Sequence Of Spaces Research Articles

Inverse Sturm–Liouville Problem with Spectral Parameter in the Boundary Conditions

In this paper, for the first time, we study the inverse Sturm–Liouville problem with polynomials of the spectral parameter in the first boundary condition and with entire analytic functions in the second one. For the investigation of this new inverse problem, we develop an approach based on the construction of a special vector functional sequence in a suitable Hilbert space. The uniqueness of recovering the potential and the polynomials of the boundary condition from a part of the spectrum is proved. Furthermore, our main results are applied to the Hochstadt–Lieberman-type problems with polynomial dependence on the spectral parameter not only in the boundary conditions but also in discontinuity (transmission) conditions inside the interval. We prove novel uniqueness theorems, which generalize and improve the previous results in this direction. Note that all the spectral problems in this paper are investigated in the general non-self-adjoint form, and our method does not require the simplicity of the spectrum. Moreover, our method is constructive and can be developed in the future for numerical solution and for the study of solvability and stability of inverse spectral problems.

Linker Length Drives Heterogeneity of Multivalent Complexes of Hub Protein LC8 and Transcription Factor ASCIZ.

LC8, a ubiquitous and highly conserved hub protein, binds over 100 proteins involved in numerous cellular functions, including cell death, signaling, tumor suppression, and viral infection. LC8 binds intrinsically disordered proteins (IDPs), and although several of these contain multiple LC8 binding motifs, the effects of multivalency on complex formation are unclear. Drosophila ASCIZ has seven motifs that vary in sequence and inter-motif linker lengths, especially within subdomain QT2-4 containing the second, third, and fourth LC8 motifs. Using isothermal-titration calorimetry, analytical-ultracentrifugation, and native mass-spectrometry of QT2-4 variants, with methodically deactivated motifs, we show that inter-motif spacing and specific motif sequences combine to control binding affinity and compositional heterogeneity of multivalent duplexes. A short linker separating strong and weak motifs results in stable duplexes but forms off-register structures at high LC8 concentrations. Contrastingly, long linkers engender lower cooperativity and heterogeneous complexation at low LC8 concentrations. Accordingly, two-mers, rather than the expected three-mers, dominate negative-stain electron-microscopy images of QT2-4. Comparing variants containing weak-strong and strong-strong motif combinations demonstrates sequence also regulates IDP/LC8 assembly. The observed trends persist for trivalent ASCIZ subdomains: QT2-4, with long and short linkers, forms heterogeneous complexes, whereas QT4-6, with similar mid-length linkers, forms homogeneous complexes. Implications of linker length variations for function are discussed.

Generative power of a protein language model trained on multiple sequence alignments.

Computational models starting from large ensembles of evolutionarily related protein sequences capture a representation of protein families and learn constraints associated to protein structure and function. They thus open the possibility for generating novel sequences belonging to protein families. Protein language models trained on multiple sequence alignments, such as MSA Transformer, are highly attractive candidates to this end. We propose and test an iterative method that directly employs the masked language modeling objective to generate sequences using MSA Transformer. We demonstrate that the resulting sequences score as well as natural sequences, for homology, coevolution, and structure-based measures. For large protein families, our synthetic sequences have similar or better properties compared to sequences generated by Potts models, including experimentally validated ones. Moreover, for small protein families, our generation method based on MSA Transformer outperforms Potts models. Our method also more accurately reproduces the higher-order statistics and the distribution of sequences in sequence space of natural data than Potts models. MSA Transformer is thus a strong candidate for protein sequence generation and protein design.

Generative modeling of short, disordered proteins with homogeneous sequence composition.

