Invariant Formula Research Articles

Особенности построения инвариантных кубатурных формул высокой степени точности для приближенного интегрирования по поверхности тора

Invariant cubature formulas of 11th and 13th degrees of accuracy for a torus in R3 are constructed. An algorithm for constructing such formulas is described.

Methods for constructing invariant cubature formulas for integrals over the surface of a torus in R^3

Methods for constructing invariant cubature formulas for integrals over the surface of a torus in R^3

Gauss diagram formulae for Vassiliev invariants from Kauffman polynomial

A state model for Kauffman polynomial of Dubrovnik-version is described using Gauss diagrams. Based on the state model, the Gauss diagram formulae for Vassiliev invariants are given from the coefficients of Kauffman polynomial following the method of Chmutov and Polyak. Some arrow diagram identities are given to simplify the Gauss diagram formulae of order 3, which give Polyak–Viro and Chmutov–Polyak formulae for the Vassiliev invariant of order 3. The models of Kauffman polynomial and HOMFLY-PT polynomial give different Gauss diagram expressions when specializing in Jones polynomial.

Relative quantum cohomology

We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov–Witten invariants of a Lagrangian submanifold L \subset X with a bounding chain. Simultaneously, we define the quantum cohomology algebra of X relative to L and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov–Witten invariants. Another result is a vanishing theorem for open Gromov–Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov–Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov–Witten invariants of (X,L) = (\mathbb{C} P^n, \mathbb{R} P^n) with n odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya–Oh–Ohta–Ono theory of bounding chains.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields, the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates. The initial growth directions depend on the Jacobians of Poincaré map on that cycle, for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincaré sections. The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems. Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration. A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

The restriction problem on the ellipsoid

Following a restriction argument in the Euclidean space, we derive a geometric invariant formula for a possible viscosity operator for an incompressible fluid flow on an ellipsoid embedded in R3. We also give an asymptotic expansion of the formula in terms of the eccentricity associated with the ellipsoid.

The wave trace and Birkhoff billiards

The purpose of this article is to develop a Hadamard–Riesz-type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in the work by Marvizi and Melrose (1982) and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works by Marvizi and Melrose (1982) and Popov (1994), is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard–Riesz-type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author’s previous work [J. Geom. Anal. 31 (2021), 2238–2295] in the special case of elliptical domains. This allows us to circumvent the symbol calculus used by Duistermaat and Guillemin (1975) and Hezari and Zelditch (2012) when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal.

Explicit formulae for Chern-Simons invariants of the hyperbolic J(2n,-2m) knot orbifolds

We calculate the Chern-Simons invariants of the hyperbolic double twist knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of double twist knots.

Invariant kinematic formulas and inequalities for chord power integrals in integral geometry

In this paper, we first introduce the history, developing process, recent progress, andworks contributed by Chinese researchers in integral and convex geometry.Then we show our recent works in invariant kinematicformulas and chord power integrals. The affine isoperimetric inequalities on the set of convex bodies, closely related to integral geometry, are also addressed.

A Local Trace Formula for p-adic Infinitesimal Symmetric Spaces: the Case of Guo–Jacquet

Abstract We establish an invariant local trace formula for the tangent space of some symmetric spaces over a non-archimedean local field of characteristic zero. These symmetric spaces are studied in Guo–Jacquet trace formulae, and our methods are inspired by works of Waldspurger and Arthur. Some other results are given during the proof including a noninvariant local trace formula, Howe’s finiteness for weighted orbital integrals, and the representability of the Fourier transform of weighted orbital integrals. These local results are preparations for the comparison of regular semi-simple terms, which are weighted orbital integrals, appearing in an infinitesimal variant of Guo–Jacquet trace formulae.

Field theory of higher-order topological crystalline response, generalized global symmetries and elasticity tetrads

I discuss aspects of higher-order topological field theory of crystalline insulators with no other symmetries. I show how the topology and geometry of the crystalline lattice is organized in terms of so-called elasticity tetrads which are ground state degrees of freedom labelling translational lattice topological charges, higher-form conservation laws and responses on sub-dimensional manifolds of the bulk insulator. The quasitopological responses obtained in this way depend on the lattice and its embedding in space, as expected for weak topology. In a topological crystalline insulator, they classify higher-order responses and global symmetries in a transparent fashion in generic dimensions. This hierarchy coincides with the dimensional hierarchy of topological terms, the multipole expansion, and anomaly inflow, related to a mixed number of elasticity tetrads and electromagnetic gauge fields. In the continuum limit of the elasticity tetrads, the semi-classical expansion in momentum space can be used to derive the higher-order or subdimensional topological responses to local U(1) symmetries, such as electromagnetic gauge fields, with explicit formulas for the higher-order quasi-topological invariants in terms of the elasticity tetrads and Green’s functions. The topological responses in arbitrary dimensions are readily generalized to parameter space to allow for e.g. multipole pumping. The simple results further bridge the recently appreciated connections between topological field theory, higher-form symmetries and gauge fields and their relation to fractonic excitations and topological defects with restricted mobility in the elasticity of crystalline insulators.

