Sequence Space Research Articles

Exact Solutions of One Nonlinear Countable-Dimensional System of Integro-Differential Equations

In the present paper, a nonlinear countable-dimensional system of integrodifferential equations is investigated, whose vector of unknowns is a countable set of functions of two variables. These variables are interpreted as spatial coordinate and time. The nonlinearity of this system is constructed from two simultaneous convolutions: first convolution is in the sense of functional analysis and the second one is in the sense of linear space of double-sided sequences. The initial condition for this system is a doublesided sequence of functions of one variable defined on the entire real axis. The system itself can be written as a single abstract equation in the linear space of double-sided sequences. As the system may be resolved with respect to the time derivative, it may be presented as a dynamical system. The solution of this abstract equation can be interpreted as an approximation of the solution of a nonlinear integro-differential equation, whose unknown function depends not only on time, but also on two spatial variables. General representation for exact solution of system under study is obtained in the paper. Also two kinds of particular examples of exact solutions are presented. The first demonstrates oscillatory spatio-temporal behavior, and the second one shows monotone in time behavior. In the paper typical graphs of the first components of these solutions are plotted. Moreover, it is demonstrated that using some procedure one can generate countable set of new exact system’s solutions from previously found solutions. From radio engineering point of view this procedure just coincides with procedure of upsampling in digital signal processing.

Certain aspects of ℐ-statistical supremum and ℐ-statistical infimum of real-valued sequences

Abstract Over the last few years, numerous researchers have contributed significantly to summability theory by connecting various notions of convergence concepts of sequences. In this paper, we introduce the concepts of ℐ {\mathcal{I}} -statistical supremum and ℐ {\mathcal{I}} -statistical infimum of a real-valued sequence and study some fundamental features of the newly introduced notions. We also introduce the concept of ℐ {\mathcal{I}} -statistical monotonicity and establish the condition under which an ℐ {\mathcal{I}} -statistical monotonic sequence is ℐ {\mathcal{I}} -statistical convergent. We end up giving a necessary and a sufficient condition for the ℐ {\mathcal{I}} -statistical convergence of a real-valued sequence.

Context-dependence of T-loop Mediated Long-range RNA Tertiary Interactions

The architecture and folding of complex RNAs is governed by a limited set of highly recurrent structural motifs that form long-range tertiary interactions. One of these motifs is the T-loop, which was first identified in tRNA but is broadly distributed across biological RNAs. While the T-loop has been examined in detail in different biological contexts, the various receptors that it interacts with are not as well defined. In this study, we use a cell-based genetic screen in concert with bioinformatic analysis to examine three different, but related, T-loop receptor motifs found in the flavin mononucleotide (FMN) and cobalamin (Cbl) riboswitches. As a host for different T-loop receptors, we employed the env8 class-II Cbl riboswitch, an RNA that uses two T-loop motifs for both folding and supporting the ligand binding pocket. A set of libraries was created in which select nucleotides that participate in the T-loop/T-loop receptor (TL/TLR) interaction were fully randomized. Library members were screened for their ability to support Cbl-dependent expression of a reporter gene. While T-loops appear to be variable in sequence, we find that the functional sequence space is more restricted in the Cbl riboswitch, suggesting that TL/TLR interactions are context dependent. Our data reveal clear sequence signatures for the different types of receptor motifs that align with phylogenic analysis of these motifs in the FMN and Cbl riboswitches. Finally, our data suggest the functional contribution of various nucleobase-mediated long-range interactions within the riboswitch subclass of TL/TLR interactions that are distinct from those found in other RNAs.

Topological Properties of Jordan Intuitionistic Fuzzy Normed Spaces

AbstractThe article delves into the task of formulation of meticulous sequence spaces whose elements’ convergence is a generalized version of the Cauchy convergence in the setting of intuitionistic fuzzy norm. The task of attaining a finite limit is attained via an infinite matrix operator, namely, Jordan totient operator. We discuss some topological and algebraic properties and establish some inclusion relations between the spaces.

Within-host genetic diversity of SARS-CoV-2 lineages in unvaccinated and vaccinated individuals

Viral and host factors can shape SARS-CoV-2 evolution. However, little is known about lineage-specific and vaccination-specific mutations that occur within individuals. Here, we analysed deep sequencing data from 2,820 SARS-CoV-2 respiratory samples with different viral lineages to describe the patterns of within-host diversity under different conditions, including vaccine-breakthrough infections. In unvaccinated individuals, variant of Concern (VOC) Alpha, Delta, and Omicron respiratory samples were found to have higher within-host diversity and were under neutral to purifying selection at the full genome level compared to non-VOC SARS-CoV-2. Breakthrough infections in 2-dose or 3-dose Comirnaty and CoronaVac vaccinated individuals did not increase levels of non-synonymous mutations and did not change the direction of selection pressure. Vaccine-induced antibody or T cell responses did not appear to have significant impact on within-host SARS-CoV-2 sequence diversification. Our findings suggest that vaccination does not increase exploration of SARS-CoV-2 protein sequence space and may not facilitate emergence of viral variants.