Protein design has seen remarkable progress in the past decade, with numerous examples of de novo proteins with novel topologies and functions being successfully synthesized. Computational tools have played a large role in the ability to rationally design proteins. Recently, there have been numerous successes applying deep learning techniques to protein design that have demonstrated comparable or significantly improved performance over traditional energy-based approaches. However, the protein design problem has been less well explored for disordered proteins or, more generally, proteins with conformational heterogeneity. In this work, we demonstrate that if one approximates the spatial output of a coarse-grained molecular dynamics simulation as a multivariate normal distribution parameterized by a mean vector (representing an ensemble-averaged pairwise distance map) and covariance matrix, one can train a generative model to learn the distribution of these parameters across a set of sequences. Specifically, we encoded the mean vector and covariance matrix for each sequence in a low-dimensional space via a fixed linear transformation and trained a masked auto-encoder to accurately learn the distribution of this low-dimensional output. Furthermore, by sampling from the masked auto-encoder and transforming the generated samples back into their original high-dimensional space, one can generate realistic, ensemble-averaged pairwise distance maps. These results were demonstrated on coarse-grained simulation data derived from approximately 2000 distinct sequences, each sequence being 24 residues in length and consisting exclusively of glycine, serine, glutamate, and lysine. Though this set of sequences is relatively homogeneous in composition, we speculate our approach can be applied to disordered sequences of longer length and more heterogeneous composition, given the appropriate training set.

Internal Auditory Canal (IAC) and Cerebellopontine Angle (CPA): Comparison between T2-weighted SPACE and 3D-CISS sequences at 1.5T

MR cisternography relies on three-dimensional (3D) heavily T2-weighted imaging techniques. It can be used to assess the internal auditory canal (IAC) and Cerebellopontine Angle (CPA) anatomy and pathology. There are two techniques usually used for this purpose; T2-weighted SPACE and Three-Dimensional Constructive Interference in Steady State (3D-CISS). The current study aims to evaluate differences in image quality between 3D-CISS and T2-weighted SPACE sequences and to compare their ability to differentiate the anatomy of the CPA and IAC structures. This study was conducted in 64 patients. Each patient was imaged with both 3D-CISS and T2-weighted SPACE sequences, then the images were also reviewed by two radiologists for anatomy evaluation. The results showed that T2-weighted SPACE outperforms 3D-CISS for the assessments of IAC, cranial nerve, cochlear anatomic structures, semicircular canals, and vestibule. The study also showed there is no statically significant difference between the two sequences according to the gender and the age of patients.

On p-convergent sequences and p-Cauchy sequences in fuzzy metric spaces

In this note, we answer a recent question posed by Gregori et al. (2022) [8]. We give a negative answer for the question by means of Example 3.2.

Ideal convergence and ideal Cauchy sequences in intuitionistic fuzzy metric spaces

The present study introduces the concepts of ideal convergence (I-convergence), ideal Cauchy (I-Cauchy) sequences, I *-convergence, and I *-Cauchy sequences in intuitionistic fuzzy metric spaces. It defines I-limit and I-cluster points as a sequence in these spaces. Afterward, it examines some of their basic properties. Lastly, the paper discusses whether phenomena should be further investigated.

New convergence definitions for double sequences in g-metric spaces

New convergence definitions for double sequences in g-metric spaces

Frames generated by double sequences in Hilbert spaces

In this paper, we introduce frames generated by double sequences (d-frame) in Hilbert spaces and describe some of their properties. Furthermore, we discuss frame operators, alternate dual frames and stability for d-frames.

A simple screening method for distillation sequence of thermally coupled dividing wall column: distillation sequence morphological analysis method (DSMAM)

AbstractBACKGROUNDDistillation is a separation process with a high energy consumption. Therefore, energy‐saving optimizations of the distillation process have become particularly significant. However, developing methods to quickly choose the best energy‐saving distillation sequence for the separation of multiple components has become a key problem.RESULTSIn this paper, the Distillation Sequence Morphometry Analysis Method (DSMAM) was established to quickly screen energy‐saving and efficient distillation sequences and was integrated with a Dividing‐wall column (DWC) device, which has more industrial significance. Meanwhile, an evaluation and screening index η of separated multicomponent energy‐saving distillation sequences was proposed. Finally, based on the DSMAM, and combined with Microsoft Visual Basic 6.0 (VB) and Aspen Plus V11 (Aspen) software, a set of distillation sequence screening and evaluation programs of four‐component separation systems was generated. The distillation column sequence with high thermodynamic efficiency and the energy‐saving coupling structure of distillation were obtained quickly and accurately.CONCLUSIONThis program was applied to the separation of the n‐butane‐benzene‐n‐heptane‐n‐nonane system. It was able to quickly obtain the first three energy‐saving distillation sequences without thermal coupling and the first three energy‐saving distillation sequences in the search space. The direct‐thermal coupling distillation sequence of the energy‐saving fluid was integrated and coupled, providing a preliminary design reference for industrial design and application. © 2022 Society of Chemical Industry (SCI).

