Specific Functional Classes Research Articles

Different aspects of blow-up property for a nonlinear wave equation

In this work, we are concerned with a nonlinear wave equation with variable exponents. In the presence of the logarithmic nonlinear source, we established a global nonexistence result with negative initial data and without imposing the Sobolev Logarithmic Inequality. The blow-up time is established with upper bound and lower bound. In addition, under some conditions on the initial data and for a specific class of relaxation functions, we established an infinite time blow-up result.

A Specific Class of Harmonic Meromorphic Functions Associated with the Mittag-Leffler Transformation

This article uses the Mittag-Leffler transformation to explore a specific category of harmonic meromorphic functions. The Mittag-Leffler transformation is a crucial tool for analysing meromorphic functions and provides essential properties and insights into their behavior. The main focus of this study is on harmonic meromorphic functions that can be represented by the Mittag-Leffler transformation. Furthermore, this research introduces an innovative derivative operator that incorporates this transformation into the domain of harmonic meromorphic functions. The Mittag-Leffler transformation is widely recognised as a powerful technique for analysing various mathematical functions, especially those with fractional order derivatives. It improves our understanding of harmonic meromorphic functions and their inherent characteristics. The research findings highlight the effectiveness of this new derivative operator in unraveling the complexities of these functions. They provide valuable insights into their behavior and fundamental traits. Additionally, the study offers coefficient inequalities, the distortion theorem, distortion bounds, extreme points, convex combinations, and convolution analyses specifically tailored to functions within this particular class.

Sufficient Conditions for Linear Operators Related to Confluent Hypergeometric Function and Generalized Bessel Function of the First Kind to Belong to a Certain Class of Analytic Functions

Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases.

Classification of bioactive peptides: A systematic benchmark of models and encodings

Bioactive peptides are short amino acid chains possessing biological activity and exerting physiological effects relevant to human health. Despite their therapeutic value, their identification remains a major problem, as it mainly relies on time-consuming in vitro tests. While bioinformatic tools for the identification of bioactive peptides are available, they are focused on specific functional classes and have not been systematically tested on realistic settings. To tackle this problem, bioactive peptide sequences and functions were here gathered from a variety of databases to generate a unified collection of bioactive peptides from microbial fermentation. This collection was organized into nine functional classes including some previously studied and some unexplored such as immunomodulatory, opioid and cardiovascular peptides. Upon assessing their sequence properties, four alternative encoding methods were tested in combination with a multitude of machine learning algorithms, from basic classifiers like logistic regression to advanced algorithms like BERT. Tests on a total of 171 models showed that, while some functions are intrinsically easier to detect, no single combination of classifiers and encoders worked universally well for all classes. For this reason, we unified all the best individual models for each class and generated CICERON (Classification of bIoaCtive pEptides fRom micrObial fermeNtation), a classification tool for the functional classification of peptides. State-of-the-art classifiers were found to underperform on our realistic benchmark dataset compared to the models included in CICERON. Altogether, our work provides a tool for real-world peptide classification and can serve as a benchmark for future model development.

Causal Unit Selection using Tractable Arithmetic Circuits

The unit selection problem aims to find objects, called units, that optimize a causal objective function which describes the objects' behavior in a causal context (e.g., selecting customers who are about to churn but would most likely change their mind if encouraged). While early studies focused mainly on bounding a specific class of counterfactual objective functions using data, more recent work allows one to find optimal units exactly by reducing the causal objective to a classical objective on a meta-model, and then applying a variant of the classical Variable Elimination (VE) algorithm to the meta-model---assuming a fully specified causal model is available. In practice, however, finding optimal units using this approach can be very expensive because the used VE algorithm must be exponential in the constrained treewidth of the meta-model, which is larger and denser than the original model. We address this computational challenge by introducing a new approach for unit selection that is not necessarily limited by the constrained treewidth. This is done through compiling the meta-model into a special class of tractable arithmetic circuits that allows the computation of optimal units in time linear in the circuit size. We finally present empirical results on random causal models that show order-of-magnitude speedups based on the proposed method for solving unit selection.

