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A proper subset X of vertices in the graph G is a captive dominating set if it is a total dominating set and each vertex in X dominates at least one vertex which does not belong to X ; also, the captive domination number of G , denoted by γ ca (G) , is the cardinality of a minimum captive dominating set of G. In [M. N. Al-Harere, A. A. Omran and A. T. Breesam, Captive domination in graphs, Discrete Math. Algorithms Appl. 12(6) (2020) 2050076], Al-Harere et al., presented some lower and upper bounds for the captive domination number of G , using the number of vertices and edges. Here, we present some upper bounds for γ ca (G) , using independence number and clique number. Also, the captive domination number of split graphs is studied. Among other results, some lower bounds for the captive domination number, in terms of maximum degree and the number of vertices with maximum degree, are presented. [ABSTRACT FROM AUTHOR]

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalizing this idea and, as illustration, construct and analyze straight-line programs for the Bruhat decomposition of d × d matrices with determinant 1 over a finite field of order q that have length O(d2log(q)) and require storing only O(log(q)) matrices during evaluation. [ABSTRACT FROM AUTHOR]

The Pólya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large class numbers by Cherubini et al., we extend the notion of consecutiveness of number fields to certain parametric families of cyclic cubic fields and bi-quadratic fields and address the question of the existence of infinitely many such consecutive fields with large Pólya groups. This extends a recent result of the second author and Saikia for totally real bi-quadratic fields. [ABSTRACT FROM AUTHOR]

In this paper, we show that each non-degenerate Lucas sequence contains only finitely many terms which can be written as products of double factorials or as sums or differences of double factorials. In particular, we find all the solutions to the products of double factorials which are sums or differences of two Fibonacci numbers, and Fibonacci numbers which are sums or differences of two double factorials. [ABSTRACT FROM AUTHOR]

Let P (x) : = a d x d + ⋯ + a 0 ∈ ℚ [ x ] , a d > 0 , be a polynomial of degree d ≥ 2. Let (x n) be a sequence of integers satisfying x n + 1 = P (x n) for all  n = 0 , 1 , 2 , ... and x n → ∞ as  n → ∞. Set α : = lim n → ∞ x n d − n . Then, under the assumption a d 1 / (d − 1) ∈ ℚ , in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J. 57 (2022) 569–581], either α is transcendental or α can be an integer or a quadratic Pisot unit with α − 1 being its conjugate over ℚ. In this paper, we study the nature of such α without the assumption that a d 1 / (d − 1) is in ℚ , and we prove that either the number α is transcendental, or α h is a Pisot number with h being the order of the torsion subgroup of the Galois closure of the number field ℚ α , a d − 1 d − 1 . Other results presented in this paper investigate the solutions of the inequality | | q 1 α 1 n + ⋯ + q k α k n + β | | < n in (n , q 1 , ... , q k) ∈ ℕ × (K ×) k , considering whether β is rational or irrational. Here, K represents a number field, and ∈ (0 , 1). The notation | | x | | denotes the distance between x and its nearest integer in ℤ. [ABSTRACT FROM AUTHOR]

For every algebraic number κ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of f κ (z) = log | z − κ | at the conjugates are essentially bounded from above by − h (κ). This completes a characterization on functions f κ initiated by Autissier and Baker–Masser, who cover the cases κ = 2 and | κ | ≠ 1 , respectively. Using the same ideas we also prove analogues in the p -adic setting. [ABSTRACT FROM AUTHOR]

In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes ⊆ ∏ i = 1 N q i over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity. [ABSTRACT FROM AUTHOR]

We fix a number field K and study statistical properties of the ring K [ γ ] ∩ K as γ varies over algebraic numbers of a fixed degree n ≥ 2. Given k ≥ 1 , we explicitly compute the density of γ for which K [ γ ] ∩ K = K [ 1 / k ] and show that this does not depend on the number field K. In particular, we show that the density of γ for which K [ γ ] ∩ K = K is ζ (n + 1) ζ (n) . In a recent paper [Singhal and Lin, Primes in denominators of algebraic numbers, Int. J. Number Theory (2023), doi:10.1142/S1793042124500167], the authors define X (K , γ) to be a certain finite subset of Spec ( K) and show that X (K , γ) determines the ring K [ γ ] ∩ K. We show that if 1 , 2 ∈ Spec ( K) satisfy 1 ∩ ℤ ≠ 2 ∩ ℤ , then the events 1 ∈ X (K , γ) and 2 ∈ X (K , γ) are independent. As t → ∞ , we study the asymptotics of the density of γ for which | X (K , γ) | = t. [ABSTRACT FROM AUTHOR]