Abstract 1691: Asp251 of the DEAF1 KDWK binding motif influences binding to flexibly spaced consensus sequences

Abstract 1691: Asp251 of the DEAF1 KDWK binding motif influences binding to flexibly spaced consensus sequences

Large-<inline-formula><tex-math id="M1">$ N $</tex-math></inline-formula> Limits of Spaces and Structures

For a sequence of spaces $ X_N $, with topological, algebraic or measure-theoretic structures, we show how a large-$ N $ limit $ X_\infty $ with corresponding structures is obtained. For example, when each space is a topological group $ G_N $, such as $ G_N = U(N) $, a limiting group $ G_\infty $ with topology results. Using the Weil-Kodaira construction, for compact topological groups $ G_N $ equipped with normalized Haar measures, we obtain a topological structure on $ G_\infty $ that also makes the group operations continuous. When each $ G_N $ is a Lie group we describe a Lie algebra associated to $ G_\infty $.

A Study on Image Distortion of 3D-T1-SPACE with Gd Contrast Agent

In order to increase the qualitative value of MRI and reduce the relative image acquisition time, the mutual intersection of Partial Fourier and Average was conducted to evaluate image distortion and acquisition time according to the effects of FID and Susceptibility Artifact in the T1 3D SPACE Sequence. As a result of analyzing the Roundness (%) of the phantom for Gd contrast agent concentration, when the Partial Fourier 8/8 was fixed, as the Average increased, the Roundness (%) increased (distortion decreased). When the Average 2.0 was fixed, as the Partial Fourier was reduced, the Roundness (%) decreased (distortion increased). As a result of Correlation analysis on Roundness (%), Partial Fourier and Average showed a high positive correlation. Experimental results on image distortion and acquisition time when using the T1 3D SPACE sequence showed that reducing the Partial Fourier than the Average reduced the image distortion and shortened the acquisition time.

Further results on strong λ-statistical convergence of sequences in probabilistic metric spaces

In this paper we study some basic properties of strong λ-statistical convergence of sequences in probabilistic metric spaces. Also introducing the concept of strong λ-statistically Cauchy sequences we study its relationship with strong λ-statistical convergence in a probabilistic metric space. Further introducing the notions of strong λ-statistical limit point and strong λ-statistical cluster point of a sequence in a probabilistic metric space we examine their interrelationship.

Asymptotically lacunary µ-statistical equivalence of generalized difference sequences in probabilistic normed spaces

The current article introduces the notion of asymptotically lacunary $(\Delta^n,\mu)$-statistical equivalent sequence in the settings of a probabilistic norm $N$. Furthermore, the article presents the concepts of asymptotically $(\Delta^n,\mu)$-strongly Ces\'{a}ro equivalent sequences and asymptotically $(\Delta^n,\mu)$-strongly Ces\'{a}ro Orlicz equivalent sequences in the theory of probabilistic normed spaces and also investigates their various properties including some inclusion relations as well as some equivalent conditions in this new settings.

High-resolution 3D T2-weighted SPACE sequence with compressed sensing for the prostate gland: diagnostic performance in comparison with conventional T2-weighted images.

