Invariant Space Research Articles

Reconstruction from convolution random sampling in local shift invariant spaces

In this paper, we consider the problem of reconstructing functions in local multiply generated shift invariant spaces from convolution random samples. The sampling set is randomly chosen with one kind of probability distribution over a bounded cube and the available data are the sampled convolution of the original function on the sampling set. We obtain an explicit reconstruction formula. This reconstruction formula succeeds with overwhelming probability when the sampling size is sufficiently large.

On closedness of law-invariant convex sets in rearrangement invariant spaces

This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space $\mathcal{X}$. In particular, we show that order closedness, $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-closedness and $\sigma(\mathcal{X},L^\infty)$-closedness of a law-invariant convex set in $\mathcal{X}$ are equivalent, where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. We also provide some application to proper quasiconvex law-invariant functionals with the Fatou property.

Spacing dependent interaction of vortex dipole and induced off-axis propagations of optical energy

Discussed analytically is the dynamics of a vortex dipole embedded in an elliptic Gaussian beam. Two opposite charged vortices of the vortex dipole can repel or attract each other, depending on whether their initial spacing larger or smaller than the critical value. While, stable vortex dipole with invariant spacing is predicated to exist at critical initial spacing. The critical spacing is dramatically related with the host elliptic beam. Furthermore, the dynamics of the vortex dipole can result in an off-axis propagation of optical beams, with propagating directions normal to the orientations of the vortex dipole. The optical beam exhibits larger off-axis velocity when the vortex dipole is parallel to the major axis of the host elliptic beam. The off-axis velocity can be further enhanced by increasing the ellipticity of host beams. The theoretical results can provide a controllable approach to change the propagating directions of optical beams by adjusting the initial spacing of vortex dipoles.

Hyperplane Arrangements and Tensor Product Invariants

In the first part of this paper, we consider, in the context of an arbitrary weighted hyperplane arrangement, the map from compactly supported cohomology to the usual cohomology of a local system. We obtain a formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map. In the second part, we apply these results to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels) the space of invariants acquires a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants and characterize the subspace of conformal blocks in Hodge theoretic terms.

Real Gromov–Witten theory in all genera and real enumerative geometry: computation

Gromov–Witten invariants of real-orientable symplectic manifolds of odd “complex” dimensions; the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants. The present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former. After describing large collections of real-orientable symplectic manifolds, we show that the genus $1$ real Gromov–Witten invariants of sufficiently positive almost Kahler threefolds are signed counts of real genus 1 curves only and, thus, provide direct lower bounds for the counts of these curves in such targets. We specify real orientations on the real-orientable complete intersections in projective spaces; the real Gromov–Witten invariants they determine are in a sense canonically determined by the complete intersection itself, (at least) in most cases. We also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections. Our results confirm Walcher’s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases. The localization data is also used to demonstrate the non-triviality of our lower bounds for real curves of genus $1$ in the present paper and of higher genera in a separate paper.

Invariant Valuations on Super-Coercive Convex Functions

AbstractAll non-negative, continuous,$\text{SL}(n)$, and translation invariant valuations on the space of super-coercive, convex functions on$\mathbb{R}^{n}$are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous,$\text{SL}(n)$, and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume, and polar volume are characterized.

The Levi-Civita equation in function classes

Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form sum _{i=1}^np_i cdot m_i, where m_1 ,ldots ,m_n are exponentials belonging to V, and p_1 ,ldots ,p_n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra {{mathcal {A}}} of complex valued functions such that whenever an exponential m belongs to {{mathcal {A}}}, then m^{-1}in {{mathcal {A}}}. As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary sigma -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.

Equivalence principle on cosmological backgrounds in scalar–tensor theories

We study the extent up to which the equivalence principle is obeyed in models of modified gravity and dark energy involving a single scalar degree of freedom. We focus on the effective field theories of dark energy describing the late time acceleration in the presence of ordinary matter species. In their covariant form these coincide with the Horndeski theories on a cosmological background with a slowly varying Hubble rate and a time-dependent scalar field. We show that in the case of an exactly de Sitter universe both the weak and strong equivalence principles hold. This occurs due to the combination of the shift symmetry of the scalar and the time translational invariance of de Sitter space. When generalized to Friedmann–Robertson–Walker cosmologies (FRW) we show that the weak equivalence principle is obeyed for test particles and extended objects in the quasi-static subhorizon limit. We do this by studying the field created by an extended object moving in an FRW background as well as its equation of motion in an external gravitational field. We also estimate the corrections to the geodesic equation of the extended object due to the approximations made.

Standard model plethystics

We study the vacuum geometry prescribed by the gauge invariant operators of the minimal supersymmetric standard model via the plethystic program. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the minimal supersymmetric standard model gauge invariants. All data in Mathematica format are also presented.

