Expressions For Invariants Research Articles

Numerical Simulation of Constrained Flows through Porous Media Employing Glimm’s Scheme

This work uses a mixture theory approach to describe kinematically constrained flows through porous media using an adequate constitutive relation for pressure that preserves the problem hyperbolicity even when the flow becomes saturated. This feature allows using the same mathematical tool for handling unsaturated and saturated flows. The mechanical model can represent the saturated–unsaturated transition and vice-versa. The constitutive relation for pressure is a continuous and differentiable function of saturation: an increasing function with a strictly convex, increasing, and positive first derivative. This significant characteristic permits the fluid to establish a tiny controlled supersaturation of the porous matrix. The associated Riemann problem’s complete solution is addressed in detail, with explicit expressions for the Riemann invariants. Glimm’s semi-analytical scheme advances from a given instant to a subsequent one, employing the associated Riemann problem solution for each two consecutive time steps. The simulations employ a variation in Glimm’s scheme, which uses the mean of four independent sequences for each considered time, ensuring computational solutions with reliable positions of rarefaction and shock waves. The results permit verifying this significant characteristic.

Physio-chemical properties of HCP ( U ) and HEX ( U ) crystal structure lattice via eccentricity dependent chemical invariants

A crystal lattice is the three-dimensional structural arrangement of atoms or molecules inside a crystalline solid as points. Each crystal lattice consists of repeating unit cells in different directions. From earlier scientific research, it was recognised that the physio-chemical properties of any crystal structure lattice are highly reliant on the molecular structure of the network. This study led us to investigate complex structures through a mathematical perspective using a chemical invariant (topological index) by a chemical graph theory that correlates a chemical graph structure and physio-chemical properties, where atoms represent the nodes and a bond between atoms is considered as an edge. In this research, we have given explicit expressions for eccentricity-based chemical invariants, namely, eccentric connectivity (EC), total eccentricity (TE), eccentricity-based Zagreb (EZ) and connective eccentric (CE) for inorganic symmetrical Hexagonal Close-Packed and Hexagonal ( crystal structure lattices.

Nonparaxial Tilted Waveobjects

Simple closed-form analytical expressions for tilted astigmatic wave beams and wavepackets that are exact solutions of the wave equation are constructed. They are obtained through two different but equivalent derivations, one is based on a complex shift in the Bateman-type solutions, the other employs their Lorentz transformation. Analytic expressions for the propagation invariants: energy, momentum and orbital angular momentum of tilted waveobjects with general astigmatism are presented.

Lock-exchange problem for Boussinesq fluids revisited: Exact shallow-water solution

An exact solution to the lock-exchange problem, which is a two-layer analogue of the classical dam-break problem, is obtained in the shallow-water (SW) approximation for two immiscible fluids with slightly different densities. The problem is solved by the method of characteristics using analytic expressions for the Riemann invariants. The obtained solution, which represents an inviscid approximation to the high-Reynolds-number limit, is, in general, discontinuous containing up to three hydraulic jumps which are due to either multivaluedness or instability of the continuous SW solution. Hydraulic jumps are resolved by applying the Rankine–Hugoniot conditions for the SW mass and generalized momentum conservation equations. The latter contains a free parameter α which defines the relative contribution of each layer to the interfacial pressure gradient. We consider a solution for α=0, which corresponds to both layers affecting the interfacial pressure gradient with equal weight coefficients. This solution is compared with the solutions resulting from the application of the classical Benjamin's front condition as well as the circulation conservation condition, which correspond to α=−1 and α→∞, respectively. The SW solution reproduces all principal features of 2D numerical solution for viscous fluids. The gravity current speed is found to agree well with experimental and numerical results when the front acquires the largest stable height which occurs at α=5−2. We show that two-layer SW equations for the mass and generalized momentum conservation can describe interfacial waves containing hydraulic jumps in a self-contained way without external closure conditions.

Virtual Segre and Verlinde numbers of projective surfaces

Recently, Marian–Oprea–Pandharipande established (a generalization of) Lehn's conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending the work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of any rank. Using Mochizuki's formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg–Witten invariants and intersection numbers on products of Hilbert schemes of points. We prove that certain canonical virtual Segre and Verlinde numbers of general type surfaces are topological invariants and we verify our conjectures in examples. The power series in our conjectures are algebraic functions, for which we find expressions in several cases and which are permuted under certain Galois actions. Our conjectures imply an algebraic analog of the Mariño–Moore conjecture for higher rank Donaldson invariants. For ranks 3 and 4, we obtain explicit expressions for Donaldson invariants in terms of Seiberg–Witten invariants.

