Representation Theorem Research Articles

Hypercontact semilattices

Boolean algebras are one of the main algebraic tools in the region-based theory of space. T. Ivanova provided strong motivations for the study of mere semilattices with a contact relation. Another significant motivation for considering an even weaker underlying structure comes from event structures with binary conflict in the theory of concurrent systems in computer science. All the above-hinted notions deal with a binary contact relation. Several authors suggested the more general study of n-ary ‘hypercontact’ relations. A similar evolution occurred in the study of the just mentioned event structures in computer science. To unify the above lines of research, in this paper, we study joining semilattices with a hypercontact relation. We provide representation theorems into Boolean algebras. With a single exception, our proofs are choice-free. We also present several examples and problems; in particular, we briefly discuss some connections with event structures and hypergraphs.

Predicting Chlorophyll-a Concentrations in the World's Largest Lakes Using Kolmogorov-Arnold Networks.

Accurate prediction of chlorophyll-a (Chl-a) concentrations, a key indicator of eutrophication, is essential for the sustainable management of lake ecosystems. This study evaluated the performance of Kolmogorov-Arnold Networks (KANs) along with three neural network models (MLP-NN, LSTM, and GRU) and three traditional machine learning tools (RF, SVR, and GPR) for predicting time-series Chl-a concentrations in large lakes. Monthly remote-sensed Chl-a data derived from Aqua-MODIS spanning September 2002 to April 2024 were used. The models were evaluated based on their forecasting capabilities from March 2024 to August 2024. KAN consistently outperformed others in both test and forecast (unseen data) phases and demonstrated superior accuracy in capturing trends, dynamic fluctuations, and peak Chl-a concentrations. Statistical evaluation using ranking metrics and critical difference diagrams confirmed KAN's robust performance across diverse study sites, further emphasizing its predictive power. Our findings suggest that the KAN, which leverages the KA representation theorem, offers improved handling of nonlinearity and long-term dependencies in time-series Chl-a data, outperforming neural network models grounded in the universal approximation theorem and traditional machine learning algorithms.

An intrusion detection model based on Convolutional Kolmogorov-Arnold Networks

The application of artificial neural networks (ANNs) can be found in numerous fields, including image and speech recognition, natural language processing, and autonomous vehicles. As well, intrusion detection, the subject of this paper, relies heavily on it. Different intrusion detection models have been constructed using ANNs. While ANNs are relatively mature to construct intrusion detection models, some challenges remain. Among the most notorious of these are the bloated models caused by the large number of parameters, and the non-interpretability of the models. Our paper presents Convolutional Kolmogorov-Arnold Networks (CKANs), which are designed to overcome these difficulties and provide an interpretable and accurate intrusion detection model. Kolmogorov-Arnold Networks (KANs) are developed from the Kolmogorov-Arnold representation theorem. Meanwhile, CKAN incorporates a convolutional computational mechanism based on KAN. The model proposed in this paper is constructed by incorporating attention mechanisms into CKAN’s computational logic. The datasets CICIoT2023 and CICIoMT2024 were used for model training and validation. From the results of evaluating the performance indicators of the experiments, the intrusion detection model constructed based on CKANs has an attractive application prospect. As compared with other methods, the model can predict a much higher level of accuracy with significantly fewer parameters. However, it is not superior in terms of memory usage, execution speed and energy consumption.

Enhancing Low-Light Images with Kolmogorov-Arnold Networks in Transformer Attention.

Low-light image enhancement (LLIE) techniques improve the performance of image sensors by enhancing visibility and details in poorly lit environments and have significantly benefited from recent research into Transformer models. This work presents a novel Transformer attention mechanism inspired by the Kolmogorov-Arnold representation theorem, incorporating learnable non-linearity and multivariate function decomposition. This innovative mechanism is the foundation of KAN-T, our proposed Transformer network. By enhancing feature flexibility and enabling the model to capture broader contextual information, KAN-T achieves superior performance. Our comprehensive experiments, both quantitative and qualitative, demonstrate that the proposed method achieves state-of-the-art performance in low-light image enhancement, highlighting its effectiveness and wide-ranging applicability. The code will be released upon publication.

