Basic Invariants Research Articles

Six-derivative Yang-Mills couplings in heterotic string theory

In this work, we present a comprehensive analysis of the structure of six-derivative bosonic couplings in heterotic string theory. First, we determine the maximal covariant and Yang-Mills gauge invariant basis, which consists of 801 independent coupling constants. By imposing T-duality constraints on the circular reduction of these terms, we obtain 468 relations between the coupling constants at the six-derivative order and the known couplings at lower derivative orders. Through the use of field redefinitions, we are able to eliminate the remaining 333 coupling constants. Remarkably, we find that the Yang-Mills field strength only appears through the trace of two field strengths or their derivatives. Finally, we perform further field redefinition to rewrite the remaining couplings in a canonical form characterized by 85 independent couplings.

Професійна спрямованість математичної підготовки студентів технічних спеціальностей у контексті фундаменталізації освітнього процесу

The article examines the peculiarities of mathematical training of future technical specialists in the context of fundamentalization of the educational process. The study of the definitions of fundamentalization, mathematical preparation was carried out, the main characteristic features of mathematical preparation, proposed by the researchers of the problem of mathematical preparation, were singled out. An analysis of the work of scientists dedicated to the selected research problem was carried out. It has been established that the mathematical training of future technical specialists is a dialectical combination of formed mathematical skills and abilities for their applied application and the direct process of forming these skills. The fundamentalization of mathematical training is considered to be the concept of improving mathematical training, in which basic mathematical invariants are selected for their further application in professional activities. To implement the approach of professional orientation of mathematical training of students of technical specialties, it is recommended to take into account the peculiarities of the construction of the course of higher mathematics and courses of special disciplines in order to enable the integration of educational material, as well as the organization of the educational process in higher education institutions. Based on the analysis of educational programs for students of various specialties of the Vinnytsia National Technical University, a conclusion was made about the difference in program results in mathematics for students of the same lecture stream. On the basis of which it is proposed to unify tasks of professional content. Examples of professional tasks from the "Theory of Probability" section are given.

Complete CP-eigen bases of meson-baryon chiral lagrangian up to p5-order

Chiral perturbation theory describes the low energy dynamics of mesons and baryons in terms of the nonlinear Goldstone boson and fermion degrees of freedom. Through the Young tensor technique, we construct the on-shell operator bases for the meson-baryon system up to p5-order, using the chiral dimension power counting and heavy baryon expansion. For the Lorentz structure, additional treatments on off-shell external sources and operators with higher derivatives are necessarily considered, while for the internal structure, the invariant tensor basis is converted into the trace basis equivalently, and Cayley-Hamilton relations are utilized to classify different CP eigen-operators. Finally we present the complete operator set of C+P+, C+P-, C-P+, and C-P- eigen-operators at the p5-order, and obtain the operator counting from the Hilbert series.

Small instanton-induced flavor invariants and the axion potential

Small instantons which increase the axion mass due to an appropriate modification of QCD at a UV scale ΛSI, can also enhance the effect of CP-violating operators to shift the axion potential minimum by an amount, θind, proportional to the flavorful couplings in the SMEFT. Since physical observables must be flavor basis independent, we construct a basis of determinant-like flavor invariants that arise from instanton calculations containing the effects of dimension-six CP-odd operators at the scale . This new basis provides a more reliable estimate of the shift θind, that is severely constrained by neutron electric dipole moment experiments. In particular, for the case of four-quark, semi-leptonic and gluon dipole operators, these invariants are then used to provide improved limits on the ratio of scales for different flavor scenarios. The CP-odd flavor invariants also provide a classification of the leading effects from Wilson coefficients, and as an example, we show that a semi-leptonic four-fermion operator is subdominant compared to the four-quark operators. More generally, the flavor invariants, together with an instanton NDA, can be used to more accurately estimate small instanton effects in the axion potential that arise from any SMEFT operator.

Theoretical Study of the Thermal Rate Coefficients of the H3+ + C2H4 Reaction: Dynamics Study on a Full-Dimensional Potential Energy Surface.

Ion-molecular reactions play a significant role in molecular evolution within the interstellar medium. In this study, the entrance channel reaction, H3+ + C2H4 → H2 + C2H5+, was investigated using classical molecular dynamic (classical MD) and ring polymer molecular dynamic (RPMD) simulation techniques. We developed an analytical potential energy surface function with a permutationally invariant polynomial basis, specifically employing the monomial symmetrized approach. Our dynamic simulations reproduced the rate coefficient of 300 K for H3+ + C2H4 → H2 + C2H5+, aligning reasonably well with the values in the kinetic database commonly utilized in astrochemistry. The thermal rate coefficients obtained using both the classical MD and RPMD techniques exhibited an increase from 100 K to 300 K as the temperature rose. Additionally, we analyzed the excess energy distribution of the C2H5+ fragment with respect to temperature to investigate the indirect reaction pathway of C2H5+ → H2 + C2H3+. This result suggests that the indirect reaction pathway of C2H5+ → H2 + C2H3+ holds minor significance, although the distribution highly depends on the collisional temperature.