On different expressions for invariants of hyperelliptic curves of genus 3

In this paper we give a passage formula between different invariants of genus 3 hyperelliptic curves: in particular between Tsuyumine and Shioda invariants. This is needed to get modular expressions for Shioda invariants, that is, for example, useful for proving the correctness of numerically computed equations of CM genus 3 hyperelliptic curves. On the other hand, we also get Shioda invariants described in terms of differences of roots of the equation defining the hyperelliptic curve, that has applications for studying the reduction type of the curve. Under certain conditions on its Jacobian, we give a criterion for determining the type of bad reduction of a genus 3 hyperelliptic curve.

The Abelian/Nonabelian Correspondence and Gromov–Witten Invariants of Blow-Ups

Abstract We prove the abelian/nonabelian correspondence with bundles for target spaces that are partial flag bundles, combining and generalising results by Ciocan-Fontanine–Kim–Sabbah, Brown, and Oh. From this, we deduce how genus-zero Gromov–Witten invariants change when a smooth projective variety X is blown up in a complete intersection defined by convex line bundles. In the case where the blow-up is Fano, our result gives closed-form expressions for certain genus-zero invariants of the blow-up in terms of invariants of X. We also give a reformulation of the abelian/nonabelian Correspondence in terms of Givental’s formalism, which may be of independent interest.

Zhang–Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang–Zhang polynomial (also known as ZZ polynomial or Clar covering polynomial) of an important class of elementary peri-condensed benzenoids Rbn1,n2,m1,m2, usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem.2020, 84, 143–176]. The discovered formula provides compact expressions for various topological invariants of Rbn1,n2,m1,m2: the number of Kekulé structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes Ok,m,n and oblate rectangles Orm,n.

From boundary data to bound states. Part II. Scattering angle to dynamical invariants (with twist)

We recently introduced in [9] a boundary-to-bound dictionary between gravitational scattering data and observables for bound states of non-spinning bodies. In this paper, we elaborate further on this holographic map. We start by deriving the following — remarkably simple — formula relating the periastron advance to the scattering angle: Delta Phi left(J,mathcal{E}right)=upchi left(J,mathcal{E}right)+upchi left(-J,mathcal{E}right) , via analytic continuation in angular momentum and binding energy. Using explicit expressions from [9], we confirm its validity to all orders in the Post-Minkowskian (PM) expansion. Furthermore, we reconstruct the radial action for the bound state directly from the knowledge of the scattering angle. The radial action enables us to write compact expressions for dynamical invariants in terms of the deflection angle to all PM orders, which can also be written as a function of the PM-expanded amplitude. As an example, we reproduce our result in [9] for the periastron advance, and compute the radial and azimuthal frequencies and redshift variable to two-loops. Agreement is found in the overlap between PM and Post-Newtonian (PN) schemes. Last but not least, we initiate the study of our dictionary including spin. We demonstrate that the same relation between deflection angle and periastron advance applies for aligned-spin contributions, with J the (canonical) total angular momentum. Explicit checks are performed to display perfect agreement using state-of-the-art PN results in the literature. Using the map between test- and two-body dynamics, we also compute the periastron advance up to quadratic order in spin, to one-loop and to all orders in velocity. We conclude with a discussion on the generalized ‘impetus formula’ for spinning bodies and black holes as ‘elementary particles’. Our findings here and in [9] imply that the deflection angle already encodes vast amount of physical information for bound orbits, encouraging independent derivations using numerical and/or self-force methodologies.

Topological strings, strips and quivers

We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.

Application of universal multipoint invariants in Euclidean and pseudo-Euclidean spaces

Examples of application in a metric space and a space with indefinite metric of the universal multipoint invariant (UMP invariant) known as the Cayley – Menger determinant for four points are considered. Such an invariant, as the results show, determines the metric structure of the Euclidean and pseudo-Euclidean spaces of dimension n=2. For these spaces, expressions for UMP invariants associated with the use of appropriate pseudo distances are also obtained and analyzed. The considered approach can be applied also in spaces with dimension n>2. Cite as: D. O. Dziakovych, “Application of universal multipoint invariants in Euclidean and pseudo-Euclidean spaces,” Prykl. Probl. Mekh. Mat. , Issue 16, 50–57 (2018) (in Ukrainian), https://doi.org/10.15407/apmm2018.16.50-57

Entanglement dynamics of two Ising-coupled qubits with nonperpendicular local driving fields

We present an approximate analytical solution to the dynamic equation of two Ising-coupled qubits with oscillating classical control fields that are nonperpendicular to the static drift fields. This is a situation that has recently arisen in some solid-state experiments. With our solution we derive the analytical expressions for the local invariants as well as the local rotations needed to isolate a purely nonlocal gate. This determines the set of parameters that are required to generate any entangling gate. Moreover, we use our results to describe a recent experimental work on capacitively coupled singlet-triplet qubits in GaAs and discuss possible differences for a similar device in silicon.

