Invariant Problem Research Articles

Invariant cones for jump-diffusions in infinite dimensions

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. We emphasize that the diffusion coefficient of the SPDE does not need to be smooth, an assumption which is frequently imposed when dealing with stochastic invariance problems. Our results apply to the positive cone of abstract L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.

A note on a Halmos problem

AbstractWe address the existence of non‐trivial closed invariant subspaces of operators on Banach spaces whenever their square have or, more generally, whether there exists a polynomial with such that the lattice of invariant subspaces of is non‐trivial. In the Hilbert space setting, the ‐problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the Invariant Subspace Problem.

The Invariant Subspace Problem for Separable Hilbert Spaces

In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem.

Morrey regularity theory of Rivière’s equation

This note is devoted to developing Morrey regularity theory for the following system of Rivière − Δ u = Ω ⋅ ∇ u + f in B 2 , \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f f belongs to some Morrey space. Our results extend the L p L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24].

New L-fuzzy fixed point techniques for studying integral inclusions

The survey of the available literature shows that a lot of important invariant point problems of Banach and Heilpern types have been examined in both metric and quasimetric spaces. However, a handful of the existing results employed the recent approaches of interpolative contractions. Therefore, based on the new idea of interpolation techniques in fixed point theory, this article studies new notions of L-fuzzy contractions and investigates conditions for the existence of L-fuzzy fixed points for such mappings. On the fact that fixed points of point-to-point mappings satisfying interpolative-type contraction are not always unique, whence making the concepts more fitted for invariant point results of crisp set-valued maps, new multi-valued analogues of the key findings put forward in this work are derived. Comparative illustrations, which indicate the preeminence of the results presented herein, are constructed. From application viewpoint, one of the theorems so obtained is employed to introduce new solvability conditions of Fredholm-type integral inclusions.

Partial-implementation invariance and claims problems

Partial-implementation invariance and claims problems

Generalized Higher Derivations and a Center Invariance Problem

Generalized Higher Derivations and a Center Invariance Problem

Two-Level Graph Neural Network.

Graph neural networks (GNNs) are recently proposed neural network structures for the processing of graph-structured data. Due to their employed neighbor aggregation strategy, existing GNNs focus on capturing node-level information and neglect high-level information. Existing GNNs, therefore, suffer from representational limitations caused by the local permutation invariance (LPI) problem. To overcome these limitations and enrich the features captured by GNNs, we propose a novel GNN framework, referred to as the two-level GNN (TL-GNN). This merges subgraph-level information with node-level information. Moreover, we provide a mathematical analysis of the LPI problem, which demonstrates that subgraph-level information is beneficial to overcoming the problems associated with LPI. A subgraph counting method based on the dynamic programming algorithm is also proposed, and this has the time complexity of O(n3) , where n is the number of nodes of a graph. Experiments show that TL-GNN outperforms existing GNNs and achieves state-of-the-art performance.

On the invariant factors and the minimal polynomial assignments of linear systems

In this note, the invariant factors assignment and the minimal polynomial assignment problems for multiple-input linear systems are studied. It is found that for any invariant factors with the geometry multiplicity of each eigenvalue less than or equal to the number of the inputs, the invariant factors assignment problems are solvable if and only if the controllability indices set of the linear system satisfies certain condition. For the minimal polynomial assignment problem, a necessary and sufficient condition on the solvability is proposed. Three numerical examples are provided to illustrate the proposed results.

Generating new gravitational solutions by matrix multiplication

Explicit solutions to the nonlinear field equations of some gravitational theories can be obtained, by means of a Riemann–Hilbert approach, from a canonical Wiener–Hopf factorization of certain matrix functions called monodromy matrices. In this paper, we describe other types of factorization from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann–Hilbert problem and allows to construct solutions that could not have been obtained by Wiener–Hopf factorization of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener–Hopf factorization, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein–Rosen wave and gravitational pulse wave solutions.

A Metric Fixed Point Theorem and Some of Its Applications

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

Tropical invariants for binary quintics and reduction types of Picard curves

Abstract In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$ .