ENVIRONMENTAL SUSTAINABLE VALUE IN AGRICULTURE REVISITED: HOW INVESTMENT SUBSIDIES FOSTER ECO-EFFICIENCY

Many researchers and policy makers argue that CAP should support small farms mainly through environmental subsidies contributing by this mean to sustainable agriculture. This study offers a methodological contribution to the value-based sustainability approach, consisting of a computing indicator of environmental sustainable value (ESV). In this study, the authors have attempted to combine the value-oriented approach with DEA frontier benchmarking. In the next step, the authors test how investment subsidies contribute to ESV using a long-term panel of FADN region-representative farms in 2004-2015 with regards to other policy measures and factor endowments. The seminal within-between specification was employed to the control time variant and time in-variant space heterogeneity of European regions. The articles main finding is that higher investment support is beneficial for ESV. Other payments exert a negative effect on ESV besides the cross-sectional impact of environmental subsidies. When it comes to factor endowment influence, there is a positive impact of the capital-labor ratio and negative impact of the capital-land ratio.

Some properties of embeddings of rearrangement invariant spaces

Let and be rearrangement invariant spaces on , and let . This embedding is said to be strict if the functions in the unit ball of the space have absolutely equicontinuous norms in . For the main classes of rearrangement invariant spaces necessary and sufficient conditions are obtained for an embedding to be strict, and also the relationships this concept has with other properties of embeddings are studied, especially the property of disjoint strict singularity. In the final part of the paper, a characterization of the property of strict embedding in terms of interpolation spaces is obtained. Bibliography: 23 titles.

Basepoint Free Cycles on 0n from Gromov–Witten Theory

Abstract Basepoint free cycles on the moduli space $\overline{\operatorname{M}}_{0,n}$ of stable $n$-pointed rational curves, defined using Gromov–Witten invariants of smooth projective homogeneous spaces are introduced and studied. Intersection formulas to find classes are given. Gromov–Witten divisors for projective space are shown equivalent to conformal blocks divisors for type A at level 1.

Approaching the environmental sustainable value in agriculture: How factor endowments foster the eco-efficiency

Abstract Small and labor-intensive farms guarantee a higher degree of environmental sustainability compared to rich, capital-intensive farms. This paper offers a methodological contribution to the value-based sustainability approach, consisting in computing a modified indicator of environmental sustainable value. Using this approach, the authors test how factor endowments such as capital, land and labor contribute to environmental sustainable value using a long-term panel of region-representative European Union farm accountancy data network farms in 2004–2015. The seminal within-between specification was employed to control time variant and time in-variant space heterogeneity of European regions. The articles main finding is that higher capital endowment and lower labor-intensity is beneficial for environmental sustainable value. The result that farming with higher labor endowment might be less eco-efficient contradicts the commonly known perception of this problem. In light of our findings, we conclude that the pro-investment direction of the common agricultural policy should be followed but one-size-fit-all policy is not working.

Scalar MSCR Codes via the Product Matrix Construction

An $(n,k,d)$ cooperative regenerating code provides the optimal-bandwidth repair for any $t~(t\!>\!1)$ node failures in a cooperative way. In particular, an MSCR (minimum storage cooperative regenerating) code retains the same storage overhead as an $(n,k)$ MDS code. Suppose each node stores $\alpha $ symbols which indicates the sub-packetization level of the code. A scalar MSCR code attains the minimum sub-packetization, i.e., $\alpha =d-k+t$ . By now, all existing constructions of scalar MSCR codes restrict to very special parameters, eg. $d=k$ or $k=2$ , etc. In a recent work, Ye and Barg construct MSCR codes for all $n,k,d,t$ , however, their construction needs $\alpha \approx \text {exp}(n^{t})$ which is almost infeasible in practice. In this paper, we give an explicit construction of scalar MSCR codes for all $d\geq \max \{2k-1-t,k\}$ , which covers all possible parameters except the case of $k\leq d\leq 2k-2-t$ when $k . Moreover, as a complementary result, for $k we prove the nonexistence of linear scalar MSCR codes that have invariant repair spaces. Our construction and most of the previous scalar MSCR codes all have invariant repair spaces and this property is appealing in practice because of convenient repair. In this sense, this work presents an almost full description of usual scalar MSCR codes.

Learning Human Pose Models from Synthesized Data for Robust RGB-D Action Recognition

We propose Human Pose Models that represent RGB and depth images of human poses independent of clothing textures, backgrounds, lighting conditions, body shapes and camera viewpoints. Learning such universal models requires training images where all factors are varied for every human pose. Capturing such data is prohibitively expensive. Therefore, we develop a framework for synthesizing the training data. First, we learn representative human poses from a large corpus of real motion captured human skeleton data. Next, we fit synthetic 3D humans with different body shapes to each pose and render each from 180 camera viewpoints while randomly varying the clothing textures, background and lighting. Generative Adversarial Networks are employed to minimize the gap between synthetic and real image distributions. CNN models are then learned that transfer human poses to a shared high-level invariant space. The learned CNN models are then used as invariant feature extractors from real RGB and depth frames of human action videos and the temporal variations are modelled by Fourier Temporal Pyramid. Finally, linear SVM is used for classification. Experiments on three benchmark human action datasets show that our algorithm outperforms existing methods by significant margins for RGB only and RGB-D action recognition.

