An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field

Let F be a non-archimedean local field of characteristic 0, and 𝔬 {{\mathfrak{o}}} the ring of integers in F. We give an explicit formula for the Siegel series of a half-integral matrix over 𝔬 {{\mathfrak{o}}} . This formula expresses the Siegel series of a half-integral matrix B explicitly in terms of the Gross–Keating invariant of B and its related invariants.

Cosmology in f(R, T) gravity with quadratic deceleration parameter

In this work, we have developed FLRW cosmological models in f(R, T) gravity. The solution of the modified field equations are obtained under the newly proposed Bakry and Shafeek, “The periodic universe with varying deceleration parameter of the second degree,” Astrophys. Space Sci., vol. 364, p. 135, 2019, quadratic form of the deceleration parameter. Further, we have discussed the state-finder parameter, om-diagnostic analysis and energy conditions of the proposed model. The variation of deceleration parameter with respect to cosmic time and red-shift is consistent with observational data.

Subtle characteristic classes for Spin-torsors

Extending (Smirnov and Vishik, Subtle Characteristic Classes, arXiv:1401.6661), we obtain a complete description of the motivic cohomology with $${{\,\mathrm{\mathbb {Z}}\,}}/2$$ Z / 2 -coefficients of the Nisnevich classifying space of the spin group $$Spin_n$$ S p i n n associated to the standard split quadratic form. This provides us with very simple relations among subtle Stiefel–Whitney classes in the motivic cohomology of Čech simplicial schemes associated to quadratic forms from $$I^3$$ I 3 , which are closely related to $$Spin_n$$ S p i n n -torsors over the point. These relations come from the action of the motivic Steenrod algebra on the second subtle Stiefel–Whitney class. Moreover, exploiting the relation between $$Spin_7$$ S p i n 7 and $$G_2$$ G 2 , we describe completely the motivic cohomology ring of the Nisnevich classifying space of $$G_2$$ G 2 . The result in topology was obtained by Quillen (Math Ann 194:197–212, 1971).

Projective Metric Geometry and Clifford Algebras

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Fourier Optimization and Quadratic Forms

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

MATHEMATICAL MODEL OF THE DYNAMICS OF A SIX-MASS SYSTEM OF A PARALLEL STRUCTURE MANIPULATOR MECHANISM

The paper is devoted to the construction of a mathematical model of the dynamics of a parallel structure manipulator with three controlled degrees of freedom, based on the reduction of the kinetic energy of the manipulator to a quadratic form relative to three independent generalized coordinates, comparative results of mathematical modeling are presented.

Algorithms for quadratic forms over global function fields of odd characteristic

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in [4] to global function fields of odd characteristics. First, we present algorithm for checking if a given non-degenerate quadratic form is isotropic or hyperbolic. Next we devise a method for computing the dimension of the anisotropic part of a quadratic form. Finally we present algorithms computing two field invariants: the level and the Pythagoras number.

The number of representations of squares by integral quaternary quadratic forms

Let [Formula: see text] be a positive definite (non-classic) integral quaternary quadratic form. We say [Formula: see text] is strongly[Formula: see text]-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly [Formula: see text] strongly [Formula: see text]-regular diagonal quaternary quadratic forms representing [Formula: see text] (see Table [Formula: see text]). In particular, we use eta-quotients to prove the strong [Formula: see text]-regularity of the quaternary quadratic form [Formula: see text], which is, in fact, of class number [Formula: see text] (see Lemma 4.5 and Proposition 4.6).