Linear Map Research Articles

The strongest Banach–Stone theorem for C0(K,ℓ22)$C_{0}(K, \ell _2^2)$ spaces

AbstractAs usual denote by the real two‐dimensional Hilbert space. We prove that if and are locally compact Hausdorff spaces and is a linear isomorphism from onto satisfying then and are homeomorphic.This theorem is the strongest of all the other vector‐valued Banach–Stone theorems known so far in the sense that in none of them the distortion of the isomorphism , denoted by , is as large as .Some remarks on the proof method developed here to prove our theorem suggest the conjecture that it is in fact very close to the optimal Banach–Stone theorem for spaces, or in more precise words, the exact value of the Banach–Stone constant of is between and .

YOLO-DCNet

Enhanced processors empower edge devices like smartphones for human detection, yet their application is constrained by algorithmic efficiency and precision. This paper introduces YOLO-DCNet, a lightweight neural network detector built upon YOLOv7-tiny. Incorporating a dynamic multi-head structural re-parameterization (DMSR) module within its backbone network enables effective processing of the features utilized in the model. To improve multi-scale feature aggregation, the model integrates a channel information compression and linear mapping (CLM) module into its feature pyramid architecture. Moreover, the optimization of training and inference performance is achieved by employing RepVGG blocks between the main computational modules of the model. Experimental data reveal that the enhanced YOLOv7-tiny model achieves a 31.7% faster inference speed and marginal gains of 0.7% in mAP@0.5 and 0.5% in mAP@0.5:0.95 over the original. This underscores the model's improved performance and applicability for real-time human detection on edge devices across diverse applications.

Antibody profiling and predictive modeling discriminate between Kaposi sarcoma and asymptomatic KSHV infection.

Protein-level immunodominance patterns against Kaposi sarcoma-associated herpesvirus (KSHV), the aetiologic agent of Kaposi sarcoma (KS), have been revealed from serological probing of whole protein arrays, however, the epitopes that underlie these patterns have not been defined. We recently demonstrated the utility of phage display in high-resolution linear epitope mapping of the KSHV latency-associated nuclear antigen (LANA/ORF73). Here, a VirScan phage immunoprecipitation and sequencing approach, employing a library of 1,988 KSHV proteome-derived peptides, was used to quantify the breadth and magnitude of responses of 59 sub-Saharan African KS patients and 22 KSHV-infected asymptomatic individuals (ASY), and ultimately to support an application of machine-learning-based predictive modeling using the peptide-level responses. Comparing anti-KSHV antibody repertoire revealed that magnitude, not breadth, increased in KS. The most targeted epitopes in both KS and ASY were in the immunodominant proteins, notably, K8.129-56 and ORF65140-168, in addition to LANA. Finally, using unbiased machine-learning-based predictive models, reactivity to a subset of 25 discriminative peptides was demonstrated to successfully classify KS patients from asymptomatic individuals. Our study provides the highest resolution mapping of antigenicity across the entire KSHV proteome to date, which is vital to discern mechanisms of viral pathogenesis, to define prognostic biomarkers, and to design effective vaccine and therapeutic strategies. Future studies will investigate the diagnostic, prognostic, and therapeutic potential of the 25 discriminative peptides.

Quantitative pressure and saturation engineering values from 4D PP and PS seismic data

AbstractFor field development and drilling decisions, production assets and reservoir engineers require dynamic reservoir properties, such as saturation and pressure changes of a reservoir from the pre‐production virgin state. To date, geophysicists have produced time‐lapse (4D) seismic attributes (mostly on stacked seismic data) rather than dynamic parameters directly. In this paper, we present a new method to estimate saturation and pressure properties from time‐lapse seismic data to provide to reservoir engineers. This new three‐step method is demonstrated over the Edvard Grieg field in the North Sea. We can realize this method thanks to advanced seismic multi‐component acquisition via PP and PS seismic data and processing that allows accurate estimation of amplitude‐variation‐with‐offset parameters P‐ and S‐impedances. With time‐lapse P‐ and S‐impedances optimally resolved, we estimate a stable set of axes identifying water saturation increase (water replacing oil), gas saturation increase (gas injection or gas out of solution), pressure increase (at injectors) and pressure decrease (at producers). Once these axes are obtained, we convert every 4D P‐ and S‐impedance data points into 4D pseudo‐saturation and pseudo‐pressure using a transformation of coordinates. We next use the rock physics relationships of this field to show that a linear relationship can be used to map any 4D change in the field from impedances to saturations and pressures. Key locations in the field with largest saturation and pressure changes are used to find calibration values at these extreme points. Next, from the pseudo‐seismic 4D data, a linear mapping is used to calculate actual reservoir property changes (fractions for saturation and bars for pressure). This allows us to obtain fieldwide dynamic values for water and gas saturations and injection and production related pressure changes. The results are shown, and dynamic changes are interpreted on the Edvard Grieg field data.