To compare the diagnostic performance of high-resolution 3D T2-weighted imaging (T2WI) with compressed sensing (CS) sampling perfection with application-optimized contrasts using different flip angle evolution (SPACE) to that of conventional T2WI with turbo spin echo (TSE) in prostate MRI. This study evaluated 179 patients (mean age 69.1 ± 9.3) who underwent prostate biopsy after prostate prebiopsy MRI, including two sets of three-plane T2WI with TSE (thickness: 3mm, scan time: 10min 4s) and CS SPACE (thickness: 0.6mm, scan time: 4min 55s). Two radiologists evaluated two sets of images with the Prostate Imaging-Reporting and Data System (PIRADS) classification and determined the extraprostatic extension (EPE) of the lesion. The diagnostic performance to detect prostate cancer (PIRADS classification) and EPE was compared between the two T2WI sets. Clinically significant cancer (CSC) was diagnosed in 103 patients (57.5%). Areas under the receiver operating characteristic curve of the PIRADS classification on both image sets with T2 TSE and T2 CS SPACE were higher than 0.7 and did not show significant differences for either radiologist (p > 0.05). EPE was confirmed in 25 of 70 patients underwent prostatectomy. For evaluating EPE on MRI, the sensitivity and specificity did not differ between the two T2WI sequences (p > 0.05). High-resolution 3D T2WI using CS SPACE, which was acquired within a shorter acquisition time than three-plane T2 TSE, showed comparable diagnostic performance to conventional T2 TSE for detecting CSC and evaluating EPE.

Two-Dimensional Block Trees

Abstract The Block Tree is a data structure for representing repetitive sequences in compressed space, which reaches space comparable with that of Lempel–Ziv compression while retaining fast direct access to any position in the sequence. In this paper, we generalize Block Trees to two dimensions, in order to exploit repetitive patterns in the representation of images, matrices and other kinds of bidimensional data. We demonstrate the practicality of the two-dimensional Block Trees (2D-BTs) in representing the adjacency matrices of Web graphs, and raster images in GIS applications. For this purpose, we integrate our 2D-BT with the $k^2$-tree—an efficient structure that exploits clustering and sparseness to compress adjacency matrices—so that it also exploits repetitive patterns. Our experiments show that this structure uses 60–80% of the space of the original $k^2$-tree, while being 30% faster to three times slower when accessing cells.

On the transfer of convergence between two sequences in Banach spaces

"Let $(X,\|\cdot\|)$ be a Banach space and $T:A\to X$ a contraction mapping, where $A\subset X$ is a closed set. Consider a sequence $\{x_n\}\subset A$ and define the sequence $\{y_n\}\subset X$, by $y_n=x_n+T\left(x_{\sigma(n)}\right)$, where $\{\sigma(n)\}$ is a sequence of natural numbers. We highlight some general conditions so that the two sequences $\{x_n\}$ and $\{y_n\}$ are simultaneously convergent. Both cases: 1) $\sigma(n)<n$, for all $n$, and 2) $\sigma(n)\ge n$, for all $n$, are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to sequences of mappings and some appropriate applications are also proposed."

Rough statistical convergence of sequences in gradual normed linear spaces

Rough statistical convergence of sequences in gradual normed linear spaces

Comparison between 2D TSE and 3D SPACE in the evaluation of craniocervical ligaments.

To evaluate the MRI findings of the craniocervical junction ligaments and compare the 2D TSE sequence and 3D SPACE sequence on cervical spine MRI. Eighty cervical spine MRIs were retrospectively analyzed, including 2D TSE and 3D SPACE sequences. The visibility and morphology of the transverse ligament (TL) and alar ligament (AL) were evaluated by two readers using a five-point grading scale for visibility (1 = not visible, 2 = barely visible, 3 = adequately visible, 4 = good visibility, 5 = excellent visibility) and a four-point grading scale for morphology (0 = homogeneous low SI with normal thickness, 1 = high SI with normal thickness, 2 = reduced thickness, 3 = full-thickness rupture or indistinguishable from surrounding structures). The grades of the two sequences were compared. Reader agreements were measured using the percentage of exact agreement. The visibility grade of TL was significantly higher in 3D, indicating better visualization. The morphology grade of TL was significantly lower in the 3D, indicating more normal-appearing ligament. The exact agreements of visibility and morphology grade of TL tended to be higher in 3D. On the other hand, there was no significant difference in visibility grade of AL between the two sequences. The morphological grade of AL was significantly higher in 3D. The agreement of the morphological grade of AL tended to be lower in 3D. The utilization of 3D sequences significantly improves the visibility of TL and could potentially reduce false-positive diagnoses of ligament injury.