Forest formulas of discrete Green's functions

AbstractThe discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this article, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests in a graph that will be used to derive further formulas for several graph invariants. For example, we show that the trace of the Green's function associated with the combinatorial Laplacian of a connected simple graph on vertices satisfies where denotes the eigenvalues of the combinatorial Laplacian, denotes the number of spanning trees, and denotes the set of rooted spanning 2‐forests in . We will prove forest formulas for discrete Green's functions for directed and weighted graphs and apply them to study random walks on graphs and digraphs. We derive a forest expression of the hitting time for digraphs, which gives combinatorial proofs to old and new results about hitting times, traces of discrete Green's functions, and other related quantities.

Cohomological DT invariants from localization

Given a quiver with potential associated with a toric Calabi–Yau threefold, the numerical Donaldson–Thomas invariants for the moduli space of framed representations can be computed by using toric localization, which reduces the problem to the enumeration of pyramid partitions. We provide a refinement of this localization procedure, which allows to compute cohomological Donaldson–Thomas invariants. Using this approach, we prove a universal formula which gives the BPS invariants of any toric quiver, up to undetermined contributions which are invariant under Poincaré duality. When the toric Calabi–Yau threefold has compact divisors, these self-Poincaré dual contributions have a complicated dependence on the stability parameters, but explicit computations suggest that they drastically simplify for the self-stability condition (also called attractor chamber). We conjecture a universal formula for the attractor invariants, which applies to any toric Calabi–Yau singularity with compact divisors.

A weighted invariant trace formula

This paper begins a new approach to the r-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic L-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur’s noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.

Znamenny Chant as a Constant of the Artistic World of Yuri Butsko: Chamber Music

The article is devoted to the chamber-instrumental works of one of the brightest representatives of the Russian school of composition of the second half of the twentieth century, Yuri Butsko, in which his religious and musical philosophy is multidimensional and vividly embodied. The focus is made on the discovery of the author's approach to the "reintoning" of the cult chant in the fabric of secular composition and its role in the processes of meaning and text formation of the artistic whole. The works created by the composer mainly in the early period (1960 – 1970s) in chamber genres are considered as a kind of creative laboratory in which the principles of dialogical poetics of religious and secular principles are formed, manifesting itself both at the conceptual level and at the level of textual relations. As a result of the analysis, the author comes to the conclusion that the church monody in the works of Yuri Butsko manifests itself as a dynamic construct, the musical vocabulary and semantic field of which, on the one hand, are amenable to reinterpretation in the process of creating new artistic samples based on it, on the other – have a transformative force that has a significant impact on the content, compositional-dramaturgical, intonation-thematic structure of the work. Both at the conceptual level, realized in the process of interactions of such structures as the symbolic invariant formula of the plot of Golgotha, the external program, the "plot" of the quoted chant, and at the level of textual relations, a multilevel original system of conjugation of Old Russian chant and individual author's poetics is formed in the chamber opuses of Yuri Butsko, which has no analogues in Russian music culture.

Asymptotic behavior of the colored Jones polynomials and Turaev–Viro invariants of the figure eight knot

In this paper we investigate the asymptotic behavior of the colored Jones polynomials and the Turaev-Viro invariants for the figure eight knot. More precisely, we consider the $M$-th colored Jones polynomials evaluated at $(N+1/2)$-th root of unity with a fixed limiting ratio, $s$, of $M$ and $(N+1/2)$. We find out the asymptotic expansion formula (AEF) of the colored Jones polynomials of the figure eight knot with $s$ close to $1$. Nonetheless, we show that the exponential growth rate of the colored Jones polynomials of the figure eight knot with $s$ close to $1/2$ is strictly less than those with $s$ close to $1$. It is known that the Turaev Viro invariant of the figure eight knot can be expressed in terms of a sum of its colored Jones polynomials. Our results show that this sum is asymptotically equal to the sum of the terms with $s$ close to 1. As an application of the asymptotic behavior of the colored Jones polynomials, we obtain the asymptotic expansion formula for the Turaev-Viro invariants of the figure eight knot. Finally, we suggest a possible generalization of our approach so as to relate the AEF for the colored Jones polynomials and the AEF for the Turaev-Viro invariants for general hyperbolic knots.

Goussarov–Polyak–Viro conjecture for degree three case

Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give [Formula: see text] Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not. 1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of [Formula: see text] formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology 39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1).

Cohomological invariants of root stacks and admissible double coverings

AbstractWe give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings.

Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions

We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.

A Klein TQFT: The local Real Gromov-Witten theory of curves

In this paper we study the Real Gromov-Witten theory of local 3-folds over Real curves. We show that this gives rise to a 2-dimensional Klein TQFT defined on an extension of the category of unorientable surfaces. We use this structure to completely solve the theory by providing a closed formula for the local RGW invariants in terms of representation theoretic data, extending earlier results of Bryan and Pandharipande. As a consequence we obtain the local version of the real Gopakumar-Vafa formula that expresses the connected real Gromov-Witten invariants in terms of integer invariants. In the case of the resolved conifold the partition function of the RGW invariants agrees with that of the SO/Sp Chern-Simons theory.