Integral Invariants Research Articles

Scattering on the worldvolume: Amplitude relations in Brower-Goddard string models

We investigate the Brower-Goddard extension of the Veneziano and Virasoro-Shapiro four-point amplitudes obtained by generalizing the Koba-Nielsen integrals to d-dimensional conformally invariant integrals. The amplitudes derived from this framework exhibit polynomial residues and can be shown to adhere to polynomial bounds at high energies. In odd dimensions, the amplitudes decompose into sums of three partial amplitudes, enabling the formulation of general amplitude relations that subsume the Kawai-Lewellen-Tye (KLT) formula as a particular case. The amplitudes contain multiple tachyons in their spectra. Still, we demonstrate that their residues comply with the positivity conditions mandated by unitarity for spacetime dimensions at or below critical values Dcrit(d), where Dcrit(6)=26 and Dcrit(∞)=10. In closing, we contemplate physical applications for membranes and potential extensions of the formalism. Published by the American Physical Society 2024

Three-dimensional fracture modes decoupling by means of invariant integrals in wood under mixed mode loading

This paper addresses the determination of the energy release rate in an orthotropic elastic material such as wood with a three-dimensional view of the problem. The mixed mode loading associated with the anisotropy of the material required mode decoupling by isolating the energy release rates in mode I, mode II and in mode III. These energy release rates are calculated using the invariant integral M of Linear Elastic Fracture Mechanics (LEFM). This decoupling has necessitated the introduction of kinetically and statically admissible three-dimensional virtual displacement and stress fields in the vicinity of the crack front. The fracture mode decoupling strategy is presented and implemented through finite element modelling of a Mixed Mode Crack Growth (MMCG) specimen. Different thicknesses are investigated to highlight the 3D effects. The modelling highlighted the effect of Mode II at the edges of free surfaces for mixed loadings including mode III (mode III, mode (II+III) and mode (I+II+III)), while mode III was the predominant at the middle of the specimen. In cases of mixed mode (I+II), the energy release rate GI was highest at the middle of the specimen and lowest at the edges of the free surfaces, regardless of the degree of mixing for a given thickness. The correlation between the results obtained using the proposed approach and the non-dependence of the integration domain, as well as with the usual approaches employed in the literature, is demonstrated.

Characterizations of Spatial Quaternionic Partner-Ruled Surfaces

In this paper, we investigate the spatial quaternionic expressions of partner-ruled surfaces. Moreover, we formulate the striction curves and dralls of these surfaces by use of the quaternionic product. Furthermore, the pitches and angles of pitches are interpreted for the spatial quaternionic ruled surfaces that are closed. Additionally, we calculate the integral invariants of these surfaces using quaternionic formulas. Finally, the partner-ruled surfaces of a given spatial quaternionic ruled surface are demonstrated as an example, and their graphics are drawn.

Stereologically adapted Crofton formulae for tensor valuations

The classical Crofton formula explains how intrinsic volumes of a convex body K in n-dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with invariantly moved affine flats. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.

The interface of gravity and dark energy

At sufficiently large radii dark energy modifies the behavior of (a) bound orbits around a galaxy and (b) virialized gas in a cluster of galaxies. Dark energy also provides a natural cutoff to a cluster’s dark matter halo. In (a) there exists a maximum circular orbit beyond which periodic motion is no longer possible, and orbital evolution near critical binding is analytically calculable using an adiabatic invariant integral. The finding implicates the study of wide galaxy pairs. In (b), dark energy necessitates the use of a generalized Virial Theorem to describe gas at the outskirts of a cluster. When coupled to the baryonic escape condition, aided by dark energy, the results is a radius beyond which the continued establishment of a hydrostatic halo of thermalized baryons is untenable. This leads to a theoretically motivated virial radius. We use this theory to probe the structure of a cluster’s baryonic halo and apply it to X-ray and weak-lensing data collected on cluster Abell 1835. We find that gas in its outskirts deviates significantly from hydrostatic equilibrium beginning at ∼1.3 Mpc , the ‘inner’ virial radius. We also define a model dependent dark matter halo cutoff radius to A1835. The dark matter cutoff gives an upper limit to the cluster’s total mass of ∼7×1015M⊙ . Moreover, it is possible to derive an ‘outer’ hydrostatic equilibrium cutoff radius given a dark matter cutoff radius. A region of cluster gas transport and turbulence occurs between the inner and outer cutoff radii.

