Dimensional Space Research Articles

Device-independent certification of desirable properties with a confidence interval

In the development of quantum technologies, a reliable means for characterizing quantum devices, be it a measurement device, a state-preparation device, or a transformation device, is crucial. However, the conventional approach based on, for example, quantum state tomography or process tomography relies on assumptions that are often not necessarily justifiable in a realistic experimental setting. Although the device-independent (DI) approach to this problem bypasses the shortcomings above by making only minimal, justifiable assumptions, most of the theoretical proposals to date only work in the idealized setting where independent and identically distributed (i.i.d.) trials are assumed. Here, we provide a versatile solution for rigorous device-independent certification that does not rely on the i.i.d. assumption. Specifically, we describe how the prediction-based ratio (PBR) protocol and martingale-based protocol developed for hypothesis testing can be applied in the present context to achieve a device-independent certification of desirable properties with confidence interval (CI). To illustrate the versatility of these methods, we demonstrate how we can use them to certify—with finite data—the underlying negativity, Hilbert space dimension, entanglement depth, and fidelity to some target pure state. In particular, we provide examples showing how the amount of certifiable negativity and fidelity scales with the number of trials and how many experimental trials one needs to certify a qutrit state space or the presence of genuine tripartite entanglement. Overall, we have found that the PBR protocol and the martingale-based protocol often offer similar performance, even though the latter does have to presuppose any witness (Bell-like inequality). In contrast, our findings also show that the performance of the martingale-based protocol may be severely affected by one’s choice of Bell-like inequality. Intriguingly, a Bell function useful for self-testing does not necessarily give the optimal confidence-gain rate for certifying the fidelity to the corresponding target state.

Inversion formulas for the attenuated conical Radon transform: Plane and cylinder case

Since the invention of Compton camera imaging, the conical Radon transform, which maps a given function defined on 3-dimensional Euclidean space to its surface integrals over cones, has been studied intensively. The problem of recovering such a function from its unrestricted conical Radon transform is overdetermined, since the set of all cones depends on the three dimensions of the vertex, the two dimensions of the central axis, and the one-dimensional opening angle. Therefore, various types of restricted conical Radon transforms have also been studied. However, most of these studies have neglected the attenuation of the medium. This article presents the study of attenuated conical Radon transforms with vertices on a plane (referred to as the plane case) or the cylinder (referred to as the cylinder case) in a 3-dimensional space. In all cases, the function is integrated over conical surfaces with an additional weight that decreases with the distance to the vertex of the cones. The main results provide explicit inversion formulas for the attenuated conical Radon transform in the plane and in the cylinder case.

Two-dimensional gravity waves at low regularity I: Energy estimates

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call balanced cubic estimates . This yields a key improvement over the earlier cubic estimates of Hunter–Ifrim–Tataru (2016), while preserving their scale-invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard–Burq–Zuily (2014, 2018), Hunter–Ifrim–Tataru (2016) and Ai (2019).

ARMNet: A Network for Image Dimensional Emotion Prediction Based on Affective Region Extraction and Multi-Channel Fusion

Compared with discrete emotion space, image emotion analysis based on dimensional emotion space can more accurately represent fine-grained emotion. Meanwhile, this high-precision representation of emotion requires dimensional emotion prediction methods to sense and capture emotional information in images as accurately and richly as possible. However, the existing methods mainly focus on emotion recognition by extracting the emotional regions where salient objects are located while ignoring the joint influence of objects and background on emotion. Furthermore, in the existing literature, when fusing multi-level features, no consideration has been given to the varying contributions of features from different levels to emotional analysis, which makes it difficult to distinguish valuable and useless features and cannot improve the utilization of effective features. This paper proposes an image emotion prediction network named ARMNet. In ARMNet, a unified affective region extraction method that integrates eye fixation detection and attention detection is proposed to enhance the combined influence of objects and backgrounds. Additionally, the multi-level features are fused with the consideration of their different contributions through an improved channel attention mechanism. In comparison to the existing methods, experiments conducted on the CGnA10766 dataset demonstrate that the performance of valence and arousal, as measured by Mean Squared Error (MSE), Mean Absolute Error (MAE), and Coefficient of Determination (R²), has improved by 4.74%, 3.53%, 3.62%, 1.93%, 6.29%, and 7.23%, respectively. Furthermore, the interpretability of the network is enhanced through the visualization of attention weights corresponding to emotional regions within the images.

