Topological models for stable motivic invariants of regular number rings

For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers and finite fields. We use this to extend Morel’s identification of the endomorphism ring of the motivic sphere with the Grothendieck–Witt ring of quadratic forms to deeper base schemes.

A New System with a Self-Excited Fully-Quadratic Strange Attractor and Its Twin Strange Repeller

In nonlinear dynamics, the study of chaotic systems has attracted the attention of many researchers around the world due to the exciting and peculiar properties of such systems. In this regard, the present paper introduces a new system with a self-excited strange attractor and its twin strange repeller. The unique characteristic of the presented system is that the system variables are all in their quadratic forms; therefore, the proposed system is called a fully-quadratic system. This paper also elaborates on the study of the bifurcation diagram, the interpretation of Lyapunov exponents, the representation of basin of attraction, and the calculation of connecting curves as the employed method for investigating the system’s dynamics. The investigation of 2D bifurcation diagrams and Lyapunov exponents indicated in this paper can better recognize the system’s dynamics since they are plotted considering simultaneous changes of two parameters. Moreover, the connecting curves of the proposed system are calculated. The system’s connecting curves help identify the system’s different behaviors by providing general information about the nature of the flows.

Ordinary Versus Novel Particle With the Example of Their Spontaneous Transition

The bicubic equation of particle limiting velocity formalism yields three solutions c1, c2 and c3, (primary, secondary and tertiary) limiting velocities in terms of the congruent parameter  which is defined in terms of m, v, and E, respectively being particle mass, velocity and energy. The bicubic equation discriminant D is given in terms of the congruent parameter z(m). When one has z2(m) ≤ 1 with the discriminant satisfying D ≤ 0 then we are talking about limiting velocities of ordinary particles. Good examples are the relativistic particles such as electron, neutrino,etc., with luminal limiting velocity c3 = c and calculated superluminal c2, and imaginary superluminal c1, all corresponding to the real particle energy. On the specific level, the situations like these, we discuss in the muon neutrino velocities with the OPERA detector and the electron velocities from the 2010 Grab Nebula Flare. The z(m) = 1 value separates the ordinary particles from novel particles, associated with D ⪰ 0 and z2 ⪰ 1 with new novel particle limiting velocity solutions c1, c2 and c3 which depend, in addition to z(m), also on the congruent angle α(m), nonlinearly related to z(m). These solutions are discussed on the newly defined sterile neutrino which here is modeled as an ordinary particle with z2 ⪯ 1 spontaneously transiting via z(m) = 1 into the modeled novel sterile neutrino with z2 ⪰ 1. All ordinary and novel particles limiting velocities carry real particle energies; the ordinary particle limiting velocity solutions being in quadratic forms, while the novel particle limiting velocity solutions being respectively, in quadratic complex form, linear complex form, and just congruent angle α complex quadratic form.

Approximation of partial differential equations on compact resistance spaces

We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed.

Composition of binary quadratic forms over number fields

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class number one. The article contains an explicit description of the correspondence. In the case of totally negative discriminants, equivalent conditions are given for a binary quadratic form to be totally positive definite.

Automated generation of turbulence shell models by the methods of computer algebra

Описана разработанная методика генерации уравнений каскадных моделей турбулентности с помощью систем компьютерной алгебры. Методика позволяет варьировать размер масштабной нелокальности модели, вид квадратичных законов сохранения и спектральных законов, знаменатель геометрической прогрессии масштабов. Ее использование позволяет быстро и безошибочно генерировать целые классы моделей. Может использоваться для разработки каскадных моделей гидродинамических, магнитогидродинамических и конвективных турбулентных систем. There is a great variety of shell turbulence models. Such models reproduce certain characteristics of turbulence. A model that could reproduce all turbulence regimes does not exist at the moment. Information about a particular model is contained in a set of persistent quantities, which are some quadratic forms of turbulent fields. These quadratic forms should be formal analogs of the exact conserved quantities. It is important to note that the main idea of Shell models presupposes a refusal to describe the geometric structure of movements. At the same time, it is well known that turbulent processes in spaces of two and three dimensions behave differently. Therefore, the provision of certain combinations of conserved quantities allows indirect introducing into the shell model the information about the dimension of the physical space in which the turbulent process develops. Purpose. The aim of this work was to create software tools that would quickly generate classes of models that satisfy one or another set of conservation laws. The choice of a specific model within these classes can then be specified using additional physical considerations, for example, the existence of a given probability distribution for the interaction of certain shells. Methods. The developed technique for generating equations of shell turbulence models is carried out using symbolic computation systems (computer algebra systems - CAS). Note that symbolic packages are used not for studying ready-made shell models, but for the automated generation of the equations of these models themselves. The technique allows varying the value of the scale nonlocality of the model, the form of the quadratic conservation laws and spectral laws, the denominator of the geometric progression of scales. It allows quickly and accurately generating the entire set of classes of the models. It can be used to develop shell models of hydrodynamic, magnetohydrodynamic and convective turbulent systems. Findings. It seems that the proposed technique will be useful for studying the properties of turbulence in the framework of cascade models

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).