Large Order-Invariant Bayesian VARs with Stochastic Volatility

Many popular specifications for Vector Autoregressions (VARs) with multivariate stochastic volatility are not invariant to the way the variables are ordered due to the use of a lower triangular parameterization of the error covariance matrix. We show that the order invariance problem in existing approaches is likely to become more serious in large VARs. We propose the use of a specification which avoids the use of this lower triangular parameterization. We show that the presence of multivariate stochastic volatility allows for identification of the proposed model and prove that it is invariant to ordering. We develop a Markov chain Monte Carlo algorithm which allows for Bayesian estimation and prediction. In exercises involving artificial and real macroeconomic data, we demonstrate that the choice of variable ordering can have non-negligible effects on empirical results when using the nonorder invariant approach. In a macroeconomic forecasting exercise involving VARs with 20 variables we find that our order-invariant approach leads to the best forecasts and that some choices of variable ordering can lead to poor forecasts using a conventional, non-order invariant, approach.

SE‐Swin: An improved Swin‐Transfomer network of self‐ensemble feature extraction framework for image retrieval

AbstractThe Swin‐Transformer is a variant of the Vision Transformer, which constructs a hierarchical Transformer that computes representations with shifted windows and window multi‐head self‐attention. This method can handle the scale invariance problem and performs well in many computer vision tasks. In image retrieval, high‐quality feature descriptors are necessary to improve retrieval accuracy. This paper proposes a self‐ensemble Swin‐Transformer network structure to fuse the features of different layers of the Swin‐Transformer network, eliminating noise points present in a single layer, and improving the retrieval effect. Two experiments were conducted, one on the In‐shop Clothes Retrieval dataset and another on the Stanford Online Product dataset. The experiments showed that the proposed method significantly increased the retrieval effect of features extracted using Vision Transformer, surpassing previous state‐of‐the‐art image retrieval methods. In the second experiment, the feature map of the trained model was visualized, revealing that the improved network significantly reduces focus on some noise points and enhances focus on image features compared to the original network.

Green supplier selection problems with data scaling and production frontier estimations in a DEA model

AbstractConsidering ecological issues in supplier evaluation and management alongside business considerations is getting more recognition among firms. Data envelopment analysis (DEA) is one of those methods, which is frequently suggested by the literature to support management decisions. However, the data requirements of the method should be an important consideration. The literature often addresses the issue of desirable outputs and undesirable input as an important data related problem in case of the ecological use of DEA. This paper will present a new solution to manage these data problems along with connecting the evaluation of management criteria, environmental criteria and total cost aspects. The proposed environmental supplier selection problem is an extension of a former paper. The new model examines the effect of inventory related costs, such as EOQ costs of inventory holding or ordering costs on the selected supplier, extended with newly introduced scaled values of input and output indicators. The usage of scaled values is motivated by the problem of invariance to data alteration. In addition to the uncertainty of the data, the paper looks for a functional relationship between the input and output criterion values and the efficiency that can be assigned to them using DEA.

A weighted composition semigroup related to three open problems

The semi-group of weighted composition operators (Wn)n≥1 whereWnf(z)=(1+z+…+zn−1)f(zn) on the classical Hardy-Hilbert space H2 of the open unit disk is related to the Riemann Hypothesis (RH) (see [16]). The semigroup (Wn)n≥1 is also closely related to the Invariant Subspace Problem (ISP) and the Periodic Dilation Completeness Problem (PDCP). We obtain results on cyclic vectors, spectra, invariant and reducing subspaces. In particular, we show that several basic questions related to the semigroup (Wn)n≥1 are equivalent to the RH and provide generalizations of the Báez-Duarte criterion for the RH (see [5]).

