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This document provides a summary of a special issue of the journal Mathematics, which features research papers in the field of algebraic, analytic, and computational number theory. The papers cover a wide range of topics, including generalized Lucas pseudoprimality, Fibonacci numbers, piecewise linear difference equations, properties of Euler's function, regularities in ordered semihypergroups, zero-density results for automorphic functions, properties of Fibonacci finite operator quaternions, and more. The authors of the special issue believe that the techniques used in these papers have the potential to yield new results in these areas of mathematics. [Extracted from the article]

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled "Monogenity and Power Integral Bases". We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. [ABSTRACT FROM AUTHOR]

The first topic covered in [[1]] is built around the Gauss hypergeometric function HT ht , the Kummer hypergeometric function HT ht , and their generalization HT ht . Finally, paper [[11]] investigates certain identities associated with HT ht -binomial coefficients and HT ht -Stirling polynomials of the second kind. Hypergeometric Bernoulli numbers and polynomials connect to the hypergeometric function HT ht which is the subject of [[4]]. [Extracted from the article]

In this paper, we study the action of diamond operators on Hilbert modular forms with coefficients in a general commutative ring. In particular, we generalize a result of Chai on the surjectivity of the constant term map for Hilbert modular forms with nebentype to the setting of group ring valued modular forms. As an application, we construct certain Hilbert modular forms required for Dasgupta–Kakde's proof of the Brumer–Stark conjecture at odd primes. Since the forms required for the Brumer–Stark conjecture live on the non-PEL Shimura variety associated to the reductive group G = Res F / Q (GL 2) , as opposed to the PEL Shimura variety associated to the subgroup G ∗ ⊂ G studied by Chai, we give a detailed explanation of theory of algebraic diamond operators for G, as well as how the theory of toroidal and minimal compactifications for G may be deduced from the analogous theory for G ∗ . [ABSTRACT FROM AUTHOR]

In this article, we find some properties of certain types of entropies of a natural number. We are studying a way of measuring the "disorder" of the divisors of a natural number. We compare two of the entropies H and H ¯ defined for a natural number. An useful property of the Shannon entropy is the additivity, H S (pq) = H S (p) + H S (q) , where pq denotes tensor product, so we focus on its study in the case of numbers and ideals. We mention that only one of the two entropy functions discussed in this paper satisfies additivity, whereas the other does not. In addition, regarding the entropy H of a natural number, we generalize this notion for ideals, and we find some of its properties. [ABSTRACT FROM AUTHOR]

This article is devoted to the even entire functions, which are the exact solutions for the Laplace and diffusion equations. These functions are considered in the algebraic number field. We guess that the functions have purely real zeros in the entire complex plane. These are proposed as new connections with algebraic number theory and mathematical physics. [ABSTRACT FROM AUTHOR]

In his biography of Emil Artin, Richard Brauer describes the years from 1931-1941 as a time when 'Artin spoke through his students and through the members of his mathematical circle' rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context. [ABSTRACT FROM AUTHOR]

Space-time block code designs based on algebraic field extension for full rate, large diversity product, and nonvanishing minimum determinant of codewords have received great attention. There are many different types of codes available for two-transmitter antennas, such as cyclotomic space-time block codes, the golden space-time block code, and rotation-based space-time block codes. In this paper, a more general space-time block code design scheme, which is called quadratic space-time block coding, is proposed for the two-transmitter antennas using quadratic field extension. The optimal design of the quadratic space-time block codes in terms of a diversity product criterion is also presented. It is shown that the optimal quadratic space-time block codes designed in this paper do not belong to the existing space-time block code family such as the cyclotomic, golden, and rotation-based space-time block codes. The simulation results demonstrate that the average codeword error rate of the optimal quadratic space-time block code attains about 0.5 dB signal to noise ratio gain over those of the optimal cyclotomic and golden space-time block codes. [ABSTRACT FROM PUBLISHER]

