Quadratic Number Research Articles

Gaussian behavior of quadratic irrationals

We study the probabilistic behavior of the continued fraction expansion of a quadratic irrational number, when weighted by some “additive” cost. We prove asymptotic Gaussian limit laws, with an optimal speed of convergence. We deal with the underlying dyn

Quadratic fields with a class group of large 3-rank

We prove that there are $\gg X^{1/30}/\!\log X$ imaginary quadratic number fields with an ideal class group of $3$-rank at least $5$ and discriminant bounded in absolute value by $X$. This improves on an earlier result of Craig, who proved the infinitude

Unraveling Genome Evolution Throughout Visual Analysis: The XCout Portal.

Due to major breakthroughs in sequencing technologies throughout the last decades, the time and cost per sequencing experiment have reduced drastically, overcoming the data generation barrier during the early genomic era. Such a shift has encouraged the scientific community to develop new computational methods that are able to compare large genomic sequences, thus enabling large-scale studies of genome evolution. The field of comparative genomics has proven itself invaluable for studying the evolutionary mechanisms and the forces driving genome evolution. In this line, a full genome comparison study between 2 species requires a quadratic number of comparisons in terms of the number of sequences (around 400 chromosome comparisons in the case of mammalian genomes); however, when studying conserved syntenies or evolutionary rearrangements, many sequence comparisons can be skipped for not all will contain significant signals. Subsequently, the scientific community has developed fast heuristics to perform multiple pairwise comparisons between large sequences to determine whether significant sets of conserved similarities exist. The data generation problem is no longer an issue, yet the limitations have shifted toward the analysis of such massive data. Therefore, we present XCout, a Web-based visual analytics application for multiple genome comparisons designed to improve the analysis of large-scale evolutionary studies using novel techniques in Web visualization. XCout enables to work on hundreds of comparisons at once, thus reducing the time of the analysis by identifying significant signals between chromosomes across multiple species. Among others, XCout introduces several techniques to aid in the analysis of large-scale genome rearrangements, particularly (1) an interactive heatmap interface to display comparisons using automatic color scales based on similarity thresholds to ease detection at first sight, (2) an overlay system to detect individual signal contributions between chromosomes, (3) a tracking tool to trace conserved blocks across different species to perform evolutionary studies, and (4) a search engine to search annotations throughout different species.

Hyperbolic tessellations and generators of for imaginary quadratic fields

AbstractWe develop methods for constructing explicit generators, modulo torsion, of the$K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic$3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for$ K_3 $of any field, predict the precise power of$2$that should occur in the Lichtenbaum conjecture at$ -1 $and prove that this prediction is valid for all abelian number fields.

Floristic Response of Herbaceous Flora to Intensive Cropping Systems: A Case of Ajibode-sasa Arable Agroecosystem, Ibadan, Southwest Nigeria

Agriculture a most significant land use types which alter natural ecosystem dynamics. Arable farming exerts much pressure on plant biodiversity, especially when practiced intensively in urban centers. There is dearth of information on floristic changes due to intensive arable farming in urban agroecosystems in developing countries. The study therefore assessed floristic changes resulting from and intensive farming practices at Ajibode-Sasa agricultural landscape. Ajibode-Sasa agroecosystem is a complex mix of arable cropping system between latitude N07°28′, E003°53′ and longitude N07°28′, E003°54. Comparative floristic surveys were conducted in 2016 and 2020 using quadrats (1 m2) systematically laid on 18 Transects ranging from 50 – 250 m long. A total of 224 and 184 quadrats were laid in 2016 and 2020 respectively. Reduction in numbers of quadrats laid resulted from physical anthropogenic development after the 2016 survey. Species identification followed standard procedures, and quantitative occurrence data were collected for determination of species composition and computation of relative importance values (RIV) and diversity indices. Land-use changes over four years period was determined using Google earth and QGIS. Herbacous plant composition with 123 cumulative number of species in both years reduced from 98 species in 2016 to 85 species in 2020 species RIV of species ranged from 0.038 – 14.803. Tridax procumbens had the highest RIV (14.803) in 2016, while it was Acmella brachyglossa (13.248) 2020. Species richness and floral diversity was high with Shannon-Weiner Index (3.081 and 3.088) and Dominance (0.09388 and 0.08746) in 2016 and 2020 respectively. Intensive cultivation favoured introduction and spread of invasive species like Tridax procumbens and Tithonia diversifolia. Eight introduced and invasive species were newly enumerated in 2020, with a total of 38 herbaceous species no longer encountered in 2020. Concerted efforts should be made to conserve native flora on the agroecosystem through sustainable practices like crop rotation and short fallow.