We point out that there is an error in the proof of Theorem 1.1 in [The discriminant of compositum of algebraic number fields, Int. J. Number Theory 15 (2019) 353–360]. We also prove that the result of this theorem holds with an additional hypothesis. However, it is an open problem whether the result of the theorem is true in general or not. [ABSTRACT FROM AUTHOR]

Let q be a prime number and K = Q () be an algebraic number field with a root of an irreducible polynomial x q s − a x − b having integer coefficients. In this paper, we provide some explicit conditions involving only s , q , a , b for which K is non-monogenic. As an application, in the special case of a = 0 and q = 2 , we show that if s ≥ 2 and 3 2 divides b − 1 , then K is not monogenic. We illustrate our results through examples. [ABSTRACT FROM AUTHOR]

Denote the set of algebraic numbers as ℚ ¯ and the set of algebraic integers as ℤ ¯. For γ ∈ ℚ ¯ , consider its irreducible polynomial in ℤ [ x ] , F γ (x) = a n x n + ⋯ + a 0 . Denote e (γ) = gcd (a n , a n − 1 , ... , a 1). Drungilas, Dubickas and Jankauskas show in a recent paper that ℤ [ γ ] ∩ ℚ = { α ∈ ℚ | { p | v p (α) < 0 } ⊆ { p : p | e (γ) } }. Given a number field K and γ ∈ ℚ ¯ , we show that there is a subset X (K , γ) ⊆ Spec ( K) , for which K [ γ ] ∩ K = { α ∈ K | { | v (α) < 0 } ⊆ X (K , γ) }. We prove that K [ γ ] ∩ K is a principal ideal domain if and only if the primes in X (K , γ) generate the class group of K . We show that given γ ∈ ℚ ¯ , we can find a finite set S ⊆ ℤ ¯ , such that for every number field K , we have X (K , γ) = { ∈ Spec ( K) | p ∩ S ≠ ∅ }. We study how this set S relates to the ring ℤ ¯ [ γ ] and the ideal γ = { a ∈ ℤ ¯ | a γ ∈ ℤ ¯ } of ℤ ¯. We also show that γ 1 , γ 2 ∈ ℚ ¯ satisfy γ 1 = γ 2 if and only if X (K , γ 1) = X (K , γ 2) for all number fields K. [ABSTRACT FROM AUTHOR]

Let K be an algebraic number field. With K we associate the ring of finite adeles K. In this paper we give a path integral formula for the propagator of a quantum mechanical system over the abelian group K n. Specifically, we consider matrix-valued Hamiltonian operators H K n = Δ K n ⊗ Id + V , where Δ K n is the Vladimirov operator and V is a non-negative definite potential. The free part of the Hamiltonian gives rise to a measure on the Skorokhod space of paths which allows us to prove the Feynman–Kac formula for the Schrödinger semigroup generated by − H K n . This formula is given in terms of the ordered time exponentials. [ABSTRACT FROM AUTHOR]

It is shown that the twisted Fermat quartic x 4 + y 4 + z 4 = 0 has no points over any odd degree extensions of ℚ (i). All points over quadratic extensions of ℚ (i) are found. [ABSTRACT FROM AUTHOR]

Let K = ℚ () be an algebraic number field with a root of an irreducible trinomial f (x) = x 5 + a x + b belonging to ℤ [ x ]. In this paper, we compute the highest power of each prime p dividing the discriminant of K in terms of powers of p dividing a , b and the discriminant of f (x) besides explicitly constructing a p -integral basis of K. These p -integral bases lead to the construction of an integral basis of K which is illustrated with examples. [ABSTRACT FROM AUTHOR]