Majorization property for specific classes of analytic functions

Majorization property for specific classes of analytic functions

Discerning Temporal Difference Learning

Temporal difference learning (TD) is a foundational concept in reinforcement learning (RL), aimed at efficiently assessing a policy's value function. TD(λ), a potent variant, incorporates a memory trace to distribute the prediction error into the historical context. However, this approach often neglects the significance of historical states and the relative importance of propagating the TD error, influenced by challenges such as visitation imbalance or outcome noise. To address this, we propose a novel TD algorithm named discerning TD learning (DTD), which allows flexible emphasis functions—predetermined or adapted during training—to allocate efforts effectively across states. We establish the convergence properties of our method within a specific class of emphasis functions and showcase its promising potential for adaptation to deep RL contexts. Empirical results underscore that employing a judicious emphasis function not only improves value estimation but also expedites learning across diverse scenarios.

Stability tests for neutral‐type delay systems: A delay Lyapunov matrix and piecewise linear approximation of the functional argument approach

AbstractNew necessary and sufficient stability tests for neutral‐type linear time delay systems are presented. They are based on functionals with prescribed derivative depending on the delay Lyapunov matrix and recent stability/instability theorems. A piecewise linear approximation of the initial function argument in the functional is employed. As a result, the functional is approximated via a quadratic form whose matrix provides stability criteria. This matrix involves the integrals based on the delay Lyapunov matrix which are handled via an efficient computational procedure. The main feature of these tests is that they are verified in a finite number of operations, which is achieved through rigorous bounding of the approximation error on a specific class of functions. Some examples which validate the obtained results are presented and discussed.

Results of Third-Order Strong Differential Subordinations

In this paper, we present and investigate the notion of third-order strong differential subordinations, unveiling several intriguing properties within the context of specific classes of admissible functions. Furthermore, we extend certain definitions, presenting novel and fascinating results. We also derive several interesting properties of the results of third-order strong differential subordinations for analytic functions associated with the Srivastava–Attiya operator.

Computational components of visual predictive coding circuitry.

If a full visual percept can be said to be a 'hypothesis', so too can a neural 'prediction' - although the latter addresses one particular component of image content (such as 3-dimensional organisation, the interplay between lighting and surface colour, the future trajectory of moving objects, and so on). And, because processing is hierarchical, predictions generated at one level are conveyed in a backward direction to a lower level, seeking to predict, in fact, the neural activity at that prior stage of processing, and learning from errors signalled in the opposite direction. This is the essence of 'predictive coding', at once an algorithm for information processing and a theoretical basis for the nature of operations performed by the cerebral cortex. Neural models for the implementation of predictive coding invoke specific functional classes of neuron for generating, transmitting and receiving predictions, and for producing reciprocal error signals. Also a third general class, 'precision' neurons, tasked with regulating the magnitude of error signals contingent upon the confidence placed upon the prediction, i.e., the reliability and behavioural utility of the sensory data that it predicts. So, what is the ultimate source of a 'prediction'? The answer is multifactorial: knowledge of the current environmental context and the immediate past, allied to memory and lifetime experience of the way of the world, doubtless fine-tuned by evolutionary history too. There are, in consequence, numerous potential avenues for experimenters seeking to manipulate subjects' expectation, and examine the neural signals elicited by surprising, and less surprising visual stimuli. This review focuses upon the predictive physiology of mouse and monkey visual cortex, summarising and commenting on evidence to date, and placing it in the context of the broader field. It is concluded that predictive coding has a firm grounding in basic neuroscience and that, unsurprisingly, there remains much to learn.

SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND MODULUS OF CONTINUITY IN THE WEIGHTED BERGMAN SPACE

Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is \(\gamma(\rho)=\rho^\alpha,\) \(\alpha>0\), some sharp inequalities between the best simultaneous approximation and an \(m\)th order modulus of continuity averaged with the given weight are proved. For a specific class of functions, the upper bound of the best simultaneous approximation in the space \(B_{2,\gamma_{1}},\) \(\gamma_{1}(\rho)=\rho^{\alpha},\) \(\alpha>0\), is found. Exact values of several \(n\)-widths are calculated for the classes of functions \(W_{p}^{(r)}(\omega_{m},q)\).

A Novel Paradigm for Enhancing Linear Model Predictions across Multivariate Functions through Integration of CNNs

Linear models have been a cornerstone in statistical data analysis, yet their inherent linearity often constrains their ability to extract nuanced patterns from complex datasets. This limitation poses a challenge in real-world applications where non-linearity prevails. In this research, we introduce an innovative methodology aimed at overcoming this deficiency. Leveraging the powerful non-linear modelling capabilities of Convolutional Neural Networks(CNNs), we augment the predictive capabilities of linear models. Through a systematic approach, we extend our findings to encompass specific classes of crucial multivariate functions. Our investigation primarily focuses on neural network architectures that integrate convolution (conv), Rectified Linear Unit (ReLU) activation functions, and max pooling layers. By harnessing the synergies of these components, we unveil a novel paradigm for enhancing linear model predictions, unlocking a wealth of potential applications across diverse domains. This research lays the foundation for a more robust and accurate predictive modelling framework that transcends the boundaries of conventional linear approaches.

A necessary condition for extremality of solutions to autonomous obstacle problems with general growth

Let us consider the autonomous obstacle problem minv∫ΩF(Dv(x))dxon a specific class of admissible functions, where we suppose the Lagrangian satisfies proper hypotheses of convexity and superlinearity at infinity. Our aim is to find a necessary condition for the extremality of the solution, which exists and it is unique, thanks to a primal–dual formulation of the problem. The proof is based on classical arguments of Convex Analysis and on Calculus of Variations’ techniques.

Some Results on Third-Order Differential Subordination and Differential Superordination for Analytic Functions Using a Fractional Differential Operator

In this study, we explore the implications of a third-order differential subordination in the context of analytic functions associated with fractional differential operators. Our investigation involves the consideration of specific admissible classes of third-order differential functions. We also extend this exploration to establish a dual principle, resulting in a sandwich-type outcome. We introduce these admissible function classes by employing the fractional derivative operator DzαSN,Sϑz and derive conditions on the normalized analytic function f that lead to sandwich-type subordination in combination with an appropriate fractional differential operator.

Boundedness in the Bloch Space of Symmetric Domain for a Class of Multi-Valent Meromorphic Functions Given by a Fractional Integral

Convolution operators have profited in various areas of science. They are utilized in the investigations of computing techniques. A new convolution operator linked to a specific class of multi-valent meromorphic functions in the punctured unit disk (symmetric domain) is formulated. This analysis uncovers certain properties on the connections as well as the power series. We study a novel class of holomorphic functions concerning the recommended new operator. The second part of the outcome concerns the boundedness of the suggested difference structure given by the proposed operator. We focus on the Bloch space of meromorphic functions in the open unit disk. In this case, we use the spherical derivative. To obtain the maximum value of the polar derivative of the polynomials created by their partial sums, we use their partial sums as applications of the suggested operator.

Nonlinearity of functions over finite fields

Abstract The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity.

Medullary Thyroid Carcinomas Classified According to the International Medullary Carcinoma Grading System and a Surveillance, Epidemiology, and End Results-Based Metastatic Risk Score: A Correlation With Genetic Profile and Angioinvasion