Droplet trampoline surface for regulating contact time of a bouncing droplet across a wide spectrum of Weber numbers verified by in-situ observation

The dynamic superhydrophobicity of metallic surfaces is crucial for anti-icing and anti-adhesion in the fields of aerospace and transportation. Generally, fabricating micro/nanostructures on metallic surfaces is an effective method for facilitating rapid detachment of droplets, and the micro/nanoscale composite structure can significantly reduce the solid–liquid contact area. Meanwhile, the exploration of the correlation between solid–liquid contact time and the spacing of microstructures across a wide range of Weber numbers is of great significance. In this study, it is observed that the contact time of a bouncing droplet on microstructures with small spacing exhibits a notable correlation with the Weber number. This observation challenges the prevailing understanding that the contact time of a bouncing droplet is commonly perceived as independent of the Weber number, equated to the impact velocity. Increasing the structure spacing appropriately proves advantageous in preventing pinning effect, consequently reducing the duration of solid–liquid contact. Subsequently, a regulatory diagram is generated to demonstrate the correlation between contact time and the dual dimensions of the structure spacing and Weber number. Specifically, a novel theory for predicting the contact time based on the Weber number has been established, which reveals the unconventional correlation and enriches the knowledge of droplet dynamics.

IBhb-Lys: Identify lysine β-hydroxybutyrylation sites using autoencoder feature representation and fuzzy SVM algorithm

Lysine β-hydroxybutyrylation (Kbhb) is newly discovered β-hydroxybutyrylate -derived histone modification which has been associated with the pathogenesis of many human diseases. To further elucidate the biological significance and molecular mechanism of Kbhb, it is necessary to accurately identify the Kbhb sites from protein sequences. In this study, a novel computational model named iBhb-Lys is developed for the identification of Kbhb sites. Four types of features are combined to encode each Kbhb site as a 3266-dimensional feature vector. And the autoencoder network is used to reduce the dimensionality of feature space, due to the high dimensionality of the combined features. In addition, to effectively reduce the influence of noise and outlier on classification, a new fuzzy support vector machine algorithm is proposed by incorporating the density around the sample into the fuzzy membership function. As illustrated by independent test, the AUC value of iBhb-Lys has increased by 2.22% compared to the existing predictor KbhbXG. Feature analysis shows that some amino acid composition features, such as the occurrence frequency of leucine and histidine residues around Kbhb sites, contribute profoundly to the identification of Kbhb sites. The conclusions drawn in this study may provide useful reference for studying the molecular mechanism of Kbhb.

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In the early nineties, R. M. Aron, B. Cole, T. W. Gamelin and W. B. Johnson initiated the study of the maximal ideal space (spectrum) of Banach algebras of holomorphic functions defined on the open unit ball of an infinite dimensional complex Banach space. Within this framework, we investigate the fibers and Gleason parts of the spectrum of the algebra of holomorphic and uniformly continuous functions on the unit ball of ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _p$$\\end{document} (1≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p<\\infty $$\\end{document}). We show that the inherent geometry of these spaces provides a fundamental ingredient for our results. We prove that whenever p∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in {\\mathbb {N}}$$\\end{document} (p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document}), the fiber of every z∈Bℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in B_{\\ell _p}$$\\end{document} contains a set of cardinal 2c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^{{\\mathfrak {c}}}$$\\end{document} such that any two elements of this set belong to different Gleason parts. For the case p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document}, we complete the known description of the fibers, showing that, for each z∈B¯ℓ1′′\\Sℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z\\in \\overline{B}_{\\ell _1''}{\\setminus } S_{\\ell _1}$$\\end{document}, the fiber over z is not a singleton. Also, we establish that different fibers over elements in Sℓ1′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_{\\ell _1''}$$\\end{document} cannot share Gleason parts.

Space as a mental toolbox in the representation of meaning.

The experience of meaning has been found to be mapped onto spatial proximity whereby coherent-in contrast to incoherent-elements in a set are mentally represented as closer together in physical space. In a series of four experiments, we show that spatial representation of coherence is malleable and can employ other meaningful concrete dimensions of space that are made salient. When given task instructions cueing verticality, participants represented coherence in the upper vertical location when making judgements about the logical validity of realistic (Experiments 1 and 4) and unrealistic syllogistic scenarios (Experiment 3). When the task instruction made the spatial proximity between the stimuli materials and the participant salient (subjective proximity), participants represented coherence as spatially close to themselves (Experiment 2). We also found that being accurate in judging the validity of syllogisms was associated with representing coherence in the upper visual field or close to oneself. Overall, our findings show that identifying semantic links between an abstract concept and a given spatial dimension involves using that dimension to create spatial metaphoric mappings of the concept.