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number α , the additive monoid M α of the evaluation semiring ℕ 0 [ α ] is atomic. The atomic structure of both the additive and the multiplicative monoids of ℕ 0 [ α ] has been the subject of several recent papers. Here we focus on the monoids M α , and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when α is less than 1 , the atoms of ℕ 0 [ α ] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of ℕ 0 [ α ] , including that when α is rational, the elasticity of M α is full (this was previously conjectured by Chapman, Gotti and Gotti). [ABSTRACT FROM AUTHOR]

In this paper, as an analog of the number field case, for an elliptic curve E defined over the algebraic numbers and for any subfield F of algebraic numbers, we say that E has the Northcott property over F if there are at most finitely many F -rational points on E of uniformly bounded height, and we say that E has the property (P) over F if for any infinite subset S of F -rational points on E , f (S) = S for an F -endomorphism f of E implies that f is an automorphism. We establish some criteria for both properties and provide typical examples. We also show that the Northcott property implies the property (P), but the converse is not true. [ABSTRACT FROM AUTHOR]

We study the continued fractions I 1 (τ) and I 2 (τ) of order sixteen by adopting the theory of modular functions. These functions are analogues of Rogers–Ramanujan continued fraction r (τ) with modularity and many interesting properties. Here we prove the modularities of I 1 (τ) and I 2 (τ) to find the relation with the generator of the field of modular functions on Γ 0 (1 6). Moreover we prove that the values 2 (I 1 (τ) 2 + 1 / I 1 (τ) 2) and 2 (I 2 (τ) 2 + 1 / I 2 (τ) 2) are algebraic integers for certain imaginary quadratic quantity τ. [ABSTRACT FROM AUTHOR]

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2). [ABSTRACT FROM AUTHOR]

A ring R is said to be clean if each element of R can be written as the sum of a unit and an idempotent. R is said to be weakly clean if each element of R is either a sum or a difference of a unit and an idempotent, and R is said to be feebly clean if every element r can be written as r = u + e 1 − e 2 , where u is a unit and e 1 , e 2 are orthogonal idempotents. Clearly, clean rings are weakly clean rings and both of them are feebly clean. In a recent paper (J. Algebra Appl.17 (2018) 1850111 (5 pp.)), McGoven characterized when the group ring ℤ (p) [ C q ] is weakly clean and feebly clean, where p , q are distinct primes. In this paper, we consider a more general setting. Let K be an algebraic number field, its ring of integers, ⊂ a nonzero prime ideal and the localization of at . We investigate when the group ring [ G ] is weakly clean and feebly clean, where G is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when K = ℚ (ζ n) is a cyclotomic field or K = ℚ (d) is a quadratic field. [ABSTRACT FROM AUTHOR]

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behavior at finitely many places. [ABSTRACT FROM AUTHOR]

Given a number field L , we define the degree of an algebraic number v ∈ L with respect to a choice of a primitive element of L. We propose the question of computing the minimal degrees of algebraic numbers in L , and examine these values in degree 4 Galois extensions over ℚ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves. [ABSTRACT FROM AUTHOR]

Computational geometry algorithms branch on the signs of predicates. Prior predicate evaluation techniques are slow on degenerate (zero sign) predicates, especially on predicates on algebraic numbers. Degeneracy is common for predicates whose arguments have common antecedents. We present three randomized algorithms for degeneracy detection. The first algorithm uses modular arithmetic to detect degenerate predicates on rational numbers. The second algorithm uses quotient rings to reduce detecting degenerate predicates on algebraic numbers to multiple rational predicates, which can be evaluated deterministically or using the first algorithm. This algorithm is impractical because it is exponential in the number of algebraic numbers, yet it is still much faster than prior work that uses root separation bounds. The third algorithm uses a perturbation to eliminate degenerate algebraic predicates that are not identical to zero and a second perturbation to detect those that are. The first and third algorithms are incorporated into an exact geometric computation library. By sampling values generated by the algorithms, the library estimates the degeneracy detection failure probability over the lifetime of every calling program. We call this approach statistical degeneracy detection (SDD). Extensive testing shows that predicate evaluation is reliable and fast. [ABSTRACT FROM AUTHOR]