Accurate neural-network-based fitting of full-dimensional N2 –Ar and N2–CH4 two-body potential energy surfaces aimed at spectral simulations

We describe the development of machine-learned (ML) potentials for flexible, weakly interacting monomers. A recently suggested permutationally invariant polynomial neural network (PIP-NN) approach is utilised to represent the full-dimensional two-body component of the molecular pair energy. To ensure the asymptotic zero-interaction limit, a tailored subset of the full invariant polynomial basis set is utilised, and their variables are modified to achieve a better fit of the correct asymptotic behaviour at a long range. This new technique is used to build full-dimensional potentials for the two-body N 2 –Ar and N 2 –CH 4 interactions by fitting databases of ab initio energies calculated at the coupled-cluster level of theory. The second virial coefficient, fully accounting for molecular flexibility, is then calculated within the classical framework using the obtained PIP-NN potential surfaces. A trajectory-based simulation of the N 2 –Ar collision-induced absorption is conducted, covering both the far- and mid-infrared ranges.

On some basic curvature invariants of screen homothetic lightlike hypersurfaces in a GRW spacetime

This study is focused on the investigation of lightlike hypersurfaces of a generalized Robertson-Walker (GRW) spacetime. Recently, Poyraz (2022) [51,52] established some basic inequalities involving various curvature invariants of screen homothetic lightlike hypersurfaces of GRW spacetimes, like k-scalar curvature and k-Ricci curvature. In this work, we consider other basic curvature invariants, namely the scalar curvature and δ-Casorati curvatures, and derive new inequalities for such hypersurfaces of a GRW spacetime. We also find the conditions for which the equality cases in these inequalities hold and give some applications in Lorentzian geometry.

Adaptive Prosthetic Trajectory Estimation Based on Key Points Constraints

Lower limb knee–ankle prostheses can effectively assist above-knee amputees in completing their basic daily activities. This study explored methods for estimating the joint kinematics of intelligent lower limb prostheses to better adapt them to the walking requirements of amputees. A method of estimating the knee and ankle joint trajectories under different speeds and slopes was raised. The joints of a prosthetic need to have a movement trajectory similar to that of the joints of healthy individuals, taking into account the person’s motion intentions and complying with the law of movement. In this study, a prosthetic kinematics estimation method was studied to realize continuous speed and slope adaptation through key points. The iterative Douglas–Peucker algorithm automatically found the key points of the kinematic trajectory curve, and an invariant basis function-fitting model with key points constraints was constructed. Finally, the radial basis function neural network was used to estimate the adaptive task function of the parts affected by the speed and slope, resulting in an overall kinematics estimation model. Our experiments verified the effectiveness of the proposed method, which accurately estimated the kinematic trajectory; the knee and ankle joint estimation errors were greatly reduced compared to those obtained with previous methods, facilitating further research on individual algorithms.

Hereditary Saturated Subsets and the Invariant Basis Number Property of the Leavitt Path Algebra of Cartesian Products

Hereditary Saturated Subsets and the Invariant Basis Number Property of the Leavitt Path Algebra of Cartesian Products

Enumerating Calabi‐Yau Manifolds: Placing Bounds on the Number of Diffeomorphism Classes in the Kreuzer‐Skarke List

AbstractThe diffeomorphism class of simply connected smooth Calabi‐Yau threefolds with torsion‐free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi‐Yau threefolds constructed from the Kreuzer‐Skarke (KS) list of reflexive polytopes up to Picard number six. The main difficulty arises from the comparison of triple intersection numbers and divisor integrals of the second Chern class up to basis transformations. By using certain basis‐independent invariants, some of which appear here for the first time, we are able to place lower bounds on the number of classes. Upper bounds are obtained by explicitly identifying basis transformations, using constraints related to the index of line bundles. Extrapolating our results, we conjecture that the favorable entries of the KS list of reflexive polytopes lead to some diffeomorphically distinct Calabi‐Yau threefolds.

Bulk-Boundary Correspondence in Point-Gap Topological Phases.

A striking feature of non-Hermitian systems is the presence of two different types of topology. One generalizes Hermitian topological phases, and the other is intrinsic to non-Hermitian systems, which are called line-gap topology and point-gap topology, respectively. Whereas the bulk-boundary correspondence is a fundamental principle in the former topology, its role in the latter has not been clear yet. This Letter establishes the bulk-boundary correspondence in the point-gap topology in non-Hermitian systems. After revealing the requirement for point-gap topology in the open boundary conditions, we clarify that the bulk point-gap topology in open boundary conditions can be different from that in periodic boundary conditions. On the basis of real space topological invariants and the K theory, we give a complete classification of the open boundary point-gap topology with symmetry and show that the nontrivial open boundary topology results in robust and exotic surface states.