Index modulation (IM) is an attractive concept for next generation communication systems. However, as an emerging technique, there are still some challenges need to be tackled in this frontier. In this paper, we consider two aspects of IM, which are diversity and detection complexity. Specifically, we propose a linear precoding assisted index modulation (LPIM) scheme for orthogonal frequency division multiplexing (OFDM) systems. We commence by analyzing the diversity and coding gains of the proposed scheme. Moreover, a detailed codebook design criterion is proposed. Then, our LPIM codebook is designed based on the maximum diversity and coding gain criteria. In contrast to the signaling model of the existing full diversity precoder designed for OFDM, we introduce a modeling method to link the zero-valued IM symbols to the origin of a Lattice for implementing our full diversity precoder designed for OFDM-IM. Both analytical and computer simulation results are provided for characterizing the attainable performance of OFDM-LPIM, demonstrating that it is capable of achieving full diversity as well as an attractive coding gain. However, the maximum-likelihood (ML) detection complexity of OFDM-LPIM is excessive, hence a low-complexity generalized iterative residual check detector (GIRCD) is proposed, which is inspired by the existing sparse recovery algorithms. Finally, computer simulation results are provided for demonstrating that GIRCD can provide a beneficial trade-off between bit error ratio performance and complexity. [ABSTRACT FROM AUTHOR]

Despite several works on secrecy coding for fading and MIMO wiretap channels from an error probability perspective, the construction of information-theoretically secure codes over such channels remains an open problem. In this paper, we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information. Our channel model includes static channels, i.i.d. block fading channels, and ergodic stationary fading with fast decay of large deviations for the eavesdropper’s channel. We extend the flatness factor criterion from the Gaussian wiretap channel to fading and MIMO wiretap channels, and establish a simple design criterion where the normalized product distance/minimum determinant of the lattice and its dual should be maximized simultaneously. Moreover, we propose concrete lattice codes satisfying this design criterion, which are built from algebraic number fields with constant root discriminant in the single-antenna case, and from division algebras centered at such number fields in the multiple-antenna case. The proposed lattice codes achieve strong secrecy and semantic security for all rates $R< C_{b}-C_{e}-\kappa $ , where $C_{b}$ and $C_{e}$ are Bob and Eve’s channel capacities, respectively, and $\kappa $ is an explicit constant gap. Furthermore, these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models. Finally, we consider a compound wiretap model with a more restricted uncertainty set, and show that rates $R< \bar {C}_{b}-\bar {C}_{e}-\kappa $ are achievable, where $\bar {C}_{b}$ is a lower bound for Bob’s capacity and $\bar {C}_{e}$ is an upper bound for Eve’s capacity for all the channels in the set. [ABSTRACT FROM AUTHOR]

This paper considers space-time coding over several independently Rayleigh faded blocks. In particular, we will concentrate on giving upper bounds for the coding gain of lattice space-time codes as the number of blocks grow. This problem was previously considered in the single antenna case by Bayer-Fluckiger et al. in 2006. Crucial to their work was Odlyzko’s bound on the discriminant of an algebraic number field, as this provides an upper bound for the normalized coding gain of number field codes. In the MIMO context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding (noncommutative) algebra. In this paper, we will develop analogues of the Odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space-time codes. These bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases. [ABSTRACT FROM AUTHOR]

This paper deals with the study of the integral representations involving the product of the Aleph function with exponential function, Gauss hypergeometrc function and Fox's H-function. The results derived in this paper are basic in nature and include a number of known and new results as special cases. [ABSTRACT FROM AUTHOR]

Traditional public key exchange protocols are based on algebraic number theory. In another perspective, neural cryptography, which is based on neural networks, has been emerging. It has been reported that two parties can exchange secret key pairs with the synchronization phenomenon in neural networks. Although there are various models of neural cryptography, called Tree Parity Machine (TPM), many of them are not suitable for practical use, considering efficiency and security. In this paper, we propose a Vector-Valued Tree Parity Machine (VVTPM), which is a generalized architecture of TPM models and can be more efficient and secure for real-life systems. In terms of efficiency and security, we show that the synchronization time of the VVTPM has the same order as the basic TPM model, and it can be more secure than previous results with the same synaptic depth. [ABSTRACT FROM AUTHOR]

This paper is about a generalization of celebrated Inoue’s surfaces. To each matrix M in SL(2n+1, Z) we associate a complex non-Kähler manifold TM of complex dimension n + 1. This manifold fibers over S¹ with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the OeljeklausToma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds. [ABSTRACT FROM AUTHOR]

