Group Invariance Research Articles

Bridge trisections and classical knotted surface theory

We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a trisection-theoretic proof of the Whitney-Massey Theorem, which bounds the possible values of this number in terms of the Euler characteristic. Second, we describe in detail how to compute the fundamental group and related invariants from a tri-plane diagram, and we use this, together with an analysis of bridge trisections of ribbon surfaces, to produce an infinite family of knotted spheres that admit non-isotopic bridge trisections of minimal complexity.

Incompressible flows: Relative scale invariance and isobaric polynomial fields

This article examines the Bouton–Lie group invariants of the Navier–Stokes equation (NSE) for incompressible fluids. The theory is applied to the general scaling transformation admitted by the NSE: it adds new partial differential equations to the Navier–Stokes system of equations and is used to derive all self-similar solutions. This method can be applied to any differential equation exhibiting scaling invariance. The solutions are found to be based on isobaric polynomials, which can be smooth and nonzero at the initial moment. The non-self-similar velocity and pressure fields in the case of constant viscosity at all scales are studied and also found to be polynomials, nonzero, and smooth at the initial moment; they vanish far away from the origin and are not increasing in time. For a subset of the solutions, the cavitation number is shown to be a conserved quantity; the invariant analysis is extended to higher-dimensioned manifolds for the purpose of finding additional conserved quantities.

Second-Order Approximate Equations of the Large-Scale Atmospheric Motion Equations and Symmetry Analysis for the Basic Equations of Atmospheric Motion

In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and eight one-parameter group invariant solutions are demonstrated. Three types of nontrivial similarity solutions and group invariants are proposed. With the help of perturbation method, we derive the second-order approximate equations for the large-scale atmospheric motion equations, including the non-dimensional equations and the dimensional equations. The second-order approximate equations of the large-scale atmospheric motion equations not only show the characteristics of physical quantities changing with time, but also describe the characteristics of large-scale atmospheric vertical motion.

Equivariant 4‐genera of strongly invertible and periodic knots

We study the equivariant genera of strongly invertible and periodic knots. Our techniques include some new strongly invertible concordance group invariants, Donaldson's theorem, and the g-signature. We find many new examples where the equivariant 4-genus is larger than the 4-genus.

Next-to-soft-virtual resummed prediction for pseudoscalar Higgs boson production at NNLO+NNLL¯

We present first results on the resummation of next-to-soft virtual (NSV) logarithms for the threshold production of pseudoscalar Higgs boson through gluon fusion at the LHC. These results are presented after resumming the NSV logarithms of the kind ${\mathrm{log}}^{i}(1\ensuremath{-}z)$ to $\overline{\mathrm{NNLL}}$ accuracy and matching them systematically to the fixed order NNLO cross sections. These results are obtained using collinear factorization, renormalization group invariance and recent developments in the NSV resummation techniques. The phenomenological implications of these NSV resummed results for 13 TeV LHC are studied and it is observed that these NSV logarithms are quite large. We also evaluate theory uncertainties and find that the renormalization scale uncertainties get reduced further with the inclusion of NSV corrections at various orders in QCD. We further study the impact of QCD corrections on mixed scalar-pseudoscalar states for different values of the mixing angle $\ensuremath{\alpha}$.

Maximal Kinematical Invariance Group of Fluid Dynamics and Applications

The maximal kinematical invariance group of the Euler equations of fluid dynamics for the standard polytropic exponent is larger than the Galilei group. Specifically, the inversion transformation (Σ:t→−1/t,x→→x→/t) leaves the Euler equation’s invariant. This duality has been used to explain the striking similarities observed in simulations of the supernova explosions and laboratory implosions induced in plasma by intense lasers. The inversion symmetry extends to discontinuous fluid flows as well. In this contribution, we provide a concise review of these ideas and discuss some applications. We also explicitly work out the implosion dual of the Sedov’s explosion solution.