Penggunaan Model Make A Match Meningkatkan Motivasi Dan Hasil Belajar Bilangan Kuadrat Siswa Kelas V SDN 4 Pare Tahun Pelajaran 2019/2020

In the process of learning mathematics, the low score is because students do not have the interest and motivation to learn mathematics. The purpose of this study was to increase motivation and learning outcomes of quadratic numbers in fifth grade students of SDN 4 Pare in the 2019/2020 academic year with the Make a Match learning model. This research is a quantitative research with correlational method. Sampling using observation and tests. The results showed that the average student motivation in the initial condition was 60%, the first cycle was 71.43% and in the second cycle it was 85.71%. The results of the square number in the initial conditions mean the value of 60 classical completeness level 57. In the first cycle, the average value of 70 classical completeness level is 71. In the second cycle, the average value is 85.71 classical completeness level is 85. Based on the actions taken, it can be concluded that the use of the learning model Make a Match increases the motivation of fifth graders at SDN 4 Pare for the 2019/2020 academic year by 26.14% and increases the learning outcomes of quadratic numbers by 28%.

On the quadratic reciprocity law

In the number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. It noticed by Euler and Legendre and proved by Gauss. In this paper, we will study the quadratic reciprocity law theorem where the Euler Criterion and Legendre Symbol are involved. The application of quadratic reciprocity law theorem is given in cryptography, where the Quadratic Residuosity Problem considered as a hard mathematical problem for Goldwasser Micali Randomized Public Key Cryptosystem. This system will be discussed with the details in this paper.

On local and global bounds for Iwasawa λ-invariants

It is an open problem whether the Iwasawa λ-invariants of the Zp-extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Zp-extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Zp-extensions.

A Bombieri-type theorem for convolution with application on number field

Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ be its ring of integers. We show that, if the arithmetic functions $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ both have level of distribution $$\vartheta$$ for some $$0<\vartheta\leq 1/2$$ then the Dirichlet convolution $$f*g$$ also has level of distribution $$\vartheta$$ . As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.

Effective simultaneous rational approximation to pairs of real quadratic numbers

Let $\xi, \zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q > c$ we have $$ \max\{\|q \xi \|, \|q \zeta\| \} > q^{-1 + \tau}, $$ where $\| \cdot \|$ denotes the distance to the nearest integer.

Number-theoretic positive entropy shifts with small centralizer and large normalizer

AbstractHigher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centralizer and normalizer of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralizers, but large normalizers. In particular, we discuss several systems where the normalizer is an infinite extension of the centralizer, including the visible lattice points and the k-free integers in some real quadratic number fields.

ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS

AbstractIn this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

Derivable quadratic forms and representation numbers

Let Fm denote the set of positive-definite primitive integral quadratic forms in m variables. Let f,g∈Fm. In this paper we introduce a new concept, namely that of g being derivable from f. This concept is based on a certain theta function identity being valid. A consequence of this concept is that if g is derivable from f then the representation number of g can be given in terms of that of f. Many examples are given, especially for diagonal ternary quadratic forms.

An analogue of Wilton's formula and values of Dedekind zeta functions

J. R. Wilton obtained an expression for the product of two Riemann zeta functions. This expression played a crucial role to find the approximate functional equation for the product of two Riemann zeta functions in the critical region. We find analogous expressions for the product of two Dedekind zeta functions and then use these expressions to find some expressions for Dedekind zeta values attached to arbitrary real as well as quadratic number fields at any positive integer.

Hermitian Theta Series and Maa\xdf Spaces Under the Action of the Maximal Discrete Extension of the Hermitian Modular Group

Let Gamma _n(mathcal {scriptstyle {O}}_{mathbb {K}}) denote the Hermitian modular group of degree n over an imaginary quadratic number field mathbb {K} and Delta _{n,mathbb {K}}^* its maximal discrete extension in the special unitary group SU(n,n;mathbb {C}). In this paper we study the action of Delta _{n,mathbb {K}}^* on Hermitian theta series and Maaß spaces. For n=2 we will find theta lattices such that the corresponding theta series are modular forms with respect to Delta _{2,mathbb {K}}^* as well as examples where this is not the case. Our second focus lies on studying two different Maaß spaces. We will see that the new found group Delta _{2,mathbb {K}}^* consolidates the different definitions of the spaces.

Distribution of Modular Symbols in ℍ3

AbstractWe introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ is its ring of integers, then for certain congruence subgroups of $\textrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and second moments of the distribution.

A continuity of cycle integrals of modular functions

In this paper, we study a continuity of the “values” of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean topology.

The fourth moment of individual Dirichlet L-functions on the critical line

We prove an asymptotic formula for the second moment of a product of two Dirichlet L-functions on the critical line, which has a power saving in the error term and which is uniform with respect to the involved Dirichlet characters. As special cases we give uniform asymptotic formulae for the fourth moment of individual Dirichlet L-functions and for the second moment of Dedekind zeta functions of quadratic number fields on the critical line.

Exploring Goldbach’s conjecture, the Pebonacci sequence and its five-dimensional vortex structure based on the logarithm of the circle

&#x0D; &#x0D; &#x0D; A method based on circle logarithm to prove Goldbach’s conjecture and Pebonacci sequence is proposed. Its essence is to deal with the real infinite series, each of the finite three elements (prime numbers, number series) has asymmetry problems, forming a basic even function one-variable quadratic equation and odd function one-variable three-dimensional number sequence; it is converted to "The irrelevant mathematical model expands latently in a closed interval of 0 to 1," forming a five-dimensional vortex space structure.&#x0D; &#x0D; &#x0D;

Higher genus theory

AbstractIn $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.