Invariant Formula Research Articles

Invariant Laplacian-involving cubature formulas

For integrals in n dimensions over a hypercube, two ninth-degree cubature formulas that are invariant with respect to a hyperoctahedral group are constructed. The cubature sums of these formulas contain the Laplacian. Examples are given in which approximate values of the formulas’ parameters are presented in the form of tables.

Testing Log K-stability by blowing up formalism

We study logarithmic K-stability for pairs by extending the formula for Donaldson-Futaki invariants to log setting. We also provide algebro-geometric counterparts of recent results of existence of Kahler-Einstein metrics with cone singularities.

Reduced matrix elements of the orthosymplectic Lie superalgebra

We utilize characteristic identities to construct eigenvalue formulae for invariants and reduced matrix elements corresponding to irreducible representations of . In presenting these results, we further develop our programme of constructive representation theory via characteristic identities.

On some computations of higher rank refined Donaldson-Thomas invariants

We present some computations of higher rank refined Donaldson-Thomas invariants on local curve geometries, corresponding to local D6-D2-D0 or D4-D2-D0 configurations. A refined wall-crossing formula for invariants with higher D6 or D4 ranks is derived and verified to agree with the existing formulas under the unrefined limit. Using the formula, refined invariants on the $(-1,-1)$ and $(-2,0)$ local rational curve with higher D6 or D4 ranks are computed.

Classes of Sequentially Cohen-Macaulay Squarefree Monomial Ideals

We compute the minimal primary decomposition for completely squarefree lexsegment ideals. We show that critical squarefree monomial ideals are sequentially Cohen-Macaulay. As an application, we give a complete characterization of the completely squarefree lexsegment ideals which are sequentially Cohen-Macaulay and we also derive formulas for some homological invariants of this class of ideals.

Resistance distance-based graph invariants of subdivisions and triangulations of graphs

We study three resistance distance-based graph invariants: the Kirchhoff index, and two modifications, namely, the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index. Recently, one of the present authors (2014) and Sun et al. (2014) independently obtained (different) formulas for the Kirchhoff index of subdivisions of graphs. Huang et al. (2014) treated the Kirchhoff index of triangulations of graphs. In our paper, first we derive formulae for the additive degree-Kirchhoff index and the multiplicative degree-Kirchhoff index of subdivisions and triangulations, as well as a new formula for the Kirchhoff index of triangulations, in terms of invariants of G. Then comparisons are made between each of our Kirchhoffian graph invariants for subdivision and triangulation. Finally, formulae for these graph invariants of iterated subdivisions and triangulations of graphs are obtained.

On Mardia skewness and kurtosis of Soules basis matrices in ROM simulation

Relying on an interesting connection between generalized Helmert–Ledermann matrices and Soules basis matrices, as well as a complete binary tree classification of the latter, a recursive determination of the Mardia skewness and kurtosis coefficients of Soules basis matrices is proposed. It is based on a convenient combinatorial representation of Soules basis matrices by restricted binary partitions of a natural number into two ordered summands with recursive repetition of this binary partitioning on the summands. The obtained simple composition and invariance formulas enable the recursive calculation of these skewness and kurtosis coefficients.

Bondi Mass Cannot Become Negative in Higher Dimensions

We prove that the Bondi mass of an asymptotically flat, vacuum spacetime cannot become negative in any even dimension \({d \geq 4}\) . The notion of Bondi mass is more subtle in d > 4 dimensions because radiating metrics have a slower decay than stationary ones, and those subtleties are reflected by a considerably more difficult proof of positivity. Our proof holds for the standard spherical infinities, but it also extends to infinities of more general type which are (d − 2)-dimensional manifolds admitting a real Killing spinor. Such manifolds typically have special holonomy and Sasakian structures. The main technical advance of the paper is an expansion technique based on “conformal Gaussian null coordinates”. This expansion helps us to understand the consequences imposed by Einstein’s equations on the asymptotic tail of the metric as well as auxiliary spinorial fields. As a by-product, we derive a coordinate expression for the geometrically invariant formula for the Bondi mass originally given by Hollands and Ishibashi.

The Futaki Invariant on the Blowup of Kahler Surfaces

We prove the expansion formula for the classical Futaki invariants on the blowup of Kähler surfaces using the localization method, which explains the balancing condition of Arezzo–Pacard in [2].

