Expressions Of Invariants Research Articles

Topological orders, braiding statistics, and mixture of two types of twisted BF theories in five dimensions

Topological orders are a prominent paradigm for describing quantum many-body systems without symmetry-breaking orders. We present a topological quantum field theoretical (TQFT) study on topological orders in five-dimensional spacetime (5D) in which topological excitations include not only point-like particles, but also two types of spatially extended objects: closed string-like loops and two-dimensional closed membranes. Especially, membranes have been rarely explored in the literature of topological orders. By introducing higher-form gauge fields, we construct exotic TQFT actions that include mixture of two distinct types of BF topological terms and many twisted topological terms. The gauge transformations are properly defined and utilized to compute level quantization and classification of TQFTs. Among all TQFTs, some are not in Dijkgraaf-Witten cohomological classification. To characterize topological orders, we concretely construct all braiding processes among topological excitations, which leads to very exotic links formed by closed spacetime trajectories of particles, loops, and membranes. For each braiding process, we construct gauge-invariant Wilson operators and calculate the associated braiding statistical phases. As a result, we obtain expressions of link invariants all of which have manifest geometric interpretation. Following Wen’s definition, the boundary theory of a topological order exhibits gravitational anomaly. We expect that the characterization and classification of 5D topological orders in this paper encode information of 4D gravitational anomaly. Further consideration, e.g., putting TQFTs on 5D manifolds with boundaries, is left to future work.

(2+1)-dimensional black holes of Einstein\u2019s theory with Born\u2013Infeld type electrodynamic sources

A new family of (2+1)-dimensional black holes are investigated in the background of Born–Infeld type theories coupled to a Riemannian curved spacetime. We know that both the scale and dual invariances are violated for these nonlinear electromagnetic theories. In this set-up, first we consider a pure magnetic source in a model of exponential electrodynamics and find a magnetically charged (2+1)-dimensional black hole solution in terms of magnetic charge q and nonlinearity parameter beta . In the second case we consider a pure electric source of gravity in the framework of arcsin electrodynamics and derive the associated (2+1)-dimensional black hole solution in terms of electric charge Q and the parameter beta . The asymptotic behaviour of the solutions at infinity as well as at rrightarrow 0 in both the frameworks is discussed. The asymptotic expressions of curvature invariants in the case of exponential electrodynamics shows that there exists a finite value of curvature at the origin, while in arcsin electrodynamics, the corresponding asymptotic behaviour shows that there is a true curvature singularity at the centre of the charged object. Furthermore, thermodynamics of the resulting charged black holes within the context of both the models is studied. It is shown that the thermodynamic quantities corresponding to these objects satisfy the first law of black hole thermodynamics.

Anomaly indicators and bulk-boundary correspondences for three-dimensional interacting topological crystalline phases with mirror and continuous symmetries

We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of $U(1)$, $SU(2)$ or $SO(3)$. In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous ten-fold-way classification, with the time-reversal symmetry replaced by the mirror symmetry and with strong interaction taken into account. For surface SETs, the most general interplay between symmetries and anyon excitations is considered. Based on the previously proposed dimension reduction and folding approaches, we re-derive the classification of bulk SPT phases and define a \emph{complete} set of bulk topological invariants for every symmetry group under consideration, and then derive explicit expressions of the bulk invariants in terms of surface topological properties (such as topological spin, quantum dimension) and symmetry properties (such as mirror fractionalization, fractional charge or spin). These expressions are our quantitative bulk-boundary correspondences. Meanwhile, the bulk topological invariants can be interpreted as \emph{anomaly indicators} for the surface SETs which carry 't Hooft anomalies of the associated symmetries whenever the bulk is topologically non-trivial. Hence, the quantitative bulk-boundary correspondences provide an easy way to compute the 't Hooft anomalies of the surface SETs. Moreover, our anomaly indicators are complete. Our derivations of the bulk-boundary correspondences and anomaly indicators are explicit and physically transparent.