Near-bipartite Leonard pairs

Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let $A,A^*$ denote a Leonard pair on $V$. Let $\{v_i\}_{i=0}^d$ denote an eigenbasis for $A^*$ on which $A$ acts in an irreducible tridiagonal fashion. For $0 \leq i \leq d$, define an $\mathbb{F}$-linear map $E^*_i : V \to V$ such that $E^*_i v_i = v_i$ and $E^*_i v_j = 0$ if $j \neq i$ $(0 \leq j \leq d)$. The map $F = \sum_{i=0}^d E^*_i A E^*_i$ is called the flat part of $A$. The Leonard pair $A,A^*$ is bipartite whenever $F=0$. The Leonard pair $A,A^*$ is said to be near-bipartite whenever the pair $A-F, A^*$ is a Leonard pair on $V$. In this case, the Leonard pair $A-F, A^*$ is bipartite and called the bipartite contraction of $A,A^*$. Let $B,B^*$ denote a bipartite Leonard pair on $V$. By a near-bipartite expansion of $B,B^*$, we mean a near-bipartite Leonard pair on $V$ with bipartite contraction $B,B^*$. In the present paper, we have three goals. Assuming $\mathbb{F}$ is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over $\mathbb{F}$; (ii) for each near-bipartite Leonard pair over $\mathbb{F}$ we describe its bipartite contraction; (iii) for each bipartite Leonard pair over $\mathbb{F}$ we describe its near-bipartite expansions. Our classification (i) is summarized as follows. We identify two families of Leonard pairs, said to have Krawtchouk type and dual $q$-Krawtchouk type. A Leonard pair of dual $q$-Krawtchouk type is said to be reinforced whenever $q^{2i} \neq -1$ for $1 \leq i \leq d-1$. A Leonard pair $A,A^*$ is said to be essentially bipartite whenever the flat part of $A$ is a scalar multiple of the identity. Assuming $\mathbb{F}$ is algebraically closed, we show that a Leonard pair $A,A^*$ over $\mathbb{F}$ with $d \geq 3$ is near-bipartite if and only if at least one of the following holds: (i) $A,A^*$ is essentially bipartite; (ii) $A,A^*$ has reinforced dual $q$-Krawtchouk type; and (iii) $A,A^*$ has Krawtchouk type.

Linear coupling of transverse betatron oscillations: Dynamic stability and invariants of motion

Based on the technique of the discrete one-turn transfer maps, the problem of linear coupling between horizontal and vertical betatron oscillations in an accelerator has been treated exactly and entirely in explicit form. The stability region in the fractional part of the horizontal and the vertical betatron tune space as a function of the linear coupling strength has been obtained, and the increment/decrement of the horizontal and the vertical betatron oscillations in the case of the linear sum resonance has been shown to be approximately equal to the half of the coupling strength. The normal form parameterization of the one-turn linear map with horizontal-to-vertical coupling has been developed in detail in the spirit of Edwards and Teng formalism. The motion in the normal mode in the new normal form coordinates is decoupled by implying that two independent Courant-Snyder invariants exist, which have been found explicitly. Published by the American Physical Society 2024

Quantum Theory in Finite Dimension Cannot Explain Every General Process with Finite Memory

Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.

Transition to anomalous dynamics in a simple random map.

The famous doubling map (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one, the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability p, and the contracting one with probability 1-p, gives a prototype of a random dynamical system. Here, we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of p. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability pc, defined by a zero Lyapunov exponent. This anomalous dynamics is characterized by an infinite invariant density, weak ergodicity breaking, and power-law correlation decay.

