For certain types of quadratic forms lying in the n-th power of the fundamental ideal, we compute upper bounds and if possible exact values for the minimal number of general n-fold Pfister forms, ...that are needed to write the Witt class of that given form as the sum of the Witt classes of those n-fold Pfister forms. We restrict ourselves mostly to the case of so called rigid fields, i.e. fields in which anisotropic binary forms represent at most 2 square classes.
In this paper we look at a probabilistic approach to a non‐local quadratic form that has lately attracted some interest. This form is related to a recently introduced non‐local normal derivative. The ...goal is to construct two Markov processes: one corresponding to that form and the other which is related to a probabilistic interpretation of the Neumann problem. We also study the Dirichlet‐to‐Neumann operator for non‐local operators.
We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of ...Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer
n
by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form.
The standard approach to Clusterwise Regression is the Clusterwise Linear Regression method. This approach can lead to data over-fitting, and it is not able to distinguish linear relationships in ...groups of observations well separated in the space of explanatory variables. This paper presents a Weighted Clusterwise Linear Regression method to obtain homogeneous clusters of observations while maintaining a proper fitting for the response variable, by the minimization of an optimization criterion that combines a k-means-like criterion (based on an adaptive quadratic form dissimilarity) in x-space and the criterion of minimum squared residuals of Regression Analysis. The adaptive metric allows automatic weighing or take into account the correlation between explanatory variables under multiple constraints types. We explore six constraints types. Experiments with synthetic and benchmark datasets corroborate the usefulness of the proposed method.
•A Weighted Clusterwise Regression to obtain homogeneous clusters.•Objective function combining a kmeans-like and a minimum SSQ criteria.•Based on adaptive quadratic form dissimilarity, in x-space.•Automatic weighing of explanatory variables under six constraints types.•Synthetic and real datasets corroborate the usefulness of the method.
In this paper, we prove that if d is sufficiently large square-free positive rational integer, then there is no positive definite universal quadratic OF-lattice of rank 7 where F=Q(d).
In this article, we prove a result concerning the infinitude of square-free integers represented by a class of polynomials in two variables. More precisely, we prove that infinitely many square-free ...positive integers are represented by a primitive integral positive-definite binary quadratic form of a given discriminant
D
. We obtain our result by deriving an asymptotic formula for the summatory function associated to it using some known
L
-functions.
Let
denote the ring of integers of a quadratic field
. In 2022, Murtuza and Garge Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian ...Journal of Pure and Applied Mathematics, Published online; December 2022. gave a necessary and sufficient condition for a diagonal quadratic form
where
for
for representing all
matrices over
. Let K denote a quadratic field such that its ring of integers
is a principal ideal domain and 2 is a product of two distinct primes. It is a well-known fact that
is the only imaginary quadratic field with the above properties. Let
denote the discriminant of K. We have
if and only if 2 is a product of two distinct primes in
. With
as above, in this paper we generalize our earlier result. We give a necessary and sufficient condition for a diagonal quadratic form
where
,
to represent all
matrices over
. This result is a conjecture stated in Murtuza N, Garge A. Universality of certain diagonal quadratic forms for matrices over a ring of integers, Indian Journal of Pure and Applied Mathematics, Published online; December 2022.
The quadratic forms are algebraic expressions that have important role in different areas of computer science, mathematics, physics, statistics and others. We deal with quadratic forms that are ...widely used in the study of algebras with finite representation type, called unit forms. Especially, we study unit forms q:Zn→Z by means of their associated edge-bipartite signed (multi)graphs without loops. A big challenge in this field is to develop efficient algorithms to recognize the type of a given unit form. Actually, most algorithms can demand a long time on this task and becomes infeasible. Our aim is to present a new strategy, do the unit form recognition through sequences of mutations in the unit form related exchange matrix P. For that, we show that the positive roots of the unit form are in the c-matrix of P. We also present an algorithm to recognize the type of a given unit form, implemented using BFS (Breadth First Search) and using powerful techniques to simplify this process.
We establish an upper bound on the number of real multiquadratic fields that admit a universal quadratic lattice of a given rank, or contain a given amount of indecomposable elements modulo totally ...positive units, obtaining density zero statements. We also study the structure of indecomposable elements in real biquadratic fields, and compute a system of indecomposable elements modulo totally positive units for some families of real biquadratic fields.
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