Maximal Synthesis for Hennessy-Milner Logic with the Box Modality

This paper presents a novel approach to adapt a behavioral model in order to satisfy a requirement in Hennessy-Milner Logic, including an additional box modality operator, expressing an invariant formula. Control system synthesis, as defined in this way, retains all non-invalidating behavior, and thereby guarantees maximal permissiveness for supervisory control. This research extends earlier work by embracing a broader synthesized logic, enabling synthesis with respect to invariant formulas for non-deterministic behavioral models. All definitions and proofs in this paper have been computer verified using the Coq proof assistant.

La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin

We study the genuine part of the Arthur-Selberg trace formula for some nonlinear covers of connected reductive groups. As a first step towards the invariant trace formula, we express the geometric side in terms of weighted orbital integrals. In particular, our results apply to the covers constructed by Brylinksi and Deligne.

Computing distance moments on graphs with transitive Djoković–Winkler relation

The cut method proved to be utmost useful to obtain fast algorithms and closed formulas for classical distance-based invariants of graphs isometrically embeddable into hypercubes. These graphs are known as partial cubes and in particular contain numerous chemically important graphs such as trees, benzenoid graphs, and phenylenes. It is proved that the cut method can be used to compute an arbitrary distance moment of all the graphs that are isometrically embeddable into Cartesian products of triangles, a class much larger than partial cubes. The method in particular covers the Wiener index, the hyper-Wiener index, and the Tratch–Stankevich–Zefirov index.

Tensor products of tautological bundles under the Bridgeland–King–Reid–Haiman equivalence

We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland–King–Reid–Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.

Double-spin asymmetryALTin open charm production

In the collinear twist-three approach, we calculate for the first time the gluon contribution to double- (longitudinal-transverse) spin asymmetry ${A}_{LT}$ for open charm production in proton-proton collisions measurable at RHIC. Utilizing the Ward-Takahashi identity for the nonpole part of the hard scattering amplitude, we derive a factorized, gauge invariant formula for the asymmetry. The result may be combined with the previous studies of single-spin asymmetry in the same channel and allows for a systematic analysis of three-gluon correlations inside a transversely polarized nucleon.

On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

Abstract In previous work we have shown that the equivariant index of multi-centered $ \mathcal{N}=2 $ black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription — the Coulomb branch formula — for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke’s result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincaré polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package “CoulombHiggs.m” implementing the Coulomb and Higgs branch formulae.

Object-Image Correspondence for Algebraic Curves under Projections

We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves. We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane.

Note on Cubature Formulae and Designs Obtained from Group Orbits

Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

La formule des traces pour les revêtements de groupes réductifs connexes III: Le développement spectral fin

We pursue Arthur's invariant trace formula for certain coverings of connected reductive groups by deducing explicit formulas for its spectral side. This is based on some results in local harmonic analysis from an earlier preprint. The arguments are due to Arthur, and we will explain the necessary fine-tunings for coverings.

PARITY AND EXOTIC COMBINATORIAL FORMULAE FOR FINITE-TYPE INVARIANTS OF VIRTUAL KNOTS

The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's recently discovered notion of parity. For a parity of flat virtual knots, the new combinatorial formulae are Kauffman finite-type invariants. However, many of the combinatorial formulae possess exotic properties. It is shown that there exists an integer-valued virtualization invariant combinatorial formula of order n for every n (i.e. it is stable under the map which changes the direction of one arrow but preserves the sign). Hence, it is not of Goussarov–Polyak–Viro finite-type. Moreover, every homogeneous Polyak–Viro combinatorial formula admits a decomposition into an "even" part and an "odd" part. For the Gaussian parity, neither part of the formula is of GPV finite-type when it is non-constant on the set of classical knots. In addition, eleven new non-trivial combinatorial formulae of order 2 are presented which are not of GPV finite-type.

Stable pairs on local $K3$ surfaces

We prove a formula which relates Euler characteristic of moduli spaces of stable pairs on local $K3$ surfaces to counting invariants of semistable sheaves on them. Our formula generalizes Kawai- Yoshioka’s formula for stable pairs with irreducible curve classes to arbitrary curve classes. We also propose a conjectural multiple cover formula of sheaf counting invariants which, combined with our main result, leads to an Euler characteristic version of Katz- Klemm-Vafa conjecture for stable pairs.