Recently, the Liu-Zhang-Wong's Toeplitz space-time block code and the Shang-Xia's Alamouti code-based overlapped space-time block code have been developed to enable coherent full diversity with the linear receivers for a wireless communication system having multiple transmitter antennas and a single receiver antenna. In this paper, such a noncoherent system is considered, in which channel state information is unavailable at both the transmitter and receiver. Using the Alamouti coding scheme and the Toeplitz matrix structure, a novel noncoherent nonunitary space-time block code, which is called an Alamouti-based Toeplitz space-time block code, is proposed. By the fundamentals of Galois theory and algebraic number theory, two important properties on the two Alamouti codes generated from a pair of coprime phase shift keying (PSK) constellations, i.e., the uniqueness of factorization itself and the shift-invariant uniqueness of factorization, are first revealed and rigorously proved. Then, it is further shown that it is these two kinds of unique factorizations that enable the unique blind identification of both the channel coefficients and the transmitted signals by only processing two block received signals, as well as noncoherent full diversity with a generalized likelihood ratio test (GLRT) receiver. In addition, a full diversity unitary code design is also proposed by simply applying the QR decomposition to the full diversity nonunitary Alamouti-based Toeplitz space-time block code. Computer simulations demonstrate that the nonunitary and unitary codes presented in this paper outperform the differential codes and the signal-to-noise-ratio (SNR)-efficient training scheme based on the coherent orthogonal space-time block codes in the current literature. [ABSTRACT FROM PUBLISHER]

The paper deals with the class of entire functions that increase not faster than exp{γ∣z∣6/5(ln∣z∣)−1} and that, together with their first derivatives, take values from a fixed field of algebraic numbers at the points of a two-dimensional lattice of general form (in this case, the values increase not too fast). It is shown that any such functions is either a polynomial or can be represented in the form e−mαzP(eαz), where m is a nonnegative integer, P is a polynomial, and α is an algebraic number. [ABSTRACT FROM AUTHOR]

In this paper we describe a genetic algorithm (GA) for the uncapacitated multiple allocation p-hub center problem (UMApHCP). Binary coding is used and genetic operators adapted to the problem are constructed and implemented in our GA. Computational results are presented for the standard hub instances from the literature. It can be seen that proposed GA approach reaches all solutions that are proved to be optimal so far. The solutions are obtained in a reasonable amount of computational time, even for problem instances of higher dimensions. [ABSTRACT FROM AUTHOR]

This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semi-parametric formulations, extending the relevance of the field much beyond the original theme of‘skewness’. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more. [ABSTRACT FROM AUTHOR]

The design requirements for space-time coding typically involves achieving the goals of good performance, high rates, and low decoding complexity. In this paper, we introduce a further constraint on space-time code design In that the code should also lead to low values of the peak-to-mean envelope power ratio (PMEPR) for each antenna. Towards that end, we propose a new class of space-time codes called the "low PMEPR space-time" (LPST) codes. The LPST codes are obtained using the properties of certain cyclotomic number fields. The LPST codes achieve a performance identical to that of the threaded algebraic space-time (TAST) codes but at a much smaller PMEPR. With M antennas and a rate of one symbol per channel use, the LPST codes lead to a decrease in PMEPR by at least a factor of M relative to a Hadamard spread version of the TAST code. For rates beyond one symbol per channel use and up to a guaranteed amount, the LPST codes have provably smaller PMEPR than the corresponding TAST codes. Additionally, with the concept of punctured LPST codes proposed in this paper, significant performance Improvement Is obtained over the full diversity TAST schemes of comparable complexity. Numerical examples are provided to illustrate the advantage of the proposed codes in terms of PMEPR reduction and performance improvement for very high rate wireless communications. [ABSTRACT FROM AUTHOR]

In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation. [ABSTRACT FROM AUTHOR]

A Gaussian integer is a complex number whose real and imaginary parts are both integers. Meanwhile, a sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This paper proposes a method for constructing sparse perfect Gaussian integer sequences (SPGISs) in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining four base sequences or their cyclic-shift equivalents using nonzero Gaussian integer coefficients of equal magnitudes. Each base sequence contains four nonzero elements belonging to the set \\pm 1, \pm j\. The number of nonzero elements of the constructed SPGISs depends on the choice of complex coefficients and cyclic shifts. However, each SPGIS has at most 16 nonzero elements, irrespective of the sequence length. A systematic investigation is performed into the properties of the SPGISs and their Fourier dual equivalents. Finally, a general expression is derived for a perfect Gaussian integer sequence (PGIS) of length $4n$, where $n$ is any positive integer and most of the sequence elements are nonzero. [ABSTRACT FROM AUTHOR]