A simple one-electron expression for electron rotational factors.

Within the context of fewest-switch surface hopping (FSSH) dynamics, one often wishes to remove the angular component of the derivative coupling between states J and K. In a previous set of papers, Shu et al. [J. Phys. Chem. Lett. 11, 1135-1140 (2020)] posited one approach for such a removal based on direct projection, while we isolated a second approach by constructing and differentiating a rotationally invariant basis. Unfortunately, neither approach was able to demonstrate a one-electron operatorÔ whose matrix element JÔK was the angular component of the derivative coupling. Here, we show that a one-electron operator can, in fact, be constructed efficiently in a semi-local fashion. The present results yield physical insight into designing new surface hopping algorithms and are of immediate use for FSSH calculations.

Maximally entangled real states and SLOCC invariants: the 3-qutrit case

The absolute values of polynomial SLOCC invariants (which always vanish on separable states) can be seen as entanglement measures. We study the case of real 3-qutrit systems and discover a new set of maximally entangled states (from the point of view of maximizing the hyperdeterminant). We also study the basic fundamental invariants and find real 3-qutrit states that maximize their absolute values. It is notable that the Aharonov state is a simultaneous maximizer for all three fundamental invariants. We also study the evaluation of these invariants on random real 3-qutrit systems and analyze their behavior using histograms and level-set plots. Finally, we show how to evaluate these invariants on any 3-qutrit state using basic matrix operations.

A conjecture of Mallows and Sloane with the universal denominator of Hilbert series

Abstract A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.

About uniformly Menger spaces

Precompact type properties - precompactness (=totally precompactness), s-precompactness, pre-Lindelöfness, (=ℵ0-boundedness), t -boundedness - belong to the basic important invariants studied in the uniform topology. The theory of these invariants is wide and continues to develop. However, in a sense, the class of uniformly Menger spaces escaped the attention of researchers. Lj.D.R. Kočinac was the first who introduced and studied the class of uniformly Menger spaces in [3, 4]. It immediately follows from the definition that the class of uniformly Menger spaces lies between the class of precompact uniform spaces and the class of pre-Lindelöf uniform spaces. Therefore, we expect it to have many good properties. In this paper some important properties of the uniformly Menger spaces are investigated. In particular, it is established that under uniformly perfect mappings, the uniformly Menger property is preserved both in the image and the preimage direction.

Оптимальные оценки количества звеньев базисных горизонтальных ломаных для 2-ступенчатых групп Карно с горизонтальным распределением коранга 1

For a 2-step Carnot group D_n, "dim" D_n=n+1, with horizontal distribution of corank 1, we proved that the minimal number N_(X_(D_n ) ) such that any two points u,v∈D_n can be joined by some basis horizontal k-broken line (i.e. a broken line consisting of k links) L_k^(X_(D_n ) ) (u,v), k≤N_(X_(D_n ) ), does not exeed n+2. The examples of D_n such that N_(X_(D_n ) )=n+i, i=1,2. were found. Here X_(D_n )={X_1,…,X_n} is the set of left invariant basis horizontal vector fields of the Lie algebra of the group D_n, and every link of L_k^(X_(D_n ) ) (u,v) has the form "exp"(asX_i)(w), s∈[0,s_0], a=const.

Reciprocal distance Laplacian spectral properties double stars and their complements

Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy.

Connectivity gaps among matroids with the same enumerative invariants

Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the G-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer n, there are pairs of matroids that have the same configuration (and so the same G-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds n, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.

Group invariants for Feynman diagrams

It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.

On the analytical description of the nonlinear stage of the Weibel instability in collisionless anisotropic plasma

On the basis of the energy invariants for the Weibel instability in a collisionless non-relativistic plasma, an analytical relation is obtained between the instantaneous values of the space-averaged energy density of the magnetic field, its dominant wavenumber and the average plasma anisotropy parameter. The relation is valid for arbitrary particle velocity distribution functions, including ones that vary with time at the nonlinear stage of instability. It is obtained under the assumption that the plasma is homogeneous along a certain axis, determined, for example, by an external magnetic field, and taking into account only the modes with wavevectors orthogonal to this axis. We give an estimate of the maximum magnetic field energy achievable during the Weibel instability and show that its ratio to the initial longitudinal energy of particles, even for a large initial plasma anisotropy parameter, cannot exceed the value approximately equal to 0.2. The obtained analytical relation is verified using two-dimensional particle-in-cell simulations.