Spectral analysis of metamaterials in curved manifolds

Abstract Negative-index metamaterials possess a negative refractive index and thus present an interesting substance for designing uncommon optical effects such as invisibility cloaking. This paper deals with operators encountered in an operator-theoretic description of metamaterials. First, we introduce an indefinite Laplacian and consider it on a compact tubular neighbourhood in constantly curved compact two-dimensional Riemannian ambient manifolds, with Euclidean rectangle in $\mathbb{R}^2$ being present as a special case. As this operator is not semi-bounded, standard form-theoretic methods cannot be applied. We show that this operator is (essentially) self-adjoint via separation of variables and find its spectral characteristics. We also provide a new method for obtaining alternative definition of the self-adjoint operator in non-critical case via a generalized form representation theorem. The main motivation is existence of essential spectrum in bounded domains.

Representation theorems for normed modules

In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann’s theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli’s differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way.

Rings of almost everywhere defined functions

The following representation theorem is proven: A partially ordered commutative ring R is a subring of a ring of almost everywhere defined continuous real-valued functions on a compact Hausdorff space X if and only if R is archimedean and localizable. Here we assume that the positive cone of R is closed under multiplication and stable under multiplication with squares, but actually one of these assumptions implies the other. An almost everywhere defined function on X is one that is defined on a dense open subset of X. A partially ordered commutative ring R is archimedean if the underlying additive partially ordered abelian group is archimedean, and R is localizable essentially if its order is compatible with the construction of a localization with sufficiently large, positive denominators. As applications we discuss the σ-bounded case, lattice-ordered commutative rings (f-rings), partially ordered fields, and commutative operator algebras.

EKLT: Kolmogorov-Arnold attention-driven LSTM with Transformer model for river water level prediction

Water level prediction is crucial for water resource management and flood warnings. Hybrid LSTM-based methods are widely used but face the following challenges: (1) The attention mechanism based on the universal approximation theorem (UAT) is the mainstream method for improving performance, but its essence is infinite approximation of functions, and the accuracy is difficult to improve; (2) LSTM struggles with modeling complex, long-term dependencies. To address these problems, we apply empirical mode decomposition (EMD) and propose input-spatial attention based on the Kolmogorov-Arnold theorem (KAT) for precise water level feature representation. Cascaded LSTM and Transformer structures capture long-term dependencies. Finally, temporal attention is embedded to achieve accurate water level prediction. Experiments show the proposed method achieves RMSE, MAE, MAPE and R2 values of 0.1870, 0.1328, 1.1228 and 0.9540 in the Liaohe River of Liaoning Province, China, and 0.3027, 0.1844, 4.6034 and 0.8659 in the Hunhe River, respectively, indicating its strong predictive performance.

Weakly compact sets and Riesz representation theorem in variable exponent Lebesgue-Bochner sequence spaces

In this paper, we establish three intrinsic criteria for weakly compact sets in variable exponent Lebesgue-Bochner sequence spaces ℓ(Φ)(X), where each criterion's conditions are both necessary and sufficient. We also present a Riesz-type representation theorem for the dual spaces of these sequence spaces. Additionally, we apply these criteria to Lebesgue-Bochner sequence spaces. We provide a necessary and sufficient condition for the boundedness criterion of weak compactness, which answers an open question in sequence spaces posed by Hernandez et al. Our findings demonstrate that weak compactness cannot be directly extrapolated from a Banach space X to ℓ(Φ)(X), such as ℓ∞(X).

On simpliciality of function spaces not containing constants

We investigate simpliciality of function spaces without constants. We prove, in particular, that several properties characterizing simpliciality in the classical case differ in this new setting. We also show that it may happen that a given point is not represented by any measure pseudosupported by the Choquet boundary, illustrating so limits of possible generalizations of the representation theorem. Moreover, we address the abstract Dirichlet problem in the new setting and establish some common points and nontrivial differences with the classical case.

Is (Independent) Subordination Relevant in Equity Derivatives?

ABSTRACTMonroe (1978) demonstrates that any local semimartingale can be represented as a time‐changed Brownian Motion (BM). A natural question arises: does this representation theorem hold when the BM and the time‐change are independent? We prove that a local semimartingale is not equivalent to a BM with a time‐change that is independent from the BM. Our result is obtained utilizing a class of additive processes: the additive normal tempered stable (ATS). This class of processes exhibits an exceptional ability to calibrate the equity volatility surface accurately. We notice that the sub‐class of additive processes that can be obtained with an independent additive subordination is incompatible with market data and shows significantly worse calibration performances than the ATS, especially on short time maturities. These results have been observed every business day in a semester on a dataset of S&P 500 and EURO STOXX 50 options.