Due to the lack of a standardized tool for risk-based stratification, the International Medullary Carcinoma Grading System (IMTCGS) has been proposed for medullary thyroid carcinomas (MTCs) based on necrosis, mitosis, and Ki67. Similarly, a risk stratification study using the Surveillance, Epidemiology, and End Results (SEER) database highlighted significant differences in MTCs in terms of clinical-pathological variables. We aimed to validate both the IMTCGS and SEER-based risk table on 66 MTC cases, with special attention to angioinvasion and the genetic profile. We found a significant association between the IMTCGS and survival because patients classified as high-grade had a lower event-free survival probability. Angioinvasion was also found to be significantly correlated with metastasis and death. Applying the SEER-based risk table, patients classified either as intermediate- or high-risk had a lower survival rate than low-risk patients. In addition, high-grade IMTCGS cases had a higher average SEER-based risk score than low-grade cases. Moreover, when we explored angioinvasion in correlation with the SEER-based risk table, patients with angioinvasion had a higher average SEER-based score than patients without angioinvasion. Deep sequencing analysis found that 10 out of 20 genes frequently mutated in MTCs belonged to a specific functional class, namely chromatin organization, and function, which may be responsible for the MTC heterogeneity. In addition, the genetic signature identified 3 main clusters; cases belonging to cluster II displayed a significantly higher number of mutations and higher tumor mutational burden, suggesting increased genetic instability, but cluster I was associated with the highest number of negative events. In conclusion, we confirmed the prognostic performance of the IMTCGS and SEER-based risk score, showing that patients classified as high-grade had a lower event-free survival probability. We also underline that angioinvasion has a significant prognostic role, which has not been incorporated in previous risk scores.

Finite-Time Convergent Primal–Dual Gradient Dynamics With Applications to Distributed Optimization

This article studies the finite-time (FT) convergence of a fast primal-dual gradient dynamics (PDGD), called FT-PDGD, for solving constrained optimization with general constraints and cost functions. Based on the nonsmooth analysis and augmented Lagrangian function, sufficient conditions are established for FT-PDGD to enable the realization of primal-dual optimization in FT. A specific class of nonsmooth sign-preserving functions is defined and analyzed for ensuring FT stability. Particularly, the matrix of linear equations is not required to have a full-row rank and the cost function is not necessary to be strictly convex. By introducing auxiliary variables for general linear inequality constraints, reduced sufficient conditions are further derived for the optimization with linear equality and inequality constraints after transformation. In addition, by the nonsmooth analysis, the switching dynamics evolved in both primal and dual variables are carefully investigated and the upper bound on the convergence time is explicitly provided. Moreover, as applications of FT-PDGD, several FT convergent distributed algorithms are designed to solve distributed optimization with separated and coupled linear equations, respectively. Finally, two case studies are conducted to show the performance of the proposed algorithms.

Strong Differential Subordination and Superordination Results for Extended q-Analogue of Multiplier Transformation

The results obtained by the authors in the present article refer to quantum calculus applications regarding the theories of strong differential subordination and superordination. The q-analogue of the multiplier transformation is extended, in order to be applied on the specific classes of functions involved in strong differential subordination and superordination theories. Using this extended q-analogue of the multiplier transformation, a new class of analytic normalized functions is introduced and investigated. The convexity of the set of functions belonging to this class is proven and the symmetry properties derive from this characteristic of the class. Additionally, due to the convexity of the functions contained in this class, interesting strong differential subordination results are proven using the extended q-analogue of the multiplier transformation. Furthermore, strong differential superordination theory is applied to the extended q-analogue of the multiplier transformation for obtaining strong differential superordinations for which the best subordinants are provided.

Designing Optimal Linear Detectors: A Bottom-Up Approach

This paper develops a systematic approach to realizing linear detectors with an optimized sensitivity, allowing for the detection of extremely weak signals. First, general constraints are derived on a specific class of input-output transfer functions of a linear detector. Then a physical realization of transfer functions in that class is found using the quantum network synthesis technique, which allows for the inference of the physical setup directly from the input-output transfer function. By exploring a minimal realization which has the minimum number of internal modes, it is shown that the optimal such detectors are internal squeezing schemes. Then, investigating nonminimal realizations, which is motivated by parity-time symmetric systems, a quantum nondemolition measurement is systematically recovered.