order of G. In this note we prove that if G is a graph of order n without star as a component then k(G)=n for n¬≡2(mod4) and k(G)=n+1 otherwise. Moreover we show that for such G for every integer t≥k(G) there exists a Zt-irregular labeling.

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NITAJ, ABDERRAHMANE. "CRYPTANALYSIS OF RSA WITH CONSTRAINED KEYS." International Journal of Number Theory 05, no. 02 (2009): 311–25. http://dx.doi.org/10.1142/s1793042109002122.

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Let n = pq be an RSA modulus with unknown prime factors of equal bit-size. Let e be the public exponent and d be the secret exponent satisfying ed ≡ 1 mod ϕ(n) where ϕ(n) is the Euler totient function. To reduce the decryption time or the signature generation time, one might be tempted to use a small private exponent d. Unfortunately, in 1990, Wiener showed that private exponents smaller than [Formula: see text] are insecure and in 1999, Boneh and Durfee improved the bound to n0.292. In this paper, we show that instances of RSA with even large private exponents can be efficiently broken if the

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Rezakhanlou, Fraydoun. "The packing measure of the graphs and level sets of certain continuous functions." Mathematical Proceedings of the Cambridge Philosophical Society 104, no. 2 (1988): 347–60. http://dx.doi.org/10.1017/s0305004100065518.

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AbstractThe relationship between the local growth of a continuous function and the packing measure of its level sets and of its graph is studied. For the Weierstrass function with b an integer such that b ≥ 2 and with 0 < α < 1, and for x ∈ Range (W) outside a set of first category, the level set W−1(x) has packing dimension at least 1 − α. Furthermore, for almost all x ∈ Range (W), the packing dimension of f is at most 1 − α. Finer results on the occupation measure and the size of the graph of a continuous function satisfying the Zygmund Λ-condition are obtained.

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Liu, Hui, and Gaosheng Zhu. "Non-hyperbolic P-Invariant Closed Characteristics on Partially Symmetric Compact Convex Hypersurfaces." Advanced Nonlinear Studies 18, no. 4 (2018): 763–74. http://dx.doi.org/10.1515/ans-2017-6050.

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AbstractLet {n\geq 2} be an integer, {P=\mathrm{diag}(-I_{n-\kappa},I_{\kappa},-I_{n-\kappa},I_{\kappa})} for some integer {\kappa\in[0,n]}, and let {\Sigma\subset{\mathbb{R}}^{2n}} be a partially symmetric compact convex hypersurface, i.e., {x\in\Sigma} implies {Px\in\Sigma}, and {(r,R)}-pinched. In this paper, we prove that when {{R/r}<\sqrt{5/3}} and {0\leq\kappa\leq[\frac{n-1}{2}]}, there exist at least {E(\frac{n-2\kappa-1}{2})+E(\frac{n-2\kappa-1}{3})} non-hyperbolic P-invariant closed characteristics on Σ. In addition, when {{R/r}<\sqrt{3/2}}, {[\frac{n+1}{2}]\leq\kappa\leq n} and

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Okoh, F., and F. Zorzitto. "Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields." Canadian Journal of Mathematics 59, no. 1 (2007): 186–210. http://dx.doi.org/10.4153/cjm-2007-008-7.

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AbstractPurely simple Kronecker modules ℳ, built from an algebraically closed field K, arise from a triplet (m, h, α) where m is a positive integer, h: K ∪ ﹛∞﹜ → ﹛∞, 0, 1, 2, 3, … ﹜ is a height function, and α is a K-linear functional on the space K(X) of rational functions in one variable X. Every pair (h, α) comes with a polynomial f in K(X)[Y] called the regulator. When the module ℳ admits nontrivial endomorphisms, f must be linear or quadratic in Y. In that case ℳ is purely simple if and only if f is an irreducible quadratic. Then the K-algebra End ℳ embeds in the quadratic function field

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34

Brown, Tom C., Bruce M. Landman, and Marni Mishna. "Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}." Canadian Mathematical Bulletin 40, no. 2 (1997): 149–57. http://dx.doi.org/10.4153/cmb-1997-018-3.

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AbstractFor positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1, 2,...,N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}.We show that f (s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden’s theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + t}. We also show that f (s, t) = 4(s + t) + 1 in many other cases (for example, whenever

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35

Yang, Hong, Pu Wu, Sakineh Nazari-Moghaddam, et al. "Bounds for signed double Roman k-domination in trees." RAIRO - Operations Research 53, no. 2 (2019): 627–43. http://dx.doi.org/10.1051/ro/2018043.

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Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this

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36

Brennan, Charlotte, Arnold Knopfmacher, Toufik Mansour, and Stephan Wagner. "Separation of the maxima in samples of geometric random variables." Applicable Analysis and Discrete Mathematics 5, no. 2 (2011): 271–82. http://dx.doi.org/10.2298/aadm110817019b.