This paper investigates the design of codes for multiple-input multiple-output (MIMO) multiple access channel (MAC). If a joint maximum-likelihood decoding is to be performed at the receiver, then every MIMO-MAC code can be regarded as a single-user code, where the minimum determinant criterion proposed by Tarokh et al. is useful for designing such codes and for upper bounding the maximum pairwise error probability (PEP), whenever the codes are of finite rate and operate in finite signal-to-noise ratio range. Unlike the case of single-user codes where the minimum determinant can be lower bounded by a fixed constant as code-rate grows, it was proved by Lahtonen et al. that the minimum determinant of MIMO-MAC codes decays as a function of the rates. This decay phenomenon is further investigated in this paper, and upper bounds for the decays of minimum determinant corresponding to each error event are provided. Lower bounds for the optimal decay are established and are based on an explicit construction of codes using algebraic number theory and Diophantine approximation. For some error profiles, the constructed codes are shown to meet the aforementioned upper bounds, hence they are optimal finite-rate codes in terms of PEPs associated with such error events. An asymptotic diversity-multiplexing gain tradeoff (DMT) analysis of the proposed codes is also given. It is shown that these codes are DMT optimal when the values of multiplexing gains are small. [ABSTRACT FROM PUBLISHER]

In this paper, we present a number field version of the celebrated result of Lenstra ( Math. Comp. 42(165) () 331-340) in 1984. Also, this result allows us to improve a result of Wikstrőm (On the l-ary GCD-algorithm in rings of integers () pp. 1189-1201). [ABSTRACT FROM AUTHOR]

Let G be a group and ω(G) = {o(g)|g ∈ G} be the set of element orders of G. Let k ∈n ω(G) and sk = |{g ∈ G |o(g) = k}|. Let nse(G) = {sk|k ∈ ω(G)}. In this paper, we prove that if G is a group and G2(4) is the Chevalley group such that nse(G) = nse(G2(4)), then G ≅ G2(4). [ABSTRACT FROM AUTHOR]

In this paper we study the groups of circular numbers and circular units in Sinnott's sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct Z-bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper [2]. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of motion-induced 3-D robot-to-robot extrinsic calibration that is based on ego-motion estimates and combinations of interrobot measurements (i.e., distance and/or bearing observations from either or both of the two robots, recorded across multiple time steps). In particular, we focus on solving minimal problems, where the unknown 6-degree-of-freedom (DOF) transformation between the two robots is determined based on the minimum number of measurements necessary to find a finite set of solutions. In order to address the very large number of possible combinations of interrobot observations, we identify symmetries in the measurement sequence and use them to prove that any extrinsic robot-to-robot calibration problem can be solved based on the solutions of only 14 (base) minimal problems. Moreover, we provide algebraic (closed-form) and efficient symbolic-numerical (analytical) solution methods to these minimal problems. Finally, we evaluate the performance of our proposed solvers through extensive simulations and experiments. [ABSTRACT FROM PUBLISHER]

In this paper, we present four limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least four limit cycles with (3,1)-distribution. This result provides a positive answer to an open question in this research area. [ABSTRACT FROM AUTHOR]

The article discusses a paper on the product of smooth valuations on manifolds. The paper examined some important materials on valuations and geometric measure theory. It described the product in terms of differential forms, Gelfand transforms and blow-up spaces. It demonstrated that the product extends partially to generalized valuations and corresponds to transversal intersections.

We introduce ROME, an algebraic modeling toolbox for a class of robust optimization problems. ROME serves as an intermediate layer between the modeler and optimization solver engines, allowing modelers to express robust optimization problems in a mathematically meaningful way. In this paper, we discuss how ROME can be used to model (1) a serviceconstrained robust inventory management problem, (2) a project-crashing problem, and (3) a robust portfolio optimization problem. Through these modeling examples, we highlight the key features of ROME that allow it to expedite the modeling and subsequent numerical analysis of robust optimization problems. ROME is freely distributed for academic use at http://www.robustopt.com. [ABSTRACT FROM AUTHOR]

This paper is concerned with an algorithm for selecting the best set of s variables out of k(> s) candidate variables in a multiple linear regression model. We employ absolute deviation as the measure of deviation and solve the resulting optimization problem by using 0-1 integer programming methodologies. In addition, we will propose a heuristic algorithm to obtain a close to optimal set of variables in terms of squared deviation. Computational results show that this method is practical and reliable for determining the best set of variables. [ABSTRACT FROM AUTHOR]