Hilbert I-Statistical Convergence on Neutrosophic Normed Spaces

In this paper, λI-statistical convergence is defined to generalize statistical convergence on Neutrosophic normed spaces. As it is known, Neutrosophic theory, which brings a new breath to daily life and complex scientific studies which we encounter with many uncertainties, is a rapidly developing field with many new study subjects. Thus, researchers show great interest in this philosophical approach and try to transfer related topics to this field quickly. For this purpose, in this study, besides the definition of λI-statistical convergence, the important features of Hilbert sequence space and λI-statistical convergence in Neutrosophic spaces are examined with the help of these defined sequences. By giving the relationship between Hilbert λI-statistical convergence and Hilbert I-statistical convergence, it has been evaluated whether the definitions contain a coverage relationship as in fuzzy and intuitionistic fuzzy. As a result, it is thought that the selected convergence type is suitable for the Neutrosophic normed space structure and is a guide for new convergence types.

Spectral Decompositions of the Difference Operator Δ^m over the Sequence Space cs

This study aims to bring together some studies on the spectra of difference operators in the literature over the cs sequence space and to provide a basis for related problems. So far, the problem has been solved up to a maximum of 2 orders on the sequence space cs. In this article, we discuss the difference operator Δ^m, represented by a (m+1) banded matrix, which generalizes the difference operators of the form Δ, Δ^2, B(r,s) and B(r,s,t) and we will give its boundedness, spectrum, fine spectrum and some spectral separations over the sequence space cs.

Large structures within the class of summing operators

We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and (α,β)-lineability. Among other results, we prove that the class of multilinear operators taking values on ℓq(0<q≤∞) that are absolutely but not multiple summing is pointwise c-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also studied.

On Some Topological and Geometric Properties of Some q-Cesáro Sequence Spaces

Mathematical concepts are aesthetic tools that are useful to create methods or solutions to real-world problems in theory and practice, and that sometimes contain symmetrical and asymmetrical structures due to the nature of the problems. In this study, we investigate whether the sequence spaces Xpq, 0≤p&lt;∞, and X∞, which are constructed by q-Cesáro matrix, satisfy some of the further properties described with respect to the bounded linear operators on them. More specifically, we answer to the question: “Which of these spaces have the Approximation, Dunford-Pettis, Radon–Riesz and Hahn–Banach extension properties?”. Furthermore, we try to investigate some geometric properties such as rotundity and smootness of these spaces.

A New Notion of Paranorm Intuitionistic Fuzzy Zweier I&lt;sub&gt;3&lt;/sub&gt;-Convergent Triple Sequence Spaces

In this paper, we define and introduce the notion of paranorm intuitionistic fuzzy Zweier I3-convergent triple sequence spaces ZI3(μ,υ) (p) and ZIO3(μ,υ) (p) for p = (pnmj ) a triple sequence of positive real numbers. Besides, the fuzzy topology on these spaces is studied.

Domain-wise dissection of thermal stability enhancement in multidomain proteins

Stability is critical for the proper functioning of all proteins. Optimization of protein thermostability is a key step in the development of industrial enzymes and biologics. Herein, we demonstrate that multidomain proteins can be stabilized significantly using domain-based engineering followed by the recombination of the optimized domains. Domain-level analysis of designed protein variants with similar structures but different thermal profiles showed that the independent enhancement of the thermostability of a constituent domain improves the overall stability of the whole multidomain protein. The crystal structure and AlphaFold-predicted model of the designed proteins via domain-recombination provided a molecular explanation for domain-based stepwise stabilization. Our study suggests that domain-based modular engineering can minimize the sequence space for calculations in computational design and experimental errors, thereby offering useful guidance for multidomain protein engineering.

An Infinite System of Fractional Sturm–Liouville Operator with Measure of Noncompactness Technique in Banach Space

In the current contribution, an appropriate quantity connected to the space of all convergent sequences is provided and shown to be a measure of noncompactness in a Banach space. Through the application of the fixed point theorems of Darbo and Meir–Keeler, this amount is used to discuss whether a solution to an infinite system of fractional Sturm–Liouville operators exists. We offer a numerical example as an application of the key finding in the study.

Positively limited sets in Banach lattices

We introduce and study the class of positively limited sets in Banach lattices, that is, sets on which every weak⁎ null sequence of positive functionals converges uniformly to zero. Moreover the relationships between the class of positively limited sets and other known classes of sets such as almost limited sets and (weakly) compact sets are discussed. Some properties of Banach lattices can be characterized in terms of positively limited sets.

Bounded Linear Operators on Function Spaces and Sequences Spaces

Bounded Linear Operators on Function Spaces and Sequences Spaces

Mappings Ideal of Type the Domain of Generalized p , q -Euler Matrix in c 0 and c

We develop new Banach sequence spaces e 0 a , b p , q and e c a , b p , q derived by the domain of generalized p , q -Euler matrix E a , b p , q in the spaces of null and convergent sequences, respectively. We investigate some topological properties and inclusion natures related to these spaces. We construct bases and obtain α , β , and γ -duals of the spaces e 0 a , b p , q and e c a , b p , q . Certain classes of matrix transformations are characterized from e 0 a , b p , q and e c a , b p , q to Z ∈ ℓ ∞ , c , c 0 , ℓ 1 , ℓ k . We obtain essential conditions of compactness of operators from e 0 a , b p , q and e c a , b p , q to Z ∈ ℓ ∞ , c , c 0 , ℓ 1 , b s , c s , c s 0 . Finally, under a definite functional ϱ and a weighted sequence of positive reals δ , we define a new sequence space e 0 a , b p , q , δ ϱ . Certain geometric and topological properties of this space along with the eigenvalue distribution of mapping ideals due to this space and s -numbers are investigated.