Invariant Brauer group of an abelian variety

We study a new object that can be attached to an abelian variety or a complex torus: the invariant Brauer group, as recently defined by Yang Cao. Over the field of complex numbers this is an elementary abelian 2-group with an explicit upper bound on the rank. We exhibit many cases in which the invariant Brauer group is zero, and construct complex abelian varieties in every dimension starting with 2, both simple and non-simple, with invariant Brauer group of order 2. We also address the situation in finite characteristic and over non-closed fields.

Next to soft corrections to Drell-Yan and Higgs boson productions

We present a framework that resums threshold enhanced large logarithms to all orders in perturbation theory for the production of a pair of leptons in Drell-Yan process and of Higgs boson in gluon fusion as well as in bottom quark annihilation. We restrict ourselves to contributions from diagonal partonic channels. These logarithms include the distributions $((1-z)^{-1} \ln^i(1-z))_+$ resulting from soft plus virtual (SV) and the logarithms $\ln^i(1-z)$ from next-to-SV (NSV) contributions. We use collinear factorisation and renormalisation group invariance to achieve this. The former allows one to define a Soft-Collinear (SC) function which encapsulates soft and collinear dynamics of the perturbative results to all orders in strong coupling constant. The logarithmic structure of these results are governed by universal infrared anomalous dimensions and process dependent functions of Sudakov differential equation that the SC satisfies. The solution to the differential equation is obtained by proposing an all-order ansatz in dimensional regularisation, owing to several state-of-the-art perturbative results available to third order. The $z$ space solutions thus obtained provide an integral representation to sum up large logarithms originating from both soft and collinear configurations, conveniently in Mellin $N$ space. We show that in $N$ space, tower of logarithms $a_s^n/N^\alpha \ln^{2n-\alpha} (N), a_s^n/N^\alpha \ln^{2n-1-\alpha}(N) \cdots $ etc for $\alpha =0,1$ are summed to all orders in $a_s$.

Going beyond soft plus virtual

We present a formalism that sums up the soft-virtual (SV) and next-to-SV (NSV) diagonal contributions to inclusive colorless productions in hadron colliders to all orders in perturbative QCD. Using factorisation theorem, renormalisation group invariance and employing the transcendental structure of perturbative results, we show the exponential behavior of soft-collinear function. This allows us to predict certain SV and NSV terms to all orders from lower order information. We also present an integral representation for the coefficient functions which is suitable for Mellin $N$-space resummation.

Scalar gauge-Higgs models with discrete Abelian symmetry groups.

We investigate the phase diagram and the nature of the phase transitions of three-dimensional lattice gauge-Higgs models obtained by gauging the Z_{N} subgroup of the global Z_{q} invariance group of the Z_{q} clock model (N is a submultiple of q). The phase diagram is generally characterized by the presence of three different phases, separated by three distinct transition lines. We investigate the critical behavior along the two transition lines characterized by the ordering of the scalar field. Along the transition line separating the disordered-confined phase from the ordered-deconfined phase, standard arguments within the Landau-Ginzburg-Wilson framework predict that the behavior is the same as in a generic ferromagnetic model with Z_{p} global symmetry, p being the ratio q/N. Thus, continuous transitions belong to the Ising and to the O(2) universality class for p=2 and p≥4, respectively, while for p=3 only first-order transitions are possible. The results of Monte Carlo simulations confirm these predictions. There is also a second transition line, which separates two phases in which gauge fields are essentially ordered. Along this line we observe the same critical behavior as in the Z_{q} clock model, as it occurs in the absence of gauge fields.

On Analysis of Temperature Based Topological Indices of Some Covid-19 Drugs

A variety of graphical invariants have been described and tested, offering lots of applications in the fields of nanochemistry, computational networks and in different scientific research areas. One commonly studied group of invariants is the topological index, which allows to research the chemical, biological, and physical properties of a chemical structure. Topological indexes are numerical quantities that can be used to describe the properties of the molecular graph. In this article, we draw from the analytically closed formulas of certain molecular structures of coronavirus such as Ribavirin, Sofosbuvir and Oseltamivir by calculating temperature based topological indices.