Cyclic division algebra (CDA) has recently become a major technique to construct space-time block codes with nonvanishing determinant (NVD). One of the key steps in this technique is the determination of non-norm elements and a sufficient condition for the determination has been given by Kiran and Rajan lately based on algebraic number theory. In this paper, based on Kiran and Rajan's condition, we present a more elementary condition for non-norm elements when signals are QAM or HEX, which is easier to check. With this elementary condition, non-norm elements with smaller absolute values than the existing ones can be found. [ABSTRACT FROM AUTHOR]

Recently, many mathematicians have studied various kinds of the q-analogue of Genocchi numbers and polynomials. In the work (New approach to q-Euler, Genocchi numbers and their interpolation functions, "Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105-112, 2009.", Kim defined new generating functions of q-Genocchi, q-Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz type q-zeta function. This function interpolates q-Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials. [ABSTRACT FROM AUTHOR]

In the present paper, a polynomial algorithm is suggested for reducing the problem of taking the discrete logarithm in the ring of algebraic integers modulo a power of a prime ideal to a similar problem with the power equal to one. Explicit formulas are obtained; instead of the Fermat quotients, in the case of residues in the ring of rational integers, these formulas use other polynomially computable logarithmic functions, like the $$\mathfrak{p}$$ -adic logarithm. [ABSTRACT FROM AUTHOR]

A new approach is presented for the design of full modulation diversity (FMD) complex lattices for the Rayleigh- fading channel. The FMD lattice design problem essentially consists of maximizing a parameter called the normalized minimum product distance dp² of the finite signal set carved out of the lattice. We approach the problem of maximizing dp² by minimizing the average energy of the signal constellation obtained from a new family of FMD lattices. The unnormalized minimum product distance for every lattice in the proposed family is lower-bounded by a nonzero constant. Minimizing the average energy of the signal set translates to minimizing the Frobenius norm of the generator matrices within the proposed family. The two strategies proposed for the Frobenius norm reduction are based on the concepts of successive minima (SM) and basis reduction of an equivalent real lattice. The lattice constructions in this paper provide significantly larger normalized minimum product distances compared to the existing lattices in certain dimensions. The proposed construction is gene eral and works for any dimension as long as a list of number fields of the same degree is available. [ABSTRACT FROM AUTHOR]

This paper introduces a new radix-2 representation with the same average weight as the width-w nonadjacent form (w-NAF). In both w-NAF and the proposed representations, each nonzero digit is an odd integer with absolute value less than M. However, for w-NAF, M is of the form 2w-1, while, for the proposed representation, it can be any positive integer. Therefore, using the proposed integer representation, we can use the available memory efficiently, which is attractive for devices with limited memory. Another advantage of the proposed representation over w-NAF is that it can be obtained by scanning the bits from left-to-right. This property is also useful for memory-constrained devices because it can reduce both the time and space complexity of fast point multiplication techniques. [ABSTRACT FROM AUTHOR]

In this paper, we present some results concerning two generalized derivations on a semiprime ring. These results are a generalization of results of M. Brešar and J. Vukman in [2], which are related to a theorem of E. Posner for the product of derivations on a prime ring. [ABSTRACT FROM AUTHOR]

Since 1970, reversible finite automata have generated interest among researcher; but till now, we have not come across a model of reversible read only one-way finite automata which accept all regular languages, In this paper, we introduce a new model of one-way reversible finite automata inspired by the Watson-Crick complementarity relation and show that our model can accept all regular languages. We further show that our model accepts a language which is not accepted by any multi-head deterministic finite automaton. [ABSTRACT FROM AUTHOR]

In 1975, Don Zagier obtained a new version of the Kronecker limit formula for a real quadratic field which involved an interesting function F (x) which is now known as the Herglotz function. As demonstrated by Zagier, and very recently by Radchenko and Zagier, F (x) satisfies beautiful properties which are of interest in both algebraic number theory as well as in analytic number theory. In this paper, we study F k , N (x) , an extension of the Herglotz function which also subsumes higher Herglotz function of Vlasenko and Zagier. We call it the extended higher Herglotz function. It is intimately connected with a certain generalized Lambert series as well as with a generalized cotangent Dirichlet series inspired from Krätzel's work. We derive two different kinds of functional equations satisfied by F k , N (x). Radchenko and Zagier gave a beautiful relation between the integral ∫ 0 1 log ⁡ (1 + t x) 1 + t d t and F (x) and used it to evaluate this integral at various rational as well as irrational arguments. We obtain a relation between F k , N (x) and a generalization of the above integral involving polylogarithm. The asymptotic expansions of F k , N (x) and some generalized Lambert series are also obtained along with other supplementary results. [ABSTRACT FROM AUTHOR]

The article offers information on preparation of manuscripts for the periodical and copying along with the table of contents.