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We consider samples of n geometric random variables W1 W2 ... Wn where P{W) = i} = pqi-l, for 1 ? j ? n, with p + q = 1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice's method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random

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37

Byun, Sang-Seon, Kimmo Kansanen, Ilangko Balasingham, and Joon-Min Gil. "Achieving Fair Spectrum Allocation and Reduced Spectrum Handoff in Wireless Sensor Networks: Modeling via Biobjective Optimization." Modelling and Simulation in Engineering 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/406462.

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This paper considers the problem of centralized spectrum allocations in wireless sensor networks towards the following goals: (1) maximizing fairness, (2) reflecting the priority among sensor data, and (3) avoiding unnecessary spectrum handoff. We cast this problem into a multiobjective mixed integer nonconvex nonlinear programming that is definitely difficult to solve at least globally without any aid of conversion or approximation. To tackle this intractability, we first convexify the original problem using arithmetic-geometric mean approximation and logarithmic change of the decision variab

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SUZUKI, AKIRA, KEI UCHIZAWA, and XIAO ZHOU. "ENERGY-EFFICIENT THRESHOLD CIRCUITS COMPUTING MOD FUNCTIONS." International Journal of Foundations of Computer Science 24, no. 01 (2013): 15–29. http://dx.doi.org/10.1142/s0129054113400029.

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We prove that the modulus function MODm of n variables can be computed by a threshold circuit C of energy e and size s = O(e(n/m)1/(e − 1)) for any integer e ≥ 2, where the energy e is defined to be the maximum number of gates outputting "1" over all inputs to C, and the size s to be the number of gates in C. Our upper bound on the size s almost matches the known lower bound s = Ω(e(n/m)1/e). We also consider an extreme case where threshold circuits have energy 1, and prove that such circuits need at least 2(n − m)/2 gates to compute MODm of n variables.

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HSU, D. F., and SANMING ZHOU. "RESOLVABLE MENDELSOHN DESIGNS AND FINITE FROBENIUS GROUPS." Bulletin of the Australian Mathematical Society 98, no. 1 (2018): 1–13. http://dx.doi.org/10.1017/s0004972718000333.

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We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v,k,1)$-Mendelsohn design for any integers $v>k\geq 2$ with $v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\unicode[STIX]{x1D719}$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K\rtimes \langle \unicode[STIX]{x1D719}\rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v=p_{1}^

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Nguyen, Dung Quang. "An Effective Approach of Approximation of Fractional Order System using Real Interpolation Method." Journal of Advanced Engineering and Computation 1, no. 1 (2017): 39. http://dx.doi.org/10.25073/jaec.201711.48.

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Fractional-order controllers are recognized to guarantee better closed-loop performance and robustness than conventional integer-order controllers. However, fractional-order transfer functions make time, frequency domain analysis and simulation significantly difficult. In practice, the popular way to overcome these difficulties is linearization of the fractional-order system. Here, a systematic approach is proposed for linearizing the transfer function of fractional-order systems. This approach is based on the real interpolation method (RIM) to approximate fractional-order transfer function (F

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ERLANDSSON, VIVEKA, and HUGO PARLIER. "Short closed geodesics with self-intersections." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 3 (2020): 623–38. http://dx.doi.org/10.1017/s030500411900032x.

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AbstractOur main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

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42

Heitzig, Jobst. "Every finite system of T1 uniformities comes from a single distance structure." Applied General Topology 3, no. 1 (2002): 65. http://dx.doi.org/10.4995/agt.2002.2113.

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<p>Using the general notion of distance function introduced in an earlier paper, a construction of the finest distance structure which induces a given quasi-uniformity is given. Moreover, when the usual defining condition xy : d(y; x) of the basic entourages is generalized to nd(y; x) n (for a fixed positive integer n), it turns out that if the value-monoid of the distance function is commutative, one gets a countably infinite family of quasi-uniformities on the underlying set. It is then shown that at least every finite system and every descending sequence of T<sub>1</sub> q

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Feng, Shaojun, Washington Ochieng, Jaron Samson, et al. "Integrity Monitoring for Carrier Phase Ambiguities." Journal of Navigation 65, no. 1 (2011): 41–58. http://dx.doi.org/10.1017/s037346331100052x.

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The determination of the correct integer number of carrier cycles (integer ambiguity) is the key to high accuracy positioning with carrier phase measurements from Global Navigation Satellite Systems (GNSS). There are a number of current methods for resolving ambiguities including the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) method, which is a combination of least-squares and a transformation to reduce the search space. The current techniques to determine the level of confidence (integrity) of the resolved ambiguities (i.e. ambiguity validation), usually involve the constructio

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44

Banks, William D., and Igor E. Shparlinski. "On values taken by the largest prime factor of shifted primes." Journal of the Australian Mathematical Society 82, no. 1 (2007): 133–47. http://dx.doi.org/10.1017/s1446788700017511.