In this paper, we show that if (X n , Y n) is the n th solution of the Pell equation X 2 − d Y 2 = ± 1 for some non-square d , then given any integer c , the equation c = X n − 2 m has at most 2 integer solutions (n , m) with n ≥ 0 and m ≥ 0 , except for the only pair (c , d) = (− 1 , 2). Moreover, we show that this bound is optimal. Additionally, we propose a conjecture about the number of solutions of Pillai's problem in linear recurrent sequences. [ABSTRACT FROM AUTHOR]

For an algebraic number field K , let ζ K (s) be the associated Dedekind zeta-function. It is conjectured that ζ K (m) is transcendental for any positive integer m > 1. The only known case of this conjecture was proved independently by Siegel and Klingen, namely that, when K is a totally real number field, ζ K (2 n) is an algebraic multiple of π 2 n [ K : ℚ ] and hence, is transcendental. If K is not totally real, the question of whether ζ K (m) is irrational or not remains open. In this paper, we prove that for a fixed integer n ≥ 1 , at most one of ζ K (2 n + 1) is rational, as K varies over all imaginary quadratic fields. We also discuss a generalization of this theorem to CM-extensions of number fields. [ABSTRACT FROM AUTHOR]

The spectral and non-spectral problems of measures have been considered in recent years. For the Cantor measure μ ρ , Hu and Lau [Spectral property of the B ernoulli convolutions, Adv. Math.219(2) (2008) 554–567] showed that L 2 (μ ρ) contains infinite orthogonal exponentials if and only if ρ becomes some type of binomial number. In this paper, we classify the spectral number of the Cantor measure μ ρ except the contraction ratio ρ being some algebraic numbers called odd-trinomial number. When ρ is an odd-trinomial number, we provide an exponential and polynomial estimations of the upper bound of the spectral number related to the algebraic degree of ρ. Some examples on odd-trinomial number via generalized Fibonacci numbers are provided such that the spectral number of them can be determined. Our study involves techniques from polynomial theory, especially the decomposition theory on trinomial. [ABSTRACT FROM AUTHOR]

For each algebraic number α ∈ ℚ ̄ , a result of Habegger [P. Habegger, Singular moduli that are algebraic units, Algebra Number Theory 9(7) (2015) 1515–1524] shows that there are only finitely many singular moduli j such that j − α is an algebraic unit. His result uses Duke's Equidistribution Theorem and is thus not effective. In this paper, we give an effective proof of Habegger's result assuming that α is not a singular modulus itself. We give an explicit bound, which depends only on α , on the discriminant Δ associated with a singular modulus j such that j − α is a unit. This implies explicit bounds on the number of these singular moduli. [ABSTRACT FROM AUTHOR]

For an algebraic number field K and a prime number p , let K ̃ / K be the maximal multiple ℤ p -extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro- p extension of K ̃ is pseudo-null over the completed group ring ℤ p [ [ Gal (K ̃ / K) ] ]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers. [ABSTRACT FROM AUTHOR]

Let Γ be a finite rank subgroup of ℚ ¯ ∗ . We prove that the multiplicative group of the field generated by all elements in the divisible hull of Γ is free abelian modulo this divisible hull. This proves that a necessary condition for Rémond's generalized Lehmer conjecture is satisfied. [ABSTRACT FROM AUTHOR]

For each algebraic number α and each positive real number t , the t -metric Mahler measure m t (α) creates an extremal problem whose solution varies depending on the value of t. The second author studied the points t at which the solution changes, called exceptional points for α. Although each algebraic number has only finitely many exceptional points, it is conjectured that, for every N ∈ ℕ , there exists a number having at least N exceptional points. In this paper, we describe a polynomial time algorithm for establishing the existence of numbers with at least N exceptional points. Our work constitutes an improvement over the best known existing algorithm which requires exponential time. We apply our main result to show that there exist numbers with at least 3 7 exceptional points, another improvement over previous work which was only able to reach 1 1 exceptional points. [ABSTRACT FROM AUTHOR]