Martingale representation on enlarged filtrations: the role of the accessible jump times

We consider a filtration G obtained as enlargement of a filtration F by a filtration H . We assume that all F -local martingales are represented by a martingale M and all H -local martingales are represented by a martingale N. M and N are not necessarily quasi-left continuous processes and their jump times may overlap. We first analyse the contribution of the accessible jump times of M and N to the Jacod dimension of the space of the H 1 ( G ) -martingales. Then we prove a martingale representation theorem on G .

Korn and Poincaré-Korn inequalities: A different perspective

We present a concise point of view on the first and the second Korn’s inequality for general exponent p p and for a class of domains that includes Lipschitz domains. Our argument is conceptually very simple and, for p = 2 p = 2 , uses only the classical Riesz representation theorem in Hilbert spaces. Moreover, the argument for the general exponent 1 > p > ∞ 1>p>\infty remains the same, the only change being invoking now the q q -Riesz representation theorem (with q q the harmonic conjugate of p p ). We also complement the analysis with elementary derivations of Poincaré-Korn inequalities in bounded and unbounded domains, which are essential tools in showing the coercivity of variational problems of elasticity but also propedeutic to the proof of the first Korn inequality.

How Resilient Are Kolmogorov–Arnold Networks in Classification Tasks? A Robustness Investigation

Kolmogorov–Arnold Networks (KANs) are a novel class of neural network architectures based on the Kolmogorov–Arnold representation theorem, which has demonstrated potential advantages in accuracy and interpretability over Multilayer Perceptron (MLP) models. This paper comprehensively evaluates the robustness of various KAN architectures—including KAN, KAN-Mixer, KANConv_KAN, and KANConv_MLP—against adversarial attacks, which constitute a critical aspect that has been underexplored in current research. We compare these models with MLP-based architectures such as MLP, MLP-Mixer, and ConvNet_MLP across three traffic sign classification datasets: GTSRB, BTSD, and CTSD. The models were subjected to various adversarial attacks (FGSM, PGD, CW, and BIM) with varying perturbation levels and were trained under different strategies, including standard training, adversarial training, and Randomized Smoothing. Our experimental results demonstrate that KAN-based models, particularly the KAN-Mixer, exhibit superior robustness to adversarial attacks compared to their MLP counterparts. Specifically, the KAN-Mixer consistently achieved lower Success Attack Rates (SARs) and Degrees of Change (DoCs) across most attack types and datasets while maintaining high accuracy on clean data. For instance, under FGSM attacks with ϵ=0.01, the KAN-Mixer outperformed the MLP-Mixer by maintaining higher accuracy and lower SARs. Adversarial training and Randomized Smoothing further enhanced the robustness of KAN-based models, with t-SNE visualizations revealing more stable latent space representations under adversarial perturbations. These findings underscore the potential of KAN architectures to improve neural network security and reliability in adversarial settings.

The Spectral Representation Method: A framework for simulation of stochastic processes, fields, and waves

The Spectral Representation Method (SRM) was developed in the 1970s to simulate Gaussian stochastic processes and fields from a Fourier series expansion according to the Spectral Representation Theorem. Since those early developments, the SRM has continuously evolved into a comprehensive framework for the simulation of stochastic processes, fields, and waves with a rigorous theoretical foundation. Its major advantages are conceptual simplicity and computational efficiency. In the 1990s, much of the theory for simulation of Gaussian stochastic processes, fields, and waves was firmly established and early methods for simulation of non-Gaussian processes, fields, and waves were introduced. In the 2000s and 2010s, methods that coupled the SRM with Translation Process Theory were improved to enable efficient and accurate simulations of stochastic processes, fields, and waves with strongly non-Gaussian marginal probability distributions. More recently, the SRM was extended for higher-order non-Gaussian processes, fields, and waves by extending the Fourier stochastic expansion to include non-linear wave interactions derived from higher-order spectra. This paper reviews the key theoretical developments related with the SRM and provides the relevant algorithms necessary for its practical implementation for the simulation of stochastic processes, fields, and waves that can be either stationary or non-stationary, homogeneous or non-homogeneous, one-dimensional or multi-dimensional, scalar or multi-variate, Gaussian or non-Gaussian, or any combination thereof. The paper concludes with some brief remarks addressing the open research challenges in SRM-based theory and simulations.