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AbstractLet P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n > 1. We show that, for every real number , there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set has relative asymptotic density one in the set of all prime numbers. Moreover, in the range , one can take c(η) = 1+ε for any fixed ε > 0. In particular, our results imply that for every real number 0.486 ≤ b.thetav; ≤ 0.531, the relation P(q − a) ≍ qθ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime

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45

Pillichshammer, F. "A maximal Gross-Stadje number in the Euclidean plane." Bulletin of the Australian Mathematical Society 61, no. 1 (2000): 109–19. http://dx.doi.org/10.1017/s0004972700022061.

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Let X be a compact, connected Hausdorff space and f a real valued, symmetric, continuous function on X × X. Then the Gross-Stadje number r (X, f) is the unique real number with the property that for each positive integer n and for all (not necessarily distinct) x1,…,xn in X, there exists some x in X such that . This paper solves the following open question in distance geometry: What is the least upper bound g2(R2) of r (X, d2), where X ranges over all compact, connected subsets of the Euclidean plane with diameter one and where d2 denotes the squared, Euclidean distance. We show: .

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46

NIKIFOROV, V. "On the Edge Distribution of a Graph." Combinatorics, Probability and Computing 10, no. 6 (2001): 543–55. http://dx.doi.org/10.1017/s0963548301004837.

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We investigate a graph function which is related to the local density, the maximal cut and the least eigenvalue of a graph. In particular it enables us to prove the following assertions.Let p [ges ] 3 be an integer, c ∈ (0, 1/2) and G be a Kp-free graph on n vertices with e [les ] cn2 edges. There exists a positive constant α = α (c, p) such that:(a) some [lfloor ]n/2[rfloor ]-subset of V (G) induces at most (c-4 − α) n2 edges (this answers a question of Paul Erdős);(b) G can be made bipartite by the omission of at most (c-2 − α) n2 edges.

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Hamada, Mitsuru. "The minimum number of rotations about two axes for constructing an arbitrarily fixed rotation." Royal Society Open Science 1, no. 3 (2014): 140145. http://dx.doi.org/10.1098/rsos.140145.

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For any pair of three-dimensional real unit vectors m ^ and n ^ with | m ^ T n ^ | < 1 and any rotation U , let N m ^ , n ^ ( U ) denote the least value of a positive integer k such that U can be decomposed into a product of k rotations about either m ^ or n ^ . This work gives the number N m ^ , n ^ ( U ) as a function of U . Here, a rotation means an element D of the special orthogonal group SO (3) or an element of the special unitary group SU (2) that corresponds to D . Decompositions of U attaining the minimum number N m ^ , n ^ ( U ) are also given explicitly.

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48

Khoeilar, R., L. Shahbazi, S. M. Sheikholeslami, and Zehui Shao. "Bounds on the signed total Roman 2-domination in graphs." Discrete Mathematics, Algorithms and Applications 12, no. 01 (2020): 2050013. http://dx.doi.org/10.1142/s1793830920500135.

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Let [Formula: see text] be an integer and [Formula: see text] be a simple and finite graph with vertex set [Formula: see text]. A signed total Roman [Formula: see text]-dominating function (STR[Formula: see text]DF) on a graph [Formula: see text] is a function [Formula: see text] such that (i) every vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text] and (ii) [Formula: see text] holds for any vertex [Formula: see text]. The weight of an STR[Formula: see text]DF [Formula: see text] is [Formula: see text] and the min

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Wu, Hongtao, Xiubin Zhao, Chunlei Pang, Liang Zhang, and Bo Feng. "Multivariate Constrained GNSS Real-time Full Attitude Determination Based on Attitude Domain Search." Journal of Navigation 72, no. 2 (2018): 483–502. http://dx.doi.org/10.1017/s0373463318000784.

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A priori attitude information can improve the success rate and reliability of Global Navigation Satellite System (GNSS) multi-antennae attitude determination. However, a priori attitude information is nonlinear, and integrating a priori information into the objective function rigorously will increase the complexity of an ambiguity domain search, such as the Multivariate Constrained-Least-squares Ambiguity Decorrelation Adjustment (MC-LAMBDA) method. In this paper, a new method based on attitude domain search is presented to make use of the a priori attitude angle information with high efficien

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Bouchou, Ahmed, Mostafa Blidia, and Mustapha Chellali. "Relations between the Roman k-domination and Roman domination numbers in graphs." Discrete Mathematics, Algorithms and Applications 06, no. 03 (2014): 1450045. http://dx.doi.org/10.1142/s1793830914500451.

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Let G = (V, E) be a graph and let k be a positive integer. A Roman k-dominating function ( R k-DF) on G is a function f : V(G) → {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices v1, v2, …, vk with f(vi) = 2 for i = 1, 2, …, k. The weight of an R k-DF is the value f(V(G)) = ∑u∈V(G) f(u) and the minimum weight of an R k-DF on G is called the Roman k-domination number γkR(G) of G. In this paper, we present relations between γkR(G) and γR(G). Moreover, we give characterizations of some classes of graphs attaining equality in these relations. Finally, we esta

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