Novel PCA-Based Lower-Dimensional Remapping of the Solution Space for a Genetic Algorithm Optimization: Estimating the Director Distribution in LC-Based SLM Devices

This work introduces a novel computational approach based on Principal Component Analysis (PCA) for dimensionality reduction of the solution space in optimisation problems with known linear interdependencies among solution variables. By creating synthetic datasets with deliberately engineered properties and applying PCA, the solution space’s remapping significantly reduces its dimensionality, leading to faster computation and more robust convergence in optimisation processes. We demonstrate this method by integrating it with a Genetic Algorithm (GA) for solving the optimal director distribution in liquid crystal (LC) devices, specifically addressing 2D and complex 3D spatial light modulator (SLM) structures such as twisted nematic liquid crystals (TN-LC) and parallel-aligned liquid crystal on silicon (PA-LCoS), respectively. The phase profiles obtained from the director vector distributions for horizontal and vertical high-frequency binary phase gratings closely match the theoretical values derived from minimising the traditional elastic Frank–Oseen functional via Euler–Lagrange equations. Beyond this specific application, our method offers a general framework for reducing computational complexity in optimisation problems by directly reducing the dimensionality of the solution space. This approach is applicable across various optimisation scenarios with well-known linear interdependencies among solution variables, enabling significant reductions in computational costs and improvements in robustness and convergence.

Involute-evolute curves with modified orthogonal frame in Galilean space G 3

UDC 514 We introduce and study an involute-evolute curve pair within a modified orthogonal frame in the context of 3-dimensional Galilean space G 3 . The proposed methodology involves the investigation of the aforementioned curve pair with respect to a modified orthogonal frame, specifically by analyzing their curvature and torsion properties. By using this approach, we derive various characterizations of these curves. The proposed findings contribute to the deeper understanding of the geometric properties and the behavior of involute-evolute curve pairs in the Galilean space, thereby offering potential applications in the differential geometry and mathematical physics.

Application of Anal Manometry in the Planning for Anal Fistula Surgery

Anal fistula does not heal spontaneously without surgery, and anal function should be preserved after muscle dissection, which is a challenge even to the most experienced surgeon. In order to develop a planner for anal fistula surgery, anal manometry used in the measurement of pressures in the anal canal to investigate the anal function is reviewed. In this review paper, current techniques are described and compared with each other, and technical and clinical challenges are discussed. There are four types of catheters used to measure anal sphincter pressure: water-perfused, solid-state, air-coupled, and fiber optic catheter. Parameters acquired by anal manometry and their relationship with fecal incontinence after anal fistula surgery are discussed. Vectormanometry can provide the pressure profile along the anal canal in three dimensional space, pressure vectorgram, and cross-sectional radial asymmetry, which have more advantages than the conventional method. Understanding the technology and development of anal manometry is critical for the anal fistula surgical planning. A novel design is highly desired for 3D pressure profile measurement along the entire anal canal simultaneously without pulling the catheter.

Habitat dimensionality and feeding strategies but not temperature as determinants of body size-trophic structure relationship in a marine food web.

Disentangling the determinants of trophic structure is central to ecology. The capacity to capture subjugate and consume a prey (i.e. gape limitation) is a relevant limitation to acquire energy for most organisms, especially those in smaller size ranges. This generates a size hierarchy of trophic positions in which large organisms consume small ones. Body size is tightly correlated to gape limitation and explains a large fraction of variance in the body size-trophic position relationship. However, a considerable fraction of variance still remains to be explained. Consumer search space dimensionality (2D or 3D) and feeding strategies, temperature and the size structure of primary producers can alter the trophic structure, but tests based on information from natural food webs are scarce. We generated specific predictions about the body size trophic position relationship and evaluated them using information from a subtropical South Atlantic coastal marine ecosystem: benthic realm (2D, rocky shore and sandy beach) and the pelagic realm (3D). We characterized this marine coastal food web based on stable isotopes of carbon and nitrogen from 256 samples from primary producers (macroalgae and phytoplankton) to large predators (sand shark) in summer and winter. Consumer body size encompassed six orders of magnitude in weight from 10-2 to 6 × 104 g. Isotopic signal corresponded to an integration of carbon sources from basal consumers to top predators. The body size-trophic position relationship showed a linear positive association with different slopes for the benthic and pelagic environments. This implies a smaller predator prey size ratio for pelagic (3D) with respect to benthic consumers (2D) as theoretically expected. No seasonal differences were found in slopes and most of the overall variance in benthic environments was largely explained by feeding strategies of the different taxonomic groups. We provide an integrated evaluation on the role of body size, consumer search space and feeding strategy to understand the determinants of trophic position. Results demonstrate that integrating gape limitation hypothesis, the dimensionality of consumer search space and feeding strategies into a formal robust framework to understand trophic structure is feasible even in complex natural ecosystems.