On Border-Collision Bifurcations in a Pulse System

Considering a piecewise smooth map describing the behavior of a pulse-modulated control system, we discuss border-collision related phenomena. We show that in the parameter space which corresponds to the domain of oscillatory mode a mapping is piecewise linear continuous. It is well known that in piecewise linear maps, classical bifurcations, for example, period doubling, tangent, fold bifurcations become degenerate (“degenerate bifurcations”), combining the properties of both smooth and border-collision bifurcations. We found unusual properties of this map, that consist in the fact that border-collision bifurcations of codimension one, including degenerate ones, occur when a pair of points of a periodic orbit simultaneously collides with two switching manifolds. This paper also discuss bifurcations of chaotic attractors such as merging and expansion (“interior”) crises, associated with homoclinic bifurcations of unstable periodic orbits.

Certain linear isomorphisms for hyperalgebras relative to a Chevalley group

Let [Formula: see text] be a simply connected and simple algebraic group defined and split over a finite prime field [Formula: see text] of [Formula: see text] elements. In this paper, using an [Formula: see text]-linear map splitting Frobenius endomorphism on a hyperalgebra relative to [Formula: see text], we obtain some [Formula: see text]-linear isomorphisms induced by multiplication in the hyperalgebra.

A Robust Zero-Watermarking Scheme in Spatial Domain by Achieving Features Similar to Frequency Domain

In recent years, there has been a substantial surge in the application of image watermarking, which has evolved into an essential tool for identifying multimedia material, ensuring security, and protecting copyright. Singular value decomposition (SVD) and discrete cosine transform (DCT) are widely utilized in digital image watermarking despite the considerable computational burden they involve. By combining block-based direct current (DC) values with matrix norm, this research article presents a novel, robust zero-watermarking approach. It generates a zero-watermark without attempting to modify the contents of the image. The image is partitioned into non-overlapping blocks, and DC values are computed without applying DCT. This sub-image is further partitioned into non-overlapping blocks, and the maximum singular value of each block is calculated by matrix norm instead of SVD to obtain the binary feature matrix. A piecewise linear chaotic map encryption technique is utilized to improve the security of the watermark image. After that, the feature image is created via XOR procedure between the encrypted watermark image and the binary feature matrix. The proposed scheme is tested using a variety of distortion attacks including noise, filter, geometric, and compression attacks. It is also compared with the other relevant image watermarking methods and outperformed them in most cases.

A Spatial–Spectral Transformer for Hyperspectral Image Classification Based on Global Dependencies of Multi-Scale Features

Vision transformers (ViTs) are increasingly utilized for HSI classification due to their outstanding performance. However, ViTs encounter challenges in capturing global dependencies among objects of varying sizes, and fail to effectively exploit the spatial–spectral information inherent in HSI. In response to this limitation, we propose a novel solution: the multi-scale spatial–spectral transformer (MSST). Within the MSST framework, we introduce a spatial–spectral token generator (SSTG) and a token fusion self-attention (TFSA) module. Serving as the feature extractor for the MSST, the SSTG incorporates a dual-branch multi-dimensional convolutional structure, enabling the extraction of semantic characteristics that encompass spatial–spectral information from HSI and subsequently tokenizing them. TFSA is a multi-head attention module with the ability to encode attention to features across various scales. We integrated TFSA with cross-covariance attention (CCA) to construct the transformer encoder (TE) for the MSST. Utilizing this TE to perform attention modeling on tokens derived from the SSTG, the network effectively simulates global dependencies among multi-scale features in the data, concurrently making optimal use of spatial–spectral information in HSI. Finally, the output of the TE is fed into a linear mapping layer to obtain the classification results. Experiments conducted on three popular public datasets demonstrate that the MSST method achieved higher classification accuracy compared to state-of-the-art (SOTA) methods.