Simultaneous conjugacy classes as combinatorial invariants of finite groups

We consider the problem of counting simultaneous conjugacy classes of n-tuples in a finite group G. Let denote the number of simultaneous conjugacy classes of n-tuples, and the number of simultaneous conjugacy classes of commuting n-tuples in G. We study the generating functions and which are rational functions of t. We show that determines and is completely determined by the class equation of G. We prove that the normalized functions and are invariants of isoclinism families. We compute exponential growth factors of and

Flow invariants in a channel obstructed by a line of inclined rods

An experiment is conducted in a rectangular channel obstructed by a transverse line of four inclined cylindrical rods. The pressure on the surface of a central rod and the pressure drop through the channel are measured varying the inclination angle of the rods. Three assemblies of rods with different diameters are tested. The measurements were analyzed applying momentum conservation principles and semi-empirical considerations. Several invariant dimensionless groups of parameters relating the pressure at key locations of the system with characteristic dimensions of the rods are produced. It was found that the independence principle holds for most of the Euler numbers characterizing the pressure at different locations, that is, the group is independent of the inclination angle provided that the inlet velocity projection normal to the rods is used to non-dimensionalize the pressure. The resulting semi-empirical correlations can be useful for designing similar hydraulic units.

Injective generation of derived categories and other applications of cohomological invariants of infinite groups

In the study of the representation theory of infinite groups, cohomological invariants play a very useful role. In a recent paper, we proved a number of properties regarding how these invariants interact with each other, extending the scope of some results in the literature. In this short article, we look into several ways in which the behavior of these invariants can be applied in various areas.

Nonlinear unknown input observability and unknown input reconstruction: The general analytical solution

Observability is a fundamental structural property of any dynamic system and describes the possibility of reconstructing the state that characterizes the system from fusing the observations of its inputs and outputs. Despite the effort made to study this property and to introduce analytical criteria capable of verifying whether or not a dynamic system satisfies this property, there is no general analytical criterion to obtain the observability of the state when the dynamics are also driven by unknown inputs. Here, we introduce the general analytical solution of this fundamental open problem, often called the unknown input observability problem. We provide the systematic procedure, based on automatic calculation (differentiation and determination of the matrix rank), which allows us to check the observability of the state even in the presence of unknown inputs. One of the fundamental ingredients to obtain this solution is the characterization of the group of invariance of observability. We have very recently introduced this group, together with a new set of tensor fields with respect to this group of transformations (Martinelli, 2020). The analytical solution of the unknown input observability problem is expressed in terms of these tensor fields. In Martinelli (2020) we provided the solution by restricting our investigation to systems that satisfy a special assumption that is called canonicity with respect to the unknown inputs. Here, after an exhaustive characterization of the concept of canonicity, we also account for the case when this assumption is not satisfied and we provide the general solution. This solution is also provided in the form of a new algorithm. In addition, even in the canonic case dealt with in Martinelli (2020), here we provide a new fundamental result that regards the convergence properties of the solution. Finally, as a consequence of the results obtained here, we also provide the condition to reconstruct the unknown inputs, and, when this condition is not met, what can be reconstructed on the unknown inputs. We illustrate the implementation of the new algorithm by studying the observability properties of a nonlinear system in the framework of visual-inertial sensor fusion, whose dynamics are driven by two unknown inputs and one known input. In particular, for this system, we follow step by step the algorithm introduced by this paper, which solves the unknown input observability problem in the most general case.

Implications and Consequences of SL(2R) as Invariance Group in the Description of Complex Systems Dynamics from a Multifractal Perspective of Motion.

