Local densities and explicit bounds for representability by a quadratic form

2004 ◽  
Vol 124 (2) ◽  
pp. 351-388 ◽  
Author(s):  
Jonathan Hanke
Keyword(s):  
Quadratic Form ◽  
Explicit Bounds ◽  
Local Densities

scholarly journals Spinor representations of positive definite ternary quadratic forms

2018 ◽  
Vol 14 (02) ◽  
pp. 581-594 ◽  
Author(s):  
Jangwon Ju ◽  
Kyoungmin Kim ◽  
Byeong-Kweon Oh
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Ternary Quadratic Form ◽  
Definite Integral ◽  
Constant Multiple ◽  
Ternary Quadratic Forms ◽  
Spinor Genus ◽  
Local Densities ◽  
Spinor Representations

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

scholarly journals The proportion of genus one curves over ℚ defined by a binary quartic that everywhere locally have a point

2020 ◽  
pp. 1-21
Author(s):  
Manjul Bhargava ◽  
John Cremona ◽  
Tom Fisher
Keyword(s):  
Quadratic Form ◽  
Rational Function ◽  
Local Density ◽  
Binary Quadratic Form ◽  
Generalized Equation ◽  
Fixed Degree ◽  
Real Place ◽  
Quartic Form ◽  
Local Densities ◽  
Genus One

We consider the proportion of genus one curves over [Formula: see text] of the form [Formula: see text] where [Formula: see text] is a binary quartic form (or more generally of the form [Formula: see text] where also [Formula: see text] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime [Formula: see text] is given by a fixed degree-[Formula: see text] rational function of [Formula: see text] for all odd [Formula: see text] (and for the generalized equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.

Spheric radial basis functions in Mahalanobis measuring for displaying local field features

2019 ◽  
Vol 952 (10) ◽  
pp. 2-9
Author(s):  
Yu.M. Neiman ◽  
L.S. Sugaipova ◽  
V.V. Popadyev
Keyword(s):  
Gravitational Field ◽  
Quadratic Form ◽  
Local Field ◽  
Euclidean Distance ◽  
Basis Functions ◽  
Spherical Functions ◽  
Local Models ◽  
Stationary Field ◽  
Modeling Process ◽  
The Earth

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

scholarly journals Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

2021 ◽  
Vol 53 (1) ◽  
pp. 162-188
Author(s):  
Krzysztof Bartoszek ◽  
Torkel Erhardsson
Keyword(s):  
Normal Distribution ◽  
Phenotypic Trait ◽  
Phenotypic Traits ◽  
Normal Distributions ◽  
Average Value ◽  
Conditional Moments ◽  
Mixture Of Normal Distributions ◽  
Explicit Bounds ◽  
Mixtures Of Normal Distributions ◽  
Parameter Values

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

scholarly journals Substrate wettability guided oriented self assembly of Janus particles

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Meneka Banik ◽  
Shaili Sett ◽  
Chirodeep Bakli ◽  
Arup Kumar Raychaudhuri ◽  
Suman Chakraborty ◽  
...  
Keyword(s):  
Self Assembly ◽  
Janus Particles ◽  
Water Molecules ◽  
Functional Coatings ◽  
Hexagonal Close Packed ◽  
Local Densities ◽  
Inhomogeneous Properties ◽  
Metal Polymer ◽  
Specific Orientation

AbstractSelf-assembly of Janus particles with spatial inhomogeneous properties is of fundamental importance in diverse areas of sciences and has been extensively observed as a favorably functionalized fluidic interface or in a dilute solution. Interestingly, the unique and non-trivial role of surface wettability on oriented self-assembly of Janus particles has remained largely unexplored. Here, the exclusive role of substrate wettability in directing the orientation of amphiphilic metal-polymer Bifacial spherical Janus particles, obtained by topo-selective metal deposition on colloidal Polymestyere (PS) particles, is explored by drop casting a dilute dispersion of the Janus colloids. While all particles orient with their polymeric (hydrophobic) and metallic (hydrophilic) sides facing upwards on hydrophilic and hydrophobic substrates respectively, they exhibit random orientation on a neutral substrate. The substrate wettability guided orientation of the Janus particles is captured using molecular dynamic simulation, which highlights that the arrangement of water molecules and their local densities near the substrate guide the specific orientation. Finally, it is shown that by spin coating it becomes possible to create a hexagonal close-packed array of the Janus colloids with specific orientation on differential wettability substrates. The results reported here open up new possibilities of substrate-wettability driven functional coatings of Janus particles, which has hitherto remained unexplored.

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

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