Let K = ℚ (𝜃) be an algebraic number field with 𝜃 an algebraic integer having minimal polynomial f (x) over the field ℚ of rational numbers and A K be the ring of algebraic integers of K. For a fixed prime number p , let f ̄ (x) = ḡ 1 (x) e 1 ⋯ ḡ r (x) e r be the factorization of f (x) modulo p as a product of powers of distinct irreducible polynomials over ℤ / p ℤ with g i (x) ∈ ℤ [ x ] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number p does not divide the index [ A K : ℤ [ 𝜃 ] ] if and only if ∏ i = 1 r ḡ i (x) e i − 1 is coprime with M ¯ (x) where M (x) = 1 p [ f (x) − g 1 (x) e 1 ⋯ g r (x) e r ]. This criterion has been widely used and generalized. In this paper, a simple proof of Generalized Dedekind Criterion [S. K. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation rings, Comm. Algebra38 (2010) 684–696] using elementary valuation theory is given. [ABSTRACT FROM AUTHOR]

Fix a weakly minimal (i.e. superstable U -rank 1) structure ℳ. Let ℳ ∗ be an expansion by constants for an elementary substructure, and let A be an arbitrary subset of the universe M. We show that all formulas in the expansion (ℳ ∗ , A) are equivalent to bounded formulas, and so (ℳ , A) is stable (or NIP) if and only if the ℳ -induced structure A ℳ on A is stable (or NIP). We then restrict to the case that ℳ is a pure abelian group with a weakly minimal theory, and A ℳ is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of (ℤ , +). Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form (ℳ , A). Most notably, we show that if (G , +) is a weakly minimal additive subgroup of the algebraic numbers, A ⊆ G is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of A is a root of unity, then (G , + , B) is superstable for any B ⊆ A. [ABSTRACT FROM AUTHOR]

Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree 4 0 2 0 , while for the 5 2   3 7 7 period-20 orbits the degree rises already to 1   0 4 7   5 4 0. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map. [ABSTRACT FROM AUTHOR]

The reanalysis method to obtain the second derivatives of static displacement is innovatively proposed in this paper. This method is based on the combined approximations method. The reanalysis formulations of the second derivative of static displacement are derived to provide a programmatic procedure of formulations construction. Besides, the normalized error and the number of algebraic operations are considered to evaluate the accuracy and efficiency, respectively. Finally, three typical numerical results verify the accuracy and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

We extend two results by Boxall and Jones on algebraic values of certain analytic functions to meromorphic functions. We obtain C (log H) η bounds for the number of algebraic points of height at most H on certain restrictions of the graphs of such functions. The constant C and exponent η depend on certain data associated with the functions and can be effectively computed from them. [ABSTRACT FROM AUTHOR]

In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions 3 F 2 , to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of 3 F 2 a , b , q a + b , q ; 1 of the generalized hypergeometric function is a ℚ ¯ -linear combination of log of algebraic numbers if rational numbers a , b , q satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it. [ABSTRACT FROM AUTHOR]

The Kirchhoff index and Laplacian-energy-like invariant of a connected graph G , denoted by Kf (G) and LEL (G) , are given by the number of vertex times the sum of the reciprocals of all nonzero Laplacian eigenvalues of G and the sum of the square roots of all Laplacian eigenvalues of G , respectively. In this paper, we have obtained the Laplacian eigenvalues of some derived graphs, such as double graph, extended double cover and Mycielskian of an r -regular graph G , in terms of the adjacency eigenvalues of G and hence, we obtain some upper bounds of Kirchhoff index and Laplacian-energy-like (LEL) invariant of those derived graphs in terms of r , number of vertices and algebraic connectivity of G. We have shown that the bounds obtained here are better than some existing bounds. We have also obtained the exact formulae for Kirchhoff index and LEL invariant of those derived graph when G is a complete graph or a complete bipartite graph. [ABSTRACT FROM AUTHOR]

Following and generalizing unpublished work of Ange, we prove a generalized version of Rémond's generalized Vojta inequality. This generalization can be applied to arbitrary products of irreducible positive-dimensional projective varieties, defined over the field of algebraic numbers, instead of powers of one fixed such variety. The proof runs closely along the lines of Rémond's proof. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Ramanujan's sum to the ring of integers of an algebraic number field. We also obtain the orthogonality properties of Ramanujan's sum in the ring of integers. [ABSTRACT FROM AUTHOR]