Linking lexically guided perceptual learning to statistical patterns in speech input

Listeners use lexical information to modify the mapping between speech acoustics and speech sound categories. Despite convention to consider lexically guided perceptual learning as a binary outcome, the magnitude of the learning effect varies in the extant literature. We hypothesize that graded learning outcomes can be linked, in part, to statistical characteristics of the to-be-learned input, consistent with the ideal adapter theory of speech adaptation. Following standard methods (i.e., waveform averaging to create ambiguous variants), a lexically guided perceptual learning stimulus set for the /ʃ/-/s/ contrast was created for each of 16 talkers, yielding variability in the statistical cues specifying this contrast across talkers. Experiment 1 will (a) measure lexically guided perceptual learning for each talker, (b) identify input characteristics that are associated with learning magnitude, and (c) examine whether a computational instantiation of the ideal adapter theory can model the input-learning link. Experiment 2 will provide a confirmatory test of the patterns observed in Experiment 1 by manipulating the to-be-learned input holding talker constant. The results will provide a critical test of the ideal adapter framework for speech adaptation, thus informing an understanding of the mechanisms that allow listeners to solve the lack of invariance problem for speech perception.

Geometric Hydrodynamics in Open Problems

Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and singular vorticity solutions, differential and Hamiltonian geometry of diffeomorphism groups, long-time behaviour of fluids, as well as mechanical models of direct and inverse cascades.

Hearing is believing: Lexically guided perceptual learning is graded to reflect the quantity of evidence in speech input

There is wide variability in the acoustic patterns that are produced for a given linguistic message, including variability that is conditioned on who is speaking. Listeners solve this lack of invariance problem, at least in part, by dynamically modifying the mapping to speech sounds in response to structured variation in the input. Here we test a primary tenet of the ideal adapter framework of speech adaptation, which posits that perceptual learning reflects the incremental updating of cue-sound mappings to incorporate observed evidence with prior beliefs. Our investigation draws on the influential lexically guided perceptual learning paradigm. During an exposure phase, listeners heard a talker who produced fricative energy ambiguous between /ʃ/ and /s/. Lexical context differentially biased interpretation of the ambiguity as either /s/ or /ʃ/, and, across two behavioral experiments (n = 500), we manipulated the quantity of evidence and the consistency of evidence that was provided during exposure. Following exposure, listeners categorized tokens from an ashi – asi continuum to assess learning. The ideal adapter framework was formalized through computational simulations, which predicted that learning would be graded to reflect the quantity, but not the consistency, of the exposure input. These predictions were upheld in human listeners; the magnitude of the learning effect monotonically increased given exposure to four, 10, or 20 critical productions, and there was no evidence that learning differed given consistent versus inconsistent exposure. These results (1) provide support for a primary tenet of the ideal adapter framework, (2) establish quantity of evidence as a key determinant of adaptation in human listeners, and (3) provide critical evidence that lexically guided perceptual learning is not a binary outcome. In doing so, the current work provides foundational knowledge to support theoretical advances that consider perceptual learning as a graded outcome that is tightly linked to input statistics in the speech stream.

One-dimensional VGGNet for high-dimensional data

We consider a deep learning model for classifying high-dimensional data and seek to achieve optimal evaluation accuracy and robustness based on multicriteria decision-making (MCDM) for high-dimensional data analysis applications during comprehensive evaluation (CE) activities. We propose a novel one-dimensional visual geometry group network (1D_VGGNet) to overcome the problem that high-dimensional data are too complicated and unstable to be feasibly applied. Then, to effectively handle one-dimensional MCDM, we present a 1D_VGGNet classifier to replace the two-dimensional convolution operation applied to image data with a one-dimensional convolution operation applied to one-dimensional MCDM. Furthermore, to solve the invariance problem of the generated feature maps, the maxpooling kernel size can be flexibly adjusted to effectively meet the requirements of reducing the feature map dimension and speeding up training and prediction on different datasets. The improvement is reasonable for various high-dimensional data application scenarios. Moreover, we propose a novel objective function to accurately evaluate network performance since the objective function includes a variety of representative performance evaluation metrics, and the average value is calculated as one of the CE metrics. The experimental results illustrate that the proposed framework outperforms a one-dimensional convolutional neural network (1D_CNN) for comprehensive classification on the Shaoxing University student achievement dataset and the MIT-BIH Arrhythmia database and achieves average gains of 36.3% and 12.1% in terms of the designated evaluation metric.