Description Generation Using Variational Auto-Encoders for Precursor microRNA

Micro RNAs (miRNA) are a type of non-coding RNA involved in gene regulation and can be associated with diseases such as cancer, cardiovascular, and neurological diseases. As such, identifying the entire genome of miRNA can be of great relevance. Since experimental methods for novel precursor miRNA (pre-miRNA) detection are complex and expensive, computational detection using Machine Learning (ML) could be useful. Existing ML methods are often complex black boxes that do not create an interpretable structural description of pre-miRNA. In this paper, we propose a novel framework that makes use of generative modeling through Variational Auto-Encoders to uncover the generative factors of pre-miRNA. After training the VAE, the pre-miRNA description is developed using a decision tree on the lower dimensional latent space. Applying the framework to miRNA classification, we obtain a high reconstruction and classification performance while also developing an accurate miRNA description.

Deep reinforcement learning-based multi-objective optimization for electricity–gas–heat integrated energy systems

With the increasing global attention on energy efficiency and carbon emissions, the optimization of integrated energy systems (IES) has become the key to improve energy efficiency and reduce pollution emissions. However, most of the existing optimization methods cannot effectively deal with the complexity of high dimensional continuous action space. Therefore, this paper focuses on a novel multi-objective optimization strategy for the electricity–gas–heat integrated energy systems (EGH-IES). Firstly, considering the absorption capacity of wind power and the emission of pollutant gases, a multi-objective optimization model is constructed based on the mechanism model and operation constraints of each device in EGH-IES, in which the integrated operation cost and the environmental factors are taken as optimization objectives. Then, the multi-objective optimization problem is designed as the optimal strategy of interaction learning between agent and environment in reinforcement learning, and the output power of the devices constitutes the action of reinforcement learning. Additionally, the Ornstein–Uhlenbeck process is introduced to enhance the training efficiency and exploration performance of the agent, and the deep deterministic policy gradients (DDPG) algorithm is employed to optimize the action, thus the output power of the appliances could be obtained. Finally, the simulation results show that compared with deep Q network (DQN) method and proximal policy optimization (PPO) method, the reward function value of the proposed method increases by 2.43% and 6.09%, respectively, which represents a reduction in economic cost and pollutant emissions. These verify the effectiveness and superiority of the proposed multi-objective optimization scheme in cost reduction and benefit improvement for the EGH-IES.

First Principles Numerical Demonstration of Emergent Decoherent Histories

Within the histories formalism the decoherence functional is a formal tool to investigate the emergence of classicality in isolated quantum systems, yet an explicit evaluation of it from first principles has not been reported. We provide such an evaluation for up to five-time histories based on exact numerical diagonalization of the Schrödinger equation. We find a robust emergence of decoherence for slow and coarse observables of a generic random matrix model and extract a finite-size scaling law by varying the Hilbert space dimension over 4 orders of magnitude. Specifically, we conjecture and observe an exponential suppression of coherent effects as a function of the particle number of the system. This suggests a solution to the preferred basis problem of the many-worlds interpretation (or the set selection problem of the histories formalism) within a minimal theoretical framework without relying on environmentally induced decoherence, quantum Darwinism, Markov approximations, low-entropy initial states, or ensemble averages. Published by the American Physical Society 2024

A hybrid finite element method for moving-habitat models in two spatial dimensions

Moving-habitat models track the density of a population whose suitable habitat shifts as a consequence of climate change. Whereas most previous studies in this area consider 1-dimensional space, we derive and study a spatially 2-dimensional moving-habitat model via reaction-diffusion equations. The population inhabits the whole space. The suitable habitat is a bounded region where population growth is positive; the unbounded complement of its closure is unsuitable with negative growth. The interface between the two habitat types moves, depicting the movement of the suitable habitat poleward. Detailed modelling of individual movement behaviour induces a nonstandard discontinuity in the density across the interface. For the corresponding semi-discretised system we prove well-posedness for a constant shifting velocity before constructing an implicit-explicit hybrid finite element method. In this method, a Lagrange multiplier weakly imposes the jump discontinuity across the interface. For a stationary interface, we derive optimal a priori error estimates over a conformal mesh with nonconformal discretisation. We demonstrate with numerical convergence tests that these results hold for the moving interface. Finally, we demonstrate the strength of our hybrid finite element method with two biologically motivated cases, one for a domain with a curved boundary and the other for non-constant shifting velocity.