Second-order approximation pseudo-rigid model of leaf flexure hinge

Leaf flexure hinges are key-components in compliant mechanisms. With their simple shape and intrinsic flexibility, they connect bulky parts and allow relative motion. In order to address the synthesis and the analysis of mechanisms embodying such components, leaf hinges are often modelled with a pseudo-rigid assembly including a revolute joint and a torsion spring. However, this approximation is only valid within a limited range of motion thus it is not completely suitable for precision mechanisms or for mechanisms undergoing large displacements. In this paper, the authors propose a novel and accurate pseudo-rigid model to approximate the relative motion of the bodies connected by a leaf hinge within a large range of relative rotation. The proposed equivalent mechanism is composed of two circular shaped bodies with pure rolling contact. The body shapes represent the polodes of relative displacement between the connected bodies and the real mechanism kinematics is approximated through an epicyclic arrangement. Hence, the concept of polodes, usually limited to rigid body motion, is herein extended to compliant mechanisms. For the definition of the relative motion between bodies coupled by means of compliant hinges, new analytical expressions of kinematic invariants are deduced. In particular, polodes’ radii of curvature and diameter of inflection circle are deduced. The reliability and accuracy of the proposed leaf hinge modelling is confirmed by the agreement, within a large range of rotation and for the case of a pure bending, by a comparison of FEM analysis results with those obtained using the herein proposed model.

Symplectic invariants for curves and integrable systems in similarity symplectic geometry

In this paper, similarity symplectic geometry for curves is proposed and studied. Explicit expressions of the symplectic invariants, Frenet frame and Frenet formulae for curves in similarity symplectic geometry are obtained by using the equivariant moving frame method. The relationships between the Euclidean symplectic invariants, Frenet frame, Frenet formulae and the similarity symplectic invariants, Frenet frame, Frenet formulae for curves are established. Invariant curve flows in four-dimensional similarity symplectic geometry are also studied. It is shown that certain intrinsic invariant curve flows in four-dimensional similarity symplectic geometry are related to the integrable Burgers and matrix Burgers equations.

Euler-Poincaré formulation of hybrid plasma models

Three different hybrid Vlasov-fluid systems are derived by applying reduction by symmetry to Hamilton's variational principle. In particular, the discussion focuses on the Euler-Poincar\'e formulation of three major hybrid MHD models, which are compared in the same framework. These are the current-coupling scheme and two different variants of the pressure-coupling scheme. The Kelvin-Noether theorem is presented explicitly for each scheme, together with the Poincar\'e invariants for its hot particle trajectories. Extensions of Ertel's relation for the potential vorticity and for its gradient are also found in each case, as well as new expressions of cross helicity invariants.

Using size for bounding expressions of graph invariants

With the help of the AutoGraphiX system, we study relations of the form $$\underline{b}_m \le i_1(G) \oplus i_2(G) \le\overline{b}_m$$ where i1(G) and i2(G) are invariants of the graph G, ⊕ is one of the operations −,+,/,×, \(\underline{b}_{m}\) and \(\overline{b}_{m}\) are best possible lower and upper bounding functions depending only one the size m of G. Specifically, we consider pairs of indices where i1(G) is a measure of distance, i.e., diameter, radius or average eccentricity, and i2(G) is a measure of connectivity, i.e., minimum degree, edge connectivity and vertex connectivity. Conjectures are obtained and then proved in almost all cases.

Fatigue of Woven Composite Laminates in Off-Axis Loading I. The Mastercurves

The behaviour of woven orthotropic composite laminates in static and fatigue off-axis loading is described. It is shown that all phenomena: linear and nonlinear deformation; accumulation of damage, measured as change of cyclic modulus or hysteresis loop; and the static and cyclic strength can be described by single master curves using the generalized stress and strain functions. These functions always contain the quadratic expressions of invariants of orthotropy, but coefficients of the invariants depend on the severity of nonlinearity in the described processes or phenomena.

Extended knots and the space of states of quantum gravity

In the loop representation the quantum constraints of gravity can be solved. This fact allowed significant progress in the understanding of the space of states of the theory. The analysis of the constraints over loop-dependent wavefunctions has been traditionally based upon geometric (in contrast to analytic) properties of the loops. The reason for this preferred way is twofold: on the one hand, the inherent difficulties associated with the analytic loop calculus, and on the other hand, our limited knowledge about the analytic properties of knots invariants. Extended loops provide a way to overcome the difficulties at both levels. On the one hand, a systematic method to construct analytic expressions of diffeomorphism invariants (the extended knots) in terms of the Chern-Simons propagators can be developed. Extended knots are simply related to ordinary knots (at least formally). The analytic expressions of knot invariants could be produced then in a generic way. On the other hand, the evaluation of the Hamiltonian over extended loop wavefunctions can be thoroughly accomplished in the extended loop framework. These two ingredients promote extended loops as a potential resort for answering important questions about quantum gravity.

A mathematical model for diamond-type crystals

The physical phenomena occurring in a diamond-type crystal are usually described in terms of the invariants of some representations of the space group Oh7. By choosing a new description, we obtain equivalent representations in which the expressions of the usual invariants are simpler. The mathematical facilities obtained by using these representations offer new possibilities for elaborating improved variants of the usual theories or extensions from perfect crystals to crystals with impurities or defects.