Determining the Inverse of a Matrix over Min-Plus Algebra

Linear algebra over the semiring R_ε with ⊗ (plus) and ⨁ (maximum) operations which is known as max-plus algebra. One of the isomorphic with this algebra is a min-plus algebra. Min-plus algebra that is the set R_(ε^' )=R∪{ε'}, with ⊗^' (plus) and ⨁' (minimum) operations. Given a matrix whose components are elements of R_(ε^' ) is called min-plus algebra matrices. Any matrix can be connected by an inverse. In conventional algebra, a square matrix is said an invertible matrix if the det⁡〖(A)〗≠0. In contrast to max-plus algebra, a matrix is said to have inverse condition if it meets certain conditions. Some concepts from the max-plus algebra can be transformed to the min-plus, because of their structural similarity. This means that the inverse matrix concept in max-plus can be constructed into a min-plus version. Thus, this study will explain the inverse of a matrix over the min-plus algebra, property of multiplying two invertible matrices, and connection between invertible matrix and linear mapping used the literature study method, with literature sources such as books, journals, articles, and theses. The data analysis technique used in this research is qualitative data analysis technique. Then, this article has a principal result that is matrix A∈R_(ε^')^(n×n) has a right inverse if and only if there are permutations of σ and the value of λ_i<ε', i∈{1,2,3,…,n} such that A=P_σ ⊗^' D(λ_i ) which is the inverse of matrices. Furthermore, if B is the correct inverse that satisfies A⊗^' B=E then B⊗^' A=E and B is uniquely determined by A.

Exposedness of elementary positive maps between matrix algebras

ABSTRACT The positive linear maps Ad s which send matrices x to s ∗ x s play important roles in quantum information theory as well as matrix theory. It was proved by [Marciniak M. Rank properties of exposed positive maps. Linear Multilinear Alg. 2013;61:970–975] that the map Ad s generates an exposed ray of the convex cone of all positive linear maps. In this note, we provide two alternative proofs, using Choi matrices and Woronowicz's method, respectively.

Applications of vector bundle to finite elasto-plasticity

A new approach to finite elasto-plasticity of crystalline materials, with a differential geometry point of view toward the material description, is proposed. In order to define the plastic and elastic distortions, traditionally it is accepted the existence of an intermediate configuration, which is endowed with a differential manifold structure. A more restrictive but physically and mathematically well-motivated structure is assigned to the intermediate configuration, namely, a vector bundle structure. Two approaches to multiplicative decomposition of the deformation gradient can be rebuilt if on the same total space (intermediate configuration) the vector bundle structures are considered, but either the reference configuration, B (i.e., plastic assumption) or the deformed configuration, B t (i.e., elastic assumption) stands for the base space. As the base spaces are homeomorphic, a pullback vector bundle over one base space is provided by the other one through the motion diffeomorphism. Thus, the total space is endowed with different differential manifold structures. The plastic distortion and the inverse elastic distortion, respectively, are defined for any material points as linear, invertible, and non-integrable maps from the tangent vector space at their material points, with the value on the attached vector fiber. The multiplicative decompositions into their non-integrable components are derived, based on the associate inverse elastic distortion and plastic distortion, respectively. When the plastic and elastic assumptions are simultaneously accepted, then a three-term multiplicative decomposition of the deformation gradient is achieved. The transition functions, which characterize the compatibility of the overlapping charts which belong to these different structures, namely of the vector bundle and of the pullback vector bundle over the same configuration, say B define the non-uniqueness of the two-term multiplicative decomposition. The material symmetry transformation is defined for elastic-type materials. The compatibility of the models and examples of the bundle vector structures are also discussed.

Minimal extension for the $\alpha$-Manhattan norm

Let \partial\mathcal{Q} be the boundary of a convex polygon in \mathbb{R}^2 , e_\alpha=(\cos\alpha,\sin\alpha) and e_{\alpha}^{\bot}=(-\sin\alpha,\cos\alpha) a basis of \mathbb{R}^2 for some \alpha\in[0,2\pi) and \varphi:\partial\mathcal{Q}\to\mathbb{R}^2 a continuous, finitely piecewise linear injective map. We construct a finitely piecewise affine homeomorphism v:\mathcal{Q}\to\mathbb{R}^2 coinciding with \varphi on \partial\mathcal{Q} such that the following property holds: |\langle Dv, e_{\alpha}\rangle|(\mathcal{Q}) (resp., \langle Dv, e_{\alpha}^{\bot}\rangle|(\mathcal{Q}) ) is as close as we want to \inf|\langle Du, e_{\alpha}\rangle|(\mathcal{Q}) (resp., \inf|\langle Du, e_{\alpha}^{\bot}\rangle|(\mathcal{Q}) ) where the infimum is meant over the class of all BV homeomorphisms u extending \varphi inside \mathcal{Q} . This result extends that already proven by Pratelli and the third author in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 511–555] in the shape of the domain.