Witt’s Hyperbola Is Both Predicted and Observed to Pass Close to the Lensing Galaxies in Quadruple Quasars

When a rectangular hyperbola is constructed from the image positions of a quadruply lensed quasar, as proposed by Witt, it passes very close to the lensing galaxy. The median measured perpendicular offset between the observed light center of the lens and Witt’s hyperbola is 0.″013 for a sample for 39 systems lensed by a relatively isolated galaxy. The family of lens models adopted by Witt predicts that the lens lies on the hyperbola, but its position is not used for its construction. The median offset corresponds to roughly 1% of the Einstein ring radius and suggests that the centers of the lensing potential are close to the light centers of the lens. By putting a restrictive prior on the perpendicular distance to Witt’s hyperbola (or on the distance between the galaxy and the potential), one reduces by one the dimensionality of a model space when fitting data. Taking the brightest pixel of a lensing galaxy as its center avoids a shortcoming of using the average light center for a constraint.

A method for in silico exploration of potential glioblastoma multiforme attractors using single-cell RNA sequencing

We presented a method to find potential cancer attractors using single-cell RNA sequencing (scRNA-seq) data. We tested our method in a Glioblastoma Multiforme (GBM) dataset, an aggressive brain tumor presenting high heterogeneity. Using the cancer attractor concept, we argued that the GBM’s underlying dynamics could partially explain the observed heterogeneity, with the dataset covering a representative region around the attractor. Exploratory data analysis revealed promising GBM’s cellular clusters within a 3-dimensional marker space. We approximated the clusters’ centroid as stable states and each cluster covariance matrix as defining confidence regions. To investigate the presence of attractors inside the confidence regions, we constructed a GBM gene regulatory network, defined a model for the dynamics, and prepared a framework for parameter estimation. An exploration of hyperparameter space allowed us to sample time series intending to simulate myriad variations of the tumor microenvironment. We obtained different densities of stable states across gene expression space and parameters displaying multistability across different clusters. Although we used our methodological approach in studying GBM, we would like to highlight its generality to other types of cancer. Therefore, this report contributes to an advance in the simulation of cancer dynamics and opens avenues to investigate potential therapeutic targets.

Quantum nonlocality of density operators and their corresponding density matrices

Abstract Quantum nonlocality represents correlations between subsystems of a composite quantum system, usually including Bell nonlocality, steerability, and entanglement. According to the hypothesis of quantum mechanics, states of a quantum system $Q$ described by a $d$-dimensional Hilbert space $\H_Q$ are denoted by density operators acting on $\H_Q$. Under a basis $e$ for the Hilbert space $\H_A\otimes \H_B$, every abstract density operator $\rho$ of the system $AB$ corresponds to a density matrix ${\rho}_{e}$, which is a state of the {$d_Ad_B$-dimensional complex Hilbert space } $\C^{d_A}\otimes\C^{d_B}$. In this work, we discuss the consistency of&amp;#xD; quantum nonlocality of density operators $\rho$ and their corresponding density matrices ${\rho}_e$ under the chosen basis $e$. It is proved that only when a basis $e$ is a product one, a density operator $\rho$ is entangled (resp., Bell nonlocal, steerable) if and only if its density matrix ${\rho}_e$ is entangled (resp., Bell nonlocal, steerable).

The solenoidal Virasoro algebra and its simple weight modules

The aim of this paper is to construct a one dimensional universal central extension of the solenoidal Lie algebra W ( n ) μ introduced by Y. Billig and V. Futorny. The obtained Lie algebra is higher rank Virasoro algebra denoted by Vir ( n ) μ . Moreover, we study Vir ( n ) μ -Harich-Chandra modules. We obtain two kinds of Harich-Chandra modules, the intermediate series modules and generalized weight modules. Furthermore, we use the triangular decomposition of Vir ( n ) μ given by lexicographic order on Z n to construct new weight modules having infinite dimensional weight spaces.