In this paper, the two-dimensional (2D) turbulence perturbed by arrays of cylinders placed both horizontally and vertically is investigated by Immersed Boundary Lattice Boltzmann Method (IB-LBM). The energy spectrum reveals the coexistence of the inverse and direct cascades in 2D grid turbulence. By observing at the distribution of fluxes in space, the energy and enstrophy fluxes have explained the physical mechanism of the double cascades where the two Kolmogorov laws for structure functions are simultaneously observed. The results of vortex statistics by the conditional analysis, which are based on a new and accurate vortex identification criteria called Liutex, show that the algebraic number density n (A) ∼ A − 2 , where A is vortex area. The time-evolving vortex number density distribution constructs a theoretical framework involving a three-part: n (A , t) = A − 0. 5 (A min ≤ A < A −) ; t − 1 A − 1 (A − < A < A +) ; t 7 A − 8 (A + < A ≤ A max) , which is satisfied with the prediction well. The relationship between the vortex circulation | Γ | (A) and vortex area A is | Γ | (A) ∼ A and the one between the kinetic energy of vortex E e (A) and A is E e (A) ∼ A in the range where n (A) ∼ A − 2 . Moreover, it has been found that vortices contain about 30% of the total energy of the flow by studying the energy ratio of all vortices to the entire flow field. What is more, it is an interesting phenomenon is that there is only a range where E (k) ∼ k − 5 / 3 in the energy spectrum for the coherent structure field which is obtained by using Liutex as the extraction of vortices. The probability density function (PDF) of the fluctuations of longitudinal velocity shows that an indication of small intermittency in the direct cascade and the absence of intermittency in the inverse cascade range. On the other hand, the scaling exponents ζ (p) of the structure function for the inverse cascade are consistent with Kr67 model, which shows the absence of intermittency. While the measured intermittency parameters are J = 0. 0 0 6 and H = 0. 9 2 1 , which explains that there is a very weak intermittent correction in the direct cascade, and ESS has verified the existence of intermittency in our 2D turbulence. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of counting algebraic numbers α of fixed degree n and bounded height Q such that the derivative of the minimal polynomial P α (x) of α is bounded, P α ′ (α) < Q 1 − v . This problem has many applications to the problems of metric theory of Diophantine approximation. We prove that the number of α defined above on the interval − 1 2 , 1 2 does not exceed c 1 (n) Q n + 1 − 1 7 v for Q > Q 0 (n) and 1. 4 ≤ v ≤ 7 1 6 (n + 1). Our result is based on an improvement to a lemma from Gelfond's monograph "Transcendental and algebraic numbers". Given an integer polynomial small enough in some point, the lemma provides an upper bound for the absolute value of its irreducible divisor. We obtain a stronger estimate which holds in real points located far enough from all algebraic numbers of bounded degree and height. This is done by considering the resultant of two polynomials represented as the determinant of the Sylvester matrix for the shifted counterparts. [ABSTRACT FROM AUTHOR]

In this paper, statistical behaviors of vortex in the wake flow of a circular cylinder placed near a plane wall in the two-dimensional (2D) channel are investigated by Immersed Boundary Lattice Boltzmann Method. Although vortices shed from the cylinder periodically, there are very complicated interactions between vortices and the solid wall. The gap ratios (L / D) are considered in three cases (L / D = 0 , L / D = 0. 1 ,   L / D = 0. 5) to study vortex statistic behaviors when flow is in steady state. It can be seen that a single row of coherent structures in the same sign varies to dipole vortex shedding as L / D increases. The results of vortex statistics by the conditional analysis, which are based on a new and accurate vortex identification criteria called Liutex, show the algebraic number density n (A) ∼ A − 2 , where A is vortex area. The relationship between the vortex circulation | Γ | (A) and vortex area A is | Γ | (A) ∼ A and the one between the kinetic energy of vortex E e (A) and A is E e (A) ∼ A in the range where n (A) ∼ A − 2 . Moreover, it has been found that vortices contain about 30% of the total energy of the flow which is almost conserved as Re varies by studying the energy ratio (η) of all vortices to the entire flow field. The statistic behaviors of energy spectrums show that the spectrums contain an inertial range in which E (k) ∼ k − 5 / 3 not only in the coherent structure field but also in the background field. More features of the vortex structure in the wake flow have been described by the empirical mode decomposition (EMD). There is a linear relationship between the frequency and the mode of the form f ¯ (m) = 1 7 4. 8 2 7 5 e − 0. 7 5 2 8 m in the semi-log coordinates. The period of each mode T ¯ (m) = 5. 7 × 1 0 − 3 e 0. 7 5 2 8 m in the semi-log coordinates. The time-dependent intrinsic correlation (TDIC) shows that there is a strong negative correlation relationship between the pressure and the flow velocity in the complex attachment vortex. [ABSTRACT FROM AUTHOR]