Donaldson-Witten theory and indefinite theta functions

We consider partition functions with insertions of surface operators of topologically twisted mathcal{N}=2 , SU(2) supersymmetric Yang-Mills theory, or Donaldson-Witten theory for short, on a four-manifold. If the metric of the compact four-manifold has positive scalar curvature, Moore and Witten have shown that the partition function is completely determined by the integral over the Coulomb branch parameter a, while more generally the Coulomb branch integral captures the wall-crossing behavior of both Donaldson polynomials and Seiberg-Witten invariants. We show that after addition of a overline{mathcal{Q}} -exact surface operator to the Moore-Witten integrand, the integrand can be written as a total derivative to the anti-holomorphic coordinate ā using Zwegers’ indefinite theta functions. In this way, we reproduce Göttsche’s expressions for Donaldson invariants of rational surfaces in terms of indefinite theta functions for any choice of metric.

Superconductivity in quantum wires: A symmetry analysis

We study properties of quantum wires with spin–orbit coupling and time reversal symmetry breaking, in normal and superconducting states. Electronic band structures are classified according to quasi-one-dimensional magnetic point groups, or magnetic classes. The latter belong to one of three distinct types, depending on the way the time reversal operation appears in the group elements. The superconducting gap functions are constructed using antiunitary operations and have different symmetry properties depending on the type of the magnetic point group. We obtain the spectrum of the Andreev boundary modes near the end of the wire in a model-independent way, using the semiclassical approach with the boundary conditions described by a phenomenological scattering matrix. Explicit expressions for the bulk topological invariants controlling the number of the boundary zero modes are presented in the general multiband case for two types of the magnetic point groups with real order parameters, corresponding to DIII and BDI symmetry classes.

On the topological contents of η–invariants

We discuss an universal bordism invariant obtained from the Atiyah-Patodi-Singer eta-invariant from the analytic and homotopy theoretic point of view. Classical invariants like the Adams e-invariant, $\rho$-invariants and $String$-bordism invariants are derived as special cases. The main results are a secondary index theorem about the coincidence of the analytic and topological constructions and intrinsic expressions for the bordism invariants.

Polynomial invariants of degree 4 for even-nqubits and their applications in entanglement classification

We develop a simple method for constructing polynomial invariants of degree 4 for even-$n$ qubits and give explicit expressions for these polynomial invariants. We demonstrate the invariance of the polynomials under stochastic local operations and classical communication and exemplify the use of the invariance in classifying entangled states. The absolute values of these polynomial invariants are entanglement monotones, thereby allowing entanglement measures to be built. Finally, we discuss the properties of these entanglement measures.

TRANSFORMATIONS AND INVARIANTS FOR DIHEDRAL GAUSS HYPERGEOMETRIC FUNCTIONS

Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen's formula and terminating double hypergeometric sums. Besides, pull-back transformations for the dihedral hypergeometric equations are presented, including Klein's pullback transformations for the equations with a finite (dihedral) monodromy group.

Chern-Simons Theory, Matrix Integrals, and Perturbative Three-Manifold Invariants

The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some previous results of Lawrence and Rozansky, the Chern-Simons partition function with arbitrary simply-laced group for these spaces is written in terms of matrix integrals. The analysis of the perturbative expansion amounts to the evaluation of averages in a Gaussian ensemble of random matrices. As a result, explicit expressions for the universal perturbative invariants of Seifert homology spheres up to order five are presented.

Link invariants from classical Chern-Simons theory

Taking as the starting point a perturbative study of the classical equations of motion of the non-Abelian Chern-Simons theory with nondynamical sources, we obtain analytical expressions for link invariants. In order to present these expressions in a manifestly diffeomorphism-invariant form, we introduce a set of differential forms associated with submanifolds in ${R}^{3}$ that are metric independent, and that allow us to consider the link invariants as a kind of surface-dependent diffeomorphism invariant that presents a certain Abelian gauge symmetry.

Towards a third-order topological invariant for magnetic fields

An expression for a third-order link integral of three magnetic fields is presented. It is a topological invariant and therefore an invariant of ideal magnetohydrodynamics. The integral generalizes existing expressions for third-order invariants which are obtained from the Massey triple product, where the three fields are restricted to isolated flux tubes. The derivation and interpretation of the invariant shows a close relationship with the well-known magnetic helicity, which is a second-order topological invariant. Using gauge fields with an SU(2) symmetry, helicity and the new third-order invariant originate from the same identity, an identity which relates the second Chern class and the Chern-Simons three-form. We present an explicit example of three magnetic fields with non-disjunct support. These fields, derived from a vacuum Yang-Mills field with a non-vanishing winding number, possess a third-order linkage detected by our invariant.