A categorical derivation of Lebesgue integration

AbstractWe identify simple universal properties that uniquely characterize the Lebesgue spaces. There are two main theorems. The first states that the Banach space , equipped with a small amount of extra structure, is initial as such. The second states that the functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. We use the universal properties to derive some of the basic elements of integration theory. We also state universal properties characterizing the sequence spaces and , as well as the functor taking values in Hilbert spaces.

Some observations on Nörlund ℐ2-statistical convergence of double sequences in intuitionistic fuzzy normed spaces

The aim of this paper is to investigate the intuitionistic Nörlund [Formula: see text]-statistically convergent sequence space. We present some intuitionistic fuzzy normed spaces (IFNS) in Nörlund convergent spaces. Moreover, we also put forward several topological and algebraic properties of these convergent sequence spaces.

Monotone ODEs with discontinuous vector fields in sequence spaces

We consider a system of ODE in a Fréchet space with unconditional Schauder basis. The right side of the ODE is a discontinuous function. Under certain monotonicity conditions we prove an existence theorem for the corresponding initial value problem. We employ an idea of the partial order which seems to be new in this field.

Predicting binding affinity changes from long-distance mutations using molecular dynamics simulations and Rosetta.

Computationally modeling how mutations affect protein-protein binding not only helps uncover the biophysics of protein interfaces, but also enables the redesign and optimization of protein interactions. Traditional high-throughput methods for estimating binding free energy changes are currently limited to mutations directly at the interface due to difficulties in accurately modeling how long-distance mutations propagate their effects through the protein structure. However, the modeling and design of such mutations is of substantial interest as it allows for greater control and flexibility in protein design applications. We have developed a method that combines high-throughput Rosetta-based side-chain optimization with conformational sampling using classical molecular dynamics simulations, finding significant improvements in our ability to accurately predict long-distance mutational perturbations to protein binding. Our approach uses an analytical framework grounded in alchemical free energy calculations while enabling exploration of a vastly larger sequence space. When comparing to experimental data, we find that our method can predict internal long-distance mutational perturbations with a level of accuracy similar to that of traditional methods in predicting the effects of mutations at the protein-protein interface. This work represents a new and generalizable approach to optimize protein free energy landscapes for desired biological functions.

Profiling DNA Ligase Substrate Specificity with a Pacific Biosciences Single-Molecule Real-Time Sequencing Assay.

DNA ligases catalyze the joining of breaks in nucleic acid backbones and are essential enzymes for in vivo genome replication and repair across all domains of life. These enzymes are also critically important to in vitro manipulation of DNA in applications such as cloning, sequencing, and molecular diagnostics. DNA ligases generally catalyze the formation of a phosphodiester bond between an adjacent 5'-phosphate and 3'-hydroxyl in DNA, but they exhibit different substrate structure preferences, sequence-dependent biases in reaction kinetics, and variable tolerance for mismatched base pairs. Information on substrate structure and sequence specificity can inform both biological roles and molecular biology applications of these enzymes. Given the high complexity of DNA sequence space, testing DNA ligase substrate specificity on individual nucleic acid sequences in parallel rapidly becomes impractical when a large sequence space is investigated. Here, we describe methods for investigating DNA ligase sequence bias and mismatch discrimination using Pacific Biosciences Single-Molecule Real-Time (PacBio SMRT) sequencing technology. Through its rolling-circle amplification methodology, SMRT sequencing can give multiple reads of the same insert. This feature permits high-quality top- and bottom-strand consensus sequences to be determined while preserving information on top-bottom strand mismatches that can be obfuscated or lost when using other sequencing methods. Thus, PacBio SMRT sequencing is uniquely suited to measuring substrate bias and enzyme fidelity through multiplexing a diverse set of sequences in a single reaction. The protocols describe substrate synthesis, library preparation, and data analysis methods suitable for measuring fidelity and bias of DNA ligases. The methods are easily adapted to different nucleic acid substrate structures and can be used to characterize many enzymes under a variety of reaction conditions and sequence contexts in a rapid and high-throughput manner. © 2023 New England Biolabs and The Authors. Current Protocols published by Wiley Periodicals LLC. Basic Protocol 1: Preparation of overhang DNA substrates for ligation Basic Protocol 2: Preparation of ligation fidelity libraries Support Protocol 1: Preparation of ligation libraries for PacBio Sequel II sequencing Support Protocol 2: Loading and sequencing of a prepared library on the Sequel II instrument Basic Protocol 3: Computational processing of ligase fidelity sequencing data.