О потенциальности, дискретизации и интегральных инвариантах бесконечномерных систем Биркгофа

О потенциальности, дискретизации и интегральных инвариантах бесконечномерных систем Биркгофа

Invariant Integrals on Coideals and Their Drinfeld Doubles

Abstract Let $A$ be a CQG Hopf *-algebra, that is, a Hopf *-algebra with a positive invariant state. Given a unital right coideal *-subalgebra $B$ of $A$, we provide conditions for the existence of a relatively invariant integral on the stabilizer coideal $B^{\perp }$ inside the dual discrete multiplier Hopf *-algebra of $A$. Given such a relatively invariant integral, we show how it can be extended to a relatively invariant integral on the Drinfeld double coideal. We moreover show that the representation category of the Drinfeld double coideal has a monoidal structure. As an application, we determine the relatively invariant integral for the Drinfeld double coideal *-algebra ${\mathcal{U}}_{q}(\mathfrak{s}\mathfrak{l}(2,{\mathbb{R}}))$ constructed from the Podleś spheres.

Bertrand Offsets of Spacelike Ruled Surfaces With Blaschke Approach

Dual parametrizaions of the Bertrand offset- spacelike ruled surfaces are assigned and sundry modern outcomes are acquired in view of their integral invariants. A modern characterization of the Bertrand offsets of spacelike developable surfaces is specified. Further, many connections among the striction curves of Bertrand offsets of spacelike ruled surfaces and their integral invariants are gained.

Chord measures in integral geometry and their Minkowski problems

AbstractTo the families of geometric measures of convex bodies (the area measures of Aleksandrov‐Fenchel‐Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.

Local non-injectivity of the exponential map at critical points in sub-Riemannian geometry

We prove that the sub-Riemannian exponential map is not injective in any neighbourhood of certain critical points. Namely that it does not behave like the injective map of reals given by f(x)=x3 near its critical point x=0. As a consequence, we characterise conjugate points in ideal sub-Riemannian manifolds in terms of the metric structure of the space. The proof uses the Hilbert invariant integral of the associated variational problem.

Bertrand Offsets of Ruled Surfaces with Blaschke Frame in Euclidean 3-Space

Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants are obtained.

New conserved integrals and invariants of radial compressible flow in n > 1 dimensions

Conserved integrals and invariants (advected scalars) are studied for the equations of radial compressible fluid/gas flow in n > 1 dimensions. Apart from entropy, which is a well-known invariant, three additional invariants are found from an explicit determination of invariants up to first order. One holds for a general equation of state, and the two others hold only for entropic equations of state. A recursion operator on invariants is presented, which produces two hierarchies of higher-order invariants. Each invariant yields a corresponding integral invariant, describing an advected conserved integral on transported radial domains. In addition, a direct determination of kinematic conserved densities uncovers two ‘hidden’ non-advected conserved integrals: one describes enthalpy-flux, holding for barotropic equations of state; the other describes entropy-weighted energy, holding for entropic equations of state. A further explicit determination of a class of first-order conserved densities shows that the corresponding non-kinematic conserved integrals on transported radial domains are equivalent to integral invariants, modulo trivial densities.

The Invariants of Dual Parallel Equidistant Ruled Surfaces

In this paper, we calculate the Gaussian curvatures of the dual spherical indicatrix curves formed on unit dual sphere by the Blaschke vectors and dual instantaneous Pfaff vectors of dual parallel equidistant ruled surfaces (DPERS) and we give the relationships between these curvatures. In addition to—in cases where the base curves of these DPERS are closed—computing the dual integral invariants of the indicatrix curves. Additionally, we show the relationships between them. Finally, we provide an example for each of these indicatrix curves.

Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let (F_j)_{j \geq 1} be a sequence of normalized homeomorphic solutions to the planar Beltrami equation \partial_{\overline z} F_j (z)=\mu_j(z,\omega) \partial_{z} F_j(z) , where the random dilatation satisfies |\mu_j|\leq k<1 and has locally periodic statistics, for example of the type \mu_j (z,\omega)=\phi(z) \sum_{n\in\mathbb{Z}^2} g(2^j z-n,X_{n}(\omega)), where g(z,\omega) decays rapidly in z , the random variables X_{n} are i.i.d., and \phi \in C^\infty_0 . We establish the almost sure and local uniform convergence as j \to \infty of the maps F_j to a deterministic quasiconformal limit F_\infty . This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let T_1,\ldots , T_{m} be translation and dilation invariant singular integrals on \mathbb{R}^d , and consider a d -dimensional version of \mu_j , e.g., as defined above or within a more general setting, see Definition 3.4 below. We then prove that there is a deterministic function f such that almost surely, \mu_j T_{m}\mu_j\ldots T_1\mu_j\to f as j\to\infty weakly in L^p , for 1 < p < \infty .