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if G is a finitely generated and torsion-free multiplicative subgroup of a number field K having rank r , then the ratio between n r and the Kummer degree [ K (ζ n , G n) : K (ζ n) ] is bounded independently of n. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered). [ABSTRACT FROM AUTHOR]

We consider a family of linear recurrence sequences T (k) : = { T n (k) } n of order k whose first k terms are 0 , 1 , ... , 1 and each term afterwards is the sum of the preceding k terms. In this paper, we study the zero–multiplicity on T (k) when the indices are extended to all integers. In particular, we give a upper bound (dependent on k) for the largest positive integer n such that T − n (k) = 0 and show that T (5) has zero–multiplicity unitary when the indices are extended to all the integers. [ABSTRACT FROM AUTHOR]

For an algebraic number field K , let d K denote the discriminant of an algebraic number field K. It is well known that if K 1 , K 2 are algebraic number fields with coprime discriminants, then K 1 , K 2 are linearly disjoint over the field ℚ of rational numbers and d K 1 K 2 = d K 1 n 2 d K 2 n 1 , n i being the degree of K i over ℚ. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields K 1 , K 2 linearly disjoint over K 1 ∩ K 2 . [ABSTRACT FROM AUTHOR]

Consider an irreducible quartic polynomial of the form X 4 + a X + b , where a , b ∈ ℤ satisfy v p (a) < 3 or v p (b) < 4 , where v p denotes the exact power of a rational prime p that divides an integer. Such a polynomial is called a p -minimal quartic. Let K be a field defined by a p -minimal quartic. In this paper, we use p -integral bases and introduce the concept of p -index forms in order to determine the field index of K via the coefficients of a defining polynomial in terms of certain congruence conditions. [ABSTRACT FROM AUTHOR]

We show that the Hecke–Mahler series, the generating function of the sequence { [ n ω ] } n = 1 ∞ for ω real, has the following property: Its values and its derivatives of any order, at any nonzero distinct algebraic numbers inside the unit circle, are algebraically independent if ω is a quadratic irrational number satisfying a suitable condition. [ABSTRACT FROM AUTHOR]

D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and P-recursive sequences. It consists of the limits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definition of the class of D-finite numbers depends on two subrings of the field of complex numbers. We investigate how different choices of these two subrings affect the class. Moreover, we show that D-finite numbers are essentially limits of D-finite functions at the point one, and evaluating D-finite functions at non-singular algebraic points typically yields D-finite numbers. This result makes it easier to recognize certain numbers to be D-finite. [ABSTRACT FROM AUTHOR]

We study the Mahler measure of generators of a Galois extension with Galois group the full symmetric group. We prove that two classical constructions of generators give always algebraic numbers of big height. These results answer a question of Smyth and provide some evidence to a conjecture which asserts that the height of such a generator grows to infinity with the degree of the extension. [ABSTRACT FROM AUTHOR]

We consider for integers the -generalized Fibonacci sequences , whose first terms are and each term afterwards is the sum of the preceding terms. We give a lower bound for the largest prime factor of the sum of two terms in . As a consequence of our main result, for every fixed finite set of primes , there are only finitely many positive integers and -integers which are a non-trivial sum of two -Fibonacci numbers, and all these are effectively computable. [ABSTRACT FROM AUTHOR]