A note on Johansen's rank conditions and the Jordan form of a matrix

This note presents insights on the Jordan structure of a matrix which are derived from an extension of the and conditions in Johansen (1996). It is first observed that these conditions not only characterize, as it is well known, the size (1 or 2) of the largest Jordan block in the Jordan form of the companion matrix but more generally the geometric multiplicities, the algebraic multiplicities and the whole Jordan structure for eigenvalues of index 1 or 2. In the context of the Granger representation theorem, this means that the Johansen rank conditions do more than determine the order of integration of the process. It is then shown that an extension of these conditions leads to the characterization of the Jordan structure of any matrix.

Approximation of $$L^\infty $$ functionals with generalized Orlicz norms

AbstractThe aim of this paper is to deal with the asymptotics of generalized Orlicz norms when the lower growth rate tends to infinity. We generalize results proven by Bertazzoni, Harjulehto and Hästö in Journ. of Math. Anal. and Appl. (2024) for integral type energies (in generalized Orlicz spaces), considering milder convexity assumptions. $$\Gamma $$ Γ -convergence results and related representation theorems in terms of $$L^\infty $$ L ∞ functionals are proven. The convexity hypotheses are completely removed in the variable exponent setting, thus extending the results in Eleuteri-Prinari in Nonlinear Anal. Real. World Appl. (2021) and Prinari-Zappale in JOTA (2020).

On optimal control of coupled mean-field forward-backward stochastic equations

Abstract We consider a control problem for a mean-field coupled forward-backward stochastic differential equations, called also McKean–Vlasov equation (MF-FBSDE). For this type of equations, the coefficients depend not only on the state of the system, but also on its marginal distributions. They arise naturally in mean-field control problems and mean-field games. We consider the relaxed control problem where admissible controls are measure-valued processes. We prove the existence of a relaxed optimal control by using a suitable form of Skorokhod representation theorem and Jakubowski’s topology, on the space of càdlàg functions. We use martingale measure to define the relaxed state process. Our results extend to MF-FBSDEs those already known for forward and backward stochastic equations of Itô type.

Modulo ℓ distinction problems

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$. We start by proving a general theorem allowing to lift supercuspidal $\overline {\mathbf {F}}_{\ell }$-representations of $\operatorname {GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline {\mathbf {Q}}_{\ell }$-representation. Given a quadratic field extension $E/F$ and an irreducible $\overline {\mathbf {F}}_{\ell }$-representation $\pi$ of $\operatorname {GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi _\pi$ is irreducible and conjugate-selfdual, then $\pi$ is either $\operatorname {GL}_n(F)$-distinguished or $(\operatorname {GL}_{n}(F),\omega _{E/F})$-distinguished (where $\omega _{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of Sécherre to the case $p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$. After that, we give a complete classification of the $\operatorname {GL}_2(F)$-distinguished representations of $\operatorname {GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for $\operatorname {PGL}_2$. We show that the ‘classical’ Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil–Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\operatorname {SL}_2(F)$-distinguished modular representations of $\operatorname {SL}_2(E)$.

Predictive modeling of flexible EHD pumps using Kolmogorov–Arnold Networks

We present a novel approach to predicting the pressure and flow rate of flexible electrohydrodynamic pumps using the Kolmogorov–Arnold Network. Inspired by the Kolmogorov–Arnold representation theorem, KAN replaces fixed activation functions with learnable spline-based activation functions, enabling it to approximate complex nonlinear functions more effectively than traditional models like Multi-Layer Perceptron and Random Forest. We evaluated KAN on a dataset of flexible EHD pump parameters and compared its performance against RF, and MLP models. KAN achieved superior predictive accuracy, with Mean Squared Errors of 12.186 and 0.012 for pressure and flow rate predictions, respectively. The symbolic formulas extracted from KAN provided insights into the nonlinear relationships between input parameters and pump performance. These findings demonstrate that KAN offers exceptional accuracy and interpretability, making it a promising alternative for predictive modeling in electrohydrodynamic pumping.