Possible implications and consequences of using SL(2R) as invariance groups in the description at any scale resolution of the dynamics of any complex system are analyzed. From this perspective and based on Jaynes’ remark (any circumstance left unspecified in the description of any complex system dynamics has the concrete expression in the existence of an invariance group), in the present paper one specifies such unspecified circumstances that result directly from the consideration of the canonical formalism induced by the SL(2R) as invariance group. It follows that both the Hamiltonian function and the Guassian distribution acquire the status of invariant group functions, the parameters that define the Hamiltonian acquire statistical significances based on a principle of maximizing informational energy, the class of statistical hypotheses specific to Gaussians of the same average acts as transitivity manifolds of the group (transitivity manifolds which can be correlated with the multifractal-non-multifractal scale transitions), joint invariant functions induced through SL(2R) groups isomorphism (the SL(2R) variables group, and the SL(2R) parameters group, etc.). For an ensemble of oscillators of the same frequency, the unspecified circumstances return to the ignorance of the amplitude and phase of each of the oscillators, which forces the recourse to a statistical ensemble traversed by the transformations of the Barbilian-type group. Finally, the model is validated based on numerical simulations and experimental results that refer to transient phenomena in ablation plasmas. The novelty of our model resides in the fact that fractalization through stochasticization is imposed through group invariance, situation in which the group’s transitivity manifolds can be correlated with the scale resolution.

Quasi-Representations of Groups and Two-Homology

The Exel-Loring formula asserts that two topological invariants associated to a pair of almost commuting unitary matrices coincide. Such a pair can be viewed as a quasi-representation of $\mathbb{Z}^2$. We give a generalization of this formula for countable discrete groups. We also show the nontriviality of the corresponding invariants for quasidiagonal groups which are coarsely embeddable in a Hilbert space and have nonvanishing second Betti number.

Relating Cut and Paste Invariants and Tqfts

Abstract In this paper, we are concerned with a relation between TQFTs and the cut and paste SKK invariants introduced by Karras, Kreck, Neumann, and Ossa. Cut and paste SKK invariants are functions on the set of smooth manifolds whose values on cut and paste equivalent manifolds differ by an error term depending only on the glueing diffeomorphisms. Here we investigate a surprisingly natural group homomorphism between the group of invertible TQFTs and the group of SKK invariants and describe how these groups fit into an exact sequence. We conclude in particular that all positive real-valued SKK invariants can be realized as restrictions of invertible TQFTs.

The construction of modular invariants

We use the k [ V ] k[V] -module generator of the dual module of the polynomial ring k [ V ] k[V] over its subring of invariants of a finite group to construct modular invariants and show that it behaves better than the transfer homomorphism.

Symmetry reductions, generalized solutions and dynamics of wave profiles for the (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations

In this article, a (2+1)-dimensional system of Broer–Kaup–Kupershmidt (BKK) equations, which describes the non-linear, and long gravity waves in a dispersive system, is investigated by applying the method of Lie group of invariance. Applying the Lie group technique, the infinitesimals, vector fields, commutator table, and adjoint table are constructed for the BKK system. Furthermore, a one-dimensional optimal system of subalgebras is determined with the help of the adjoint transformation matrix; thereafter, the BKK system of equations is reduced into many systems of ordinary differential equations (ODEs) with respect to the similarity variables obtained through the symmetry reduction. These systems of ODEs are solved under some parametric constraints to obtain the exact closed form solutions. The obtained results are interpreted physically via graphical representation. Thus, dark–bright solitary waves, multi-peak mixed waves, breather type waves, periodic waves, multi-peakon solitons, and multi-solitons profiles of the obtained solutions are presented to make this research physically significant. A comparison is also presented between the solutions obtained in this article and the solutions reported by Kassem and Rashed in Kassem and Rashed (2019). The solutions obtained in this article are more general, and completely different in view of solutions obtained by Kassem and Rashed. The resulting solutions are found to be useful to understand the dynamics of BKK model and represent the authenticity as well as the effectiveness of the Lie group method. Therefore, the obtained solutions, and their dynamical wave structures are quite significant for understanding the propagation of the long gravity waves in a dispersive system. To obtain the infinitesimal generators and wave profiles of obtained solutions, symbolic computation is performed in the software package MAPLE and MATHEMATICA.