The O(3) gauge transformation and 3-state vertex models

The authors consider the O(3) gauge transformation for three-state vertex models on lattices of coordination number three. Using an explicit mapping between O(3) and SL(2), they establish that there exist exactly six polynomials of the vertex weights, which are fundamentally invariant under the O(3) transformation. Explicit expressions of these fundamental invariants are obtained in the case of symmetric vertex weights.

Affordances, affective behaviours, attachments and assistance

This paper consists of four sections. It opens by suggesting parents offer affordances, which are behavioural expressions of invariants, to their infants. Next, nine classes of relationship affordance are identified. Results from separate groups of respondents providing semantic‐scale ratings for these classes revealed a single sequence with “aggressive” at one extreme and “tender contact” at the other. Third, descriptions of parental behaviours corresponding to the four infant attachment classification groups were gleaned from the literature. When these descriptions were aligned with the nine affordance terms a strong correspondence emerged; infant attachment classifications could also be arranged into a single sequence, each with distinct peak performance indicators. In the final section it is suggested this sequence has as a major invariant “parental mourning” since parallels between the separation‐reunion process, attachment classifications and parental affordances occur.

Algebraic invariants of the O(2) gauge transformation

The authors consider the O(2) gauge transformation for a two-state vertex model on a lattice, and derive its fundamental algebraic invariants, the minimal set of homogeneous polynomials of the vertex weights which are invariant under O(2) transformations. Explicit expressions of the fundamental invariants are given for symmetric vertex models on lattices with coordination number p=2, 3, 4, 5, 6, generalising p=3 results obtained previously from more elaborate considerations.

XXXIII. A memoir on the conditions for the existence of given systems of equalities among the roots of an equation

It is well known that there is a symmetric function of the roots of an equation, viz. the product of the squares of the differences of the roots, which vanishes when any two roots are put equal to each other, and that consequently such function expressed in terms of the coefficients and equated to zero, gives the condition for the existence of a pair of equal roots. And it was remarked long ago by Professor Sylvester, in some of his earlier papers in the ‘Philosophical Magazine,’ that the like method could be applied to finding the conditions for the existence of other systems of equalities among the roots, viz. that it was possible to form symmetric functions, each of them a sum of terms containing the product of a certain number of the differences of the roots, and such that the entire function might vanish for the particular system of equalities in question; and that such functions expressed in terms of the coefficients and equated to zero would give the required conditions. The object of the present memoir is to extend this theory and render it exhaustive, by showing how to form a series of types of all the different functions which vanish for one or more systems of equalities among the roots; and in particular to obtain by the method distinctive conditions for all the different systems of equalities between the roots of a quartic or a quintic equation, viz. for each system conditions which are satisfied for the particular system, and are not satisfied for any other systems, except, of course, the more special systems included in the particular system. The question of finding the conditions for any particular system of equalities is essentially an indeterminate one, for given any set of functions which vanish, a function syzygetically connected with these will also vanish; the discussion of the nature of the syzygetic relations between the different functions which vanish for any particular system of equalities, and of the order of the system composed of the several conditions for the particular system of equalities, does not enter into the plan of the present memoir. I have referred here to the indeterminateness of the question for the sake of the remark that I have availed myself thereof, to express by means of invariants or covariants the different systems of conditions obtained in the sequel of the memoir; the expressions of the different invariants and covariants referred to are given in my ‘Second Memoir upon Quantics,’ Philosophical Transactions, vol. cxlvi. (1856). 1. Suppose, to fix the ideas, that the equation is one of the fifth order, and call the roots α,β,γ,δ,ε . Write 12 = ∑ ϕ ( α — β ) l , 12.13 = ∑ ϕ ( α — β ) l ( α — γ ) m , 12.34 = ∑ ϕ ( α — β ) l γ — δ ) n &c., where ϕ is an arbitrary function and l, m , &c. are positive integers. It is hardly necessary to remark that similar types, such as 12, 13, 45, &c., or as 12.13 and 23. 25, &c., denote identically the same sums. Two types, such as 12.13 and 14. 15. 23. 24. 25. 34. 35. 45, may be said to be complementary to each other. A particular product ( α — β )( γ — δ ) does or does not enter as a term (or factor of a term) in one of the above-mentioned sums, according as the type 12. 34 of the product, or some similar type, does or does not form part of the type of the sum; for instance, the product is a term ( α — β )( γ — δ ) (or factor of a term) of each of the sums 12. 34, 13. 45. 24, &c., but not of the sums 12. 13. 14. 15, &c.