DL-Reg: A deep learning regularization technique using linear regression

Regularization is an essential aspect in the context of deep learning as it mitigates the risk of overfitting in deep neural networks. This study presents a novel deep learning regularization method, referred to DL-Reg, which effectively reduces the nonlinearity of deep networks by enforcing linearity to a certain extent. The proposed method is based on the incorporation of a linear constraint into the objective function of deep neural networks, which is defined as the error of a linear mapping from the inputs to the outputs of the model. Specifically, DL-Reg imposes a linear constraint on the network, which is further adjusted by a regularization factor, thereby preventing the network from overfitting. The effectiveness of DL-Reg is evaluated by training state-of-the-art deep network models on several benchmark datasets. The results of the experiments demonstrate that the proposed regularization method provides significant improvements over existing regularization techniques and enhances the performance of deep neural networks, particularly when dealing with small-sized training datasets. The main code of DL-Reg written in PyTorch is also available here: https://github.com/m2dgithub/DL-Reg.git.

Frequency effects in linear discriminative learning.

Word frequency is a strong predictor in most lexical processing tasks. Thus, any model of word recognition needs to account for how word frequency effects arise. The Discriminative Lexicon Model (DLM) models lexical processing with mappings between words' forms and their meanings. Comprehension and production are modeled via linear mappings between the two domains. So far, the mappings within the model can either be obtained incrementally via error-driven learning, a computationally expensive process able to capture frequency effects, or in an efficient, but frequency-agnostic solution modeling the theoretical endstate of learning (EL) where all words are learned optimally. In the present study we show how an efficient, yet frequency-informed mapping between form and meaning can be obtained (Frequency-informed learning; FIL). We find that FIL well approximates an incremental solution while being computationally much cheaper. FIL shows a relatively low type- and high token-accuracy, demonstrating that the model is able to process most word tokens encountered by speakers in daily life correctly. We use FIL to model reaction times in the Dutch Lexicon Project by means of a Gaussian Location Scale Model and find that FIL predicts well the S-shaped relationship between frequency and the mean of reaction times but underestimates the variance of reaction times for low frequency words. FIL is also better able to account for priming effects in an auditory lexical decision task in Mandarin Chinese, compared to EL. Finally, we used ordered data from CHILDES to compare mappings obtained with FIL and incremental learning. We show that the mappings are highly correlated, but that with FIL some nuances based on word ordering effects are lost. Our results show how frequency effects in a learning model can be simulated efficiently, and raise questions about how to best account for low-frequency words in cognitive models.

Dynamical instability in multi-orbiter systems with gas friction

ABSTRACT Closely packed multiplanet systems are known to experience dynamical instability if the spacings between the planets are too small. Such instability can be tempered by the frictional forces acting on the planets from gaseous discs. A similar situation applies to stellar-mass black holes embedded in active galactic nuclei discs around supermassive black holes. We use N-body integrations to evaluate how the frictional damping of orbital eccentricity affects the growth of dynamical instability for a wide range of K (the difference in the planetary semimajor axes in units of the mutual Hill radius) and (unequal) planet masses. We find that, in general, the stable region (large K) and unstable region (small K) are separated by a “grey zone”, where the (in)stability is not guaranteed. We report the numerical values of the critical spacing for stability Kcrit and the “grey zone” range in different systems, and provide fitting formulae for arbitrary frictional forcing strength. We show that the stability of a system depends on the damping time-scale τ relative to the zero-friction instability growth time-scale tinst: two-planet systems are stable if tinst ≳ τ; three-planet systems require tinst ≳ 10τ−100τ. When K is sufficiently small, tinst can be less than the synodic period between the planets, which makes frictional stabilization unlikely to occur. As K increases, tinst tends to grow exponentially, but can also fluctuate by a few orders of magnitude. We also devise a linear map to analyse the dynamical instability of the “planet + test mass” system, and find qualitative agreement with N-body simulations.

Commutators on linear spaces

We prove, among other things, that every linear map T on any infinite-dimensional linear space E (over any field K) is a commutator i.e. T is of the form AB−BA for some linear maps A,B on E.