A three-invariant cap-viscoplastic rate-dependent-damage model for cementitious materials with return mapping integration in Haigh-Westergaard coordinate space

We present a new strain-rate dependent elasto-viscoplastic damage constitutive model for cementitious materials by incorporating Duvaut-Lions viscoplasticity generalized to multi-surface plasticity (c.f. Simo et al., 1988b; Simo and Hughes, 1998) followed by strain-rate-dependent damage initiation and evolution under multiaxial loading, to our previous inviscid elastoplastic-damage model (Paliwal et al., 2017). We also present a unique numerical scheme in which the integration algorithm for modeling elastoplastic response utilizes fully implicit, backward Euler method in which the governing system of non-linear equations are solved in the space of invariants based on Haigh-Westergaard coordinate system, hardening parameter and discretized plastic multiplier. It uses the framework by (Matzenmiller and Taylor, 1994), which is modified in this work. This is particularly useful as the yield criteria, flow rule, and the non-linear hardening utilize functions that are dependent on the pressure, effective shear stress, and Lode angle components of the stress tensor. To the best of authors’ knowledge, such integration algorithm in Haigh-Westergaard coordinate space has never been developed for Lode angle dependent yield criteria and the flow rule, and non-linear hardening plasticity. Furthermore, we also develop an accurate form for the algorithmic elastoplastic-damage tangent modulus consistent with the proposed return mapping technique. Numerical simulations of uniaxial tension and compression cases demonstrate that both the local integration point algorithm as well as the global algorithm using the algorithmic elastoplastic-damage tangent modulus exhibit quadratic convergence. Finally, the predictive capability of the proposed model is demonstrated by virtue of several numerical examples under various loading conditions covering different stress paths and different dynamic strain-rates. These predictions are also compared against experimental results and experimentally observed features from tests on various concrete specimens, and demonstrate excellent agreement.

Energy bounds and vanishing results for the Gromov–Witten invariants of the projective space

We describe generating functions for arbitrary-genus Gromov–Witten invariants of the projective space with any number of marked points explicitly. The structural portion of this description gives rise to uniform energy bounds and vanishing results for these invariants. They suggest deep conjectures relating Gromov–Witten invariants of symplectic manifolds to the energy of pseudo-holomorphic maps and the expected dimension of their moduli space.

Invariant theory and orientational phase transitions.

The Landau theory of phase transitions has been productively applied to phase transitions that involve rotational symmetry breaking, such as the transition from an isotropic fluid to a nematic liquid crystal. It even can be applied to the orientational symmetry breaking of simple atomic or molecular clusters that are not true phase transitions. In this paper, we address fundamental problems that arise with the Landau theory when it is applied to rotational symmetry breaking transitions of more complex particle clusters that involve order parameters characterized by larger values of the l index of the dominant spherical harmonic that describes the broken symmetry state. The problems are twofold. First, one may encounter a thermodynamic instability of the expected ground state with respect to states with lower symmetry. A second problem concerns the proliferation of quartic invariants that may or may not be physical. We show that the combination of a geometrical method based on the analysis of the space of invariants, developed by Kim to study symmetry breaking of the Higgs potential, with modern visualization tools provides a resolution to these problems. The approach is applied to the outcome of numerical simulations of particle ordering on a spherical surface and to the ordering of protein shells.

Lacunary Walsh series in rearrangement invariant spaces

We prove that the classical results by Rodin and Semenov and by Lindenstrauss and Tzafriri on the subspace generated by the Rademacher system in rearrangement invariant spaces also hold for lacunary Walsh series.

Well-Posedness and Ill-Posedness Problems of the Stationary Navier–Stokes Equations in Scaling Invariant Besov Spaces

We consider the stationary Navier–Stokes equations in $$\mathbb {R}^n$$ for $$n\geqq 3$$ in the scaling invariant Besov spaces. It is proved that if $$n<p\leqq \infty $$ and $$1\leqq q\leqq \infty $$ , or $$p=n$$ and $$2<q\leqq \infty $$ , then some sequence of external forces converging to zero in $$\dot{B}^{-3+\frac{n}{p}}_{p,q}$$ can admit a sequence of solutions which never converges to zero in $$\dot{B}^{-1}_{\infty ,\infty }$$ , especially in $$\dot{B}^{-1+\frac{n}{p}}_{p,q}$$ . Our result may be regarded as showing the borderline case between ill-posedness and well-posedness, the latter of which Kaneko–Kozono–Shimizu proved when $$1\leqq p<n$$ and $$1\leqq q\leqq \infty $$ .