A primitive completely normal element for the extension E / F of Galois fields is a generator of the multiplicative group of E, which simultaneously is normal in E / K for every intermediate field K of E / F. Letting $$F=\mathbb {F}_q$$ und $$E=\mathbb {F}_{q^n}$$ , we denote by $$P_n(q)$$ the number of primitive elements of E, and by $$PCN_n(q)$$ the number of primitive completely normal elements for E / F. We prove that $$PCN_n(q) >0$$ provided that $$\ln (2) q \ge (t(n)-1)\cdot (\ln (2)+n\ln (q))$$ , where $$t(n):=\sum _{d|n}d$$ denotes the sum of all positive divisors of n and $$\ln $$ is the natural logarithm. Based on this estimate, we are able to prove that, for every fixed n, the quotient $$PCN_n(q)/P_n(q)$$ converges to 1 as q goes to infinity. We also achieve effective versions of the first result: $$PCN_n(q)>0$$ whenever $$q\ge n^{7/2}$$ and $$n\ge 7$$ , or when $$q\ge n^3$$ and $$n\ge 37$$ . The paper is self-contained and the proofs rely only on elementary combinatorics and number theory, as well as some basic calculus. [ABSTRACT FROM AUTHOR]

We call an abelian variety over a finite field F q super-isolated if its (F q -rational) isogeny class contains a single isomorphism class. In this paper, we use the Honda-Tate theorem to characterize super-isolated ordinary simple abelian varieties by certain algebraic integers. Our main result is that for a fixed dimension g ≥ 3 , there are finitely many such varieties over all finite fields. [ABSTRACT FROM AUTHOR]

Dedekind sum first occurred naturally in Dedekind's transformation law of his eta-function. In analogy, Hardy sums are encountered in the transformation formula of the theta function. Up to now, they have many of remarkable applications in analytic number theory (Dedekind's η-function), algebraic number theory (class numbers), lattice point problems, topology and algebraic geometry. Miscellaneous arithmetical properties of these sums have been analyzed by many scholars. Recently, considering the characteristics of Hardy sums and other celebrated sums such as Ramanujan sum and Kloosterman sum, interesting and meaningful identities have been obtained. In this paper, we continue to focus on arithmetical aspects of Hardy sums and Ramanujan sum. More precisely, we consider a mean value problem of these sums and Ramanujan sum with the help of the features of Dirichlet L-functions and present some computational formulas for them. [ABSTRACT FROM AUTHOR]

In this paper we provide a generalized integral representation of additive interval function based on a fundamental integral representation of additive interval function given in Zorich's textbook, Mathematical Analysis, Vol I. Then we use it to give a rigorous proof of the method of cylindrical shells for the evaluation of volume of solid of revolution about vertical line. [ABSTRACT FROM AUTHOR]

Let K be a finite extension of ℚ and let S = {v} denote the collection of normalized absolute values on K. Let VK+ denote the additive group of adeles over K and let c : VK+ → ℝ ≥ 0 denote the content map defined as c({av}) = Πv∈S (av) for {av} ∈ VK+. A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K with the following property: if {av} ∈ VK+ with c({av}) > c, then there exists a non-zero element b ∈ K for which v(b) ≤ v(av), ...v ∈ S. Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang's bound for various real quadratic extensions and find that our new bound is better than Lang's in many instances. [ABSTRACT FROM AUTHOR]

Abstract In this paper we describe how to use the algorithmic methods provided by Hunter and Pohst in order to give a complete classification of number fields of degree 8 and signature (2 , 3) with absolute discriminant less than a certain bound. The choice of this bound comes from the local corrections given by prime ideals to the lower estimates for discriminants obtained with the Odlyzko–Poitou–Serre method. [ABSTRACT FROM AUTHOR]