Anomalies in (2+1)D Fermionic Topological Phases and (3+1)D Path Integral State Sums for Fermionic SPTs

Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form $\mathbb{Z}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the $\mathbb{Z}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${\bf T}^2 = (-1)^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $\mathbb{Z}_{16}$ $\mathrm{Pin}^+$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin$^+$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.

Metric Gravity in the Hamiltonian Form—Canonical Transformations—Dirac’s Modifications of the Hamilton Method and Integral Invariants of the Metric Gravity

Two different Hamiltonian formulations of the metric gravity are discussed and applied to describe a free gravitational field in the d dimensional Riemann space-time. Theory of canonical transformations, which relates equivalent Hamiltonian formulations of the metric gravity, is investigated in detail. In particular, we have formulated the conditions of canonicity for transformation between the two sets of dynamical variables used in our Hamiltonian formulations of the metric gravity. Such conditions include the ordinary condition of canonicity known in classical Hamilton mechanics, i.e., the exact coincidence of the Poisson (or Laplace) brackets which are determined for both the new and old dynamical Hamiltonian variables. However, in addition to this, any true canonical transformations defined in the metric gravity, which is a constrained dynamical system, must also guarantee the exact conservation of the total Hamiltonians Ht (in both formulations) and preservation of the algebra of first-class constraints. We show that Dirac’s modifications of the classical Hamilton method contain a number of crucial advantages, which provide an obvious superiority of this method in order to develop various non-contradictory Hamiltonian theories of many physical fields, when a number of gauge conditions are also important. Theory of integral invariants and its applications to the Hamiltonian metric gravity are also discussed. For Hamiltonian dynamical systems with first-class constraints this theory leads to a number of peculiarities some of which have been investigated.

Application of Integral Invariants to Apictorial Jigsaw Puzzle Assembly

We present a method for the automatic assembly of apictorial jigsaw puzzles. This method relies on integral area invariants for shape matching and an optimization process to aggregate shape matches into a final puzzle assembly. Assumptions about individual piece shape or arrangement are not necessary. We illustrate our method by solving example puzzles of various shapes and sizes.

The Characterizations of Parallel q-Equidistant Ruled Surfaces

In this paper, parallel q-equidistant ruled surfaces are defined such that the binormal vectors of given two differentiable curves are parallel along the striction curves of their corresponding binormal ruled surfaces, and the distance between the asymptotic planes is constant at proper points, which is related to symmetry. The characterizations and some other useful relations are drawn for these surfaces as well. If the surfaces are considered to be closed, then the integral invariants such as the pitch, the angle of the pitch, and the drall of them are given. Finally, some examples are presented to indicate that the distance between the proper points on the corresponding asymptotic planes is always constant.

A Kontsevich integral of order 1

We define a 1-cocycle in the space of long knots that is a natural generalization of the Kontsevich integral seen as a 0-cocycle. It involves a 2-form that generalizes the Knizhnik--Zamolodchikov connection. We show that the well-known close relationship between the Kontsevich integral and Vassiliev invariants (via the algebra of chord diagrams and 1T-4T relations) is preserved between our integral and Vassiliev 1-cocycles, via a change of variable similar to the one that led Birman--Lin to discover the 4T relations. We explain how this construction is related to Cirio--Faria Martins' categorification of the Knizhnik--Zamolodchikov connection.

The complexity of higher Chow groups

AbstractLet $X/{\mathbb C}$ be a smooth projective variety. We consider two integral invariants, one of which is the level of the Hodge cohomology algebra $H^*(X,{\mathbb C})$ and the other involving the complexity of the higher Chow groups ${\mathrm {CH}}^*(X,m;{\mathbb Q})$ for $m\geq 0$ . We conjecture that these two invariants are the same and accordingly provide some strong evidence in support of this.