Stable Mappings Research Articles

Neighborhoods of Harmonic and Stable Harmonic Mappings

Neighborhoods of Harmonic and Stable Harmonic Mappings

On the stability of canonical correlation analysis and partial least squares with application to brain-behavior associations

Associations between datasets can be discovered through multivariate methods like Canonical Correlation Analysis (CCA) or Partial Least Squares (PLS). A requisite property for interpretability and generalizability of CCA/PLS associations is stability of their feature patterns. However, stability of CCA/PLS in high-dimensional datasets is questionable, as found in empirical characterizations. To study these issues systematically, we developed a generative modeling framework to simulate synthetic datasets. We found that when sample size is relatively small, but comparable to typical studies, CCA/PLS associations are highly unstable and inaccurate; both in their magnitude and importantly in the feature pattern underlying the association. We confirmed these trends across two neuroimaging modalities and in independent datasets with n ≈ 1000 and n = 20,000, and found that only the latter comprised sufficient observations for stable mappings between imaging-derived and behavioral features. We further developed a power calculator to provide sample sizes required for stability and reliability of multivariate analyses. Collectively, we characterize how to limit detrimental effects of overfitting on CCA/PLS stability, and provide recommendations for future studies.

On Bohr's inequality for special subclasses of stable starlike harmonic mappings

Abstract The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao (Some basic properties of certain subclass of harmonic univalent functions, Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z ∈ D z\in {\mathbb{D}} , where 0 < M ≤ 1 0\lt M\le 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.

The Relationship Between Music-Related Types of Synesthesia and Mental Imagery in Synesthete Musicians

Synesthesia is a perceptual and conceptual phenomenon that has been linked to a heightened capacity for mental imagery. However, our understanding of music-related types of synesthesia and mental imagery is still in its infancy. This study therefore aims to examine the relationship between music-related types of synesthesia and mental imagery in a sample of synesthete musicians. Four forms of data collection were employed: an online survey, a synesthesia test battery, an absolute pitch test, and a semi-structured interview. This article focuses on results obtained in response to open-ended questions relating to synesthete musicians’ mental imagery. An analysis of the interview data revealed heightened vividness of auditory imagery (AI), including the ability to control the speed, dynamics, and intensity of the imagery. A bidirectional analysis of the functional relationship between AI and synesthesia ascertained that multimodal mental imagery (MMI) could be triggered by AI, thus bypassing the need for sensory stimulation. Furthermore, cases of complex visual imagery (CVI) were confirmed to meet the synesthetic criteria of being automatic and involuntary, as well as being consistent and stable mappings between specific musical stimuli and CVI. These findings are noteworthy as they call into question the general assumption that synesthetic percepts are generic and simple in nature. Together, these findings contribute to our understanding of music-related types of synesthesia and mental imagery, challenge contemporary understanding of certain categorical aspects of music-related synesthetic percepts, and provide us with a deeper understanding of the nature of cross-modal associations in general.

Self-Organizing Control Systems for Nonlinear Spacecraft in the Class of Structurally Stable Mappings

Self-Organizing Control Systems for Nonlinear Spacecraft in the Class of Structurally Stable Mappings

Non-density of stable mappings on non-compact manifolds

Non-density of stable mappings on non-compact manifolds

On the Bohr’s Inequality for Stable Mappings

We consider the class of stable harmonic mappings $$f=h+\overline{g}$$ introduced by Martin, Hernandez and the class of stable logharmonic mappings $$f=zh\overline{g}$$ introduced by AbdulHadi, El-Hajj. We determine Bohr’s radius for the classes of stable univalent harmonic mappings, stable convex harmonic mappings and stable univalent logharmonic mappings. We also consider improved and refined versions of Bohr’s inequality and discuss the Bohr–Rogosinski’s radius for this family of mappings.

On stable harmonic mappings

On stable harmonic mappings

Context-sensitive computational mechanistic explanation in cognitive neuroscience.

Mainstream cognitive neuroscience aims to build mechanistic explanations of behavior by mapping abilities described at the organismal level via the subpersonal level of computation onto specific brain networks. We provide an integrative review of these commitments and their mismatch with empirical research findings. Context-dependent neural tuning, neural reuse, degeneracy, plasticity, functional recovery, and the neural correlates of enculturated skills each show that there is a lack of stable mappings between organismal, computational, and neural levels of analysis. We furthermore highlight recent research suggesting that task context at the organismal level determines the dynamic parcellation of functional components at the neural level. Such instability prevents the establishment of specific computational descriptions of neural function, which remains a central goal of many brain mappers – including those who are sympathetic to the notion of many-to-many mappings between organismal and neural levels. This between-level instability presents a deep epistemological challenge and requires a reorientation of methodological and theoretical commitments within cognitive neuroscience. We demonstrate the need for change to brain mapping efforts in the face of instability if cognitive neuroscience is to maintain its central goal of constructing computational mechanistic explanations of behavior; we show that such explanations must be contextual at all levels.

Reference and morphology*

AbstractThe dominant tradition in analytic philosophy of language views reference as paradigmatically enabled by the acquisition of words from other speakers. Via chains of transmission, these words connect the referrer to the referent. Such a picture assumes the notion of a word as a stable mapping between sound and meaning. Utterances are constructed out of such stable mappings. While this picture of language is both intuitive and historically distinguished, various trends and programs that have developed over the last few decades in theoretical linguistics suggest an alternative. According to these approaches, the word, conceived of as a linguistic ‘building block’, has no special theoretical significance. While natural language systems generate structures mapping sounds onto meanings, they need not do so by composing elements which themselves specify such mappings. I shall describe some of these developments, and show how they pose a problem for traditional philosophical views of language and communication, before identifying an alternative approach to reference which does not rely on this common‐sense picture of words.

ОЦЕНКИ МОЩНОСТИ КЛАССОВ ОТОБРАЖЕНИЙ, ПРИМЕНЯЕМЫХ В ПРОТОКОЛАХ КВАНТОВОГО РАСПРЕДЕЛЕНИЯ КЛЮЧЕЙ

Introduction: In the near future, quantum cryptography will play an important role in maintaining a sufficient level of information security of modern telecommunication networks in the conditions of a quantum challenge, which refers to the emergence of quantum computers that will be able to effectively solve the mathematical problems on which, for example, modern key distribution systems are based. Now quantum cryptography is actively used by commercial and government agencies around the world and, in particular, in the Russian Federation. At the same time, a large amount of research is being carried out in the field of development and implementation of quantum key distribution systems, as the main part of quantum cryptography. In this regard, the task of developing new and refining existing protocols for quantum key distribution, as well as studying various mathematical and physical objects that are associated with these protocols, is an urgent task. In particular, one of the stages of the classical BB84 protocol implement ed in a noisy quantum channel is associated with the problem of studying correlation immune and stable mappings, part of which is the problem of estimating their number, which has not been completely solved. Purpose: to find mathematical expressions for exact and asymptotic estimates of the cardinalities of classes of (n,m,k) stable and correlation immune of order k boolean mappings. Results: The best currently asymptotic upper and lower bounds for the number of such classes of mappings with the number of outputs greater than or equal to five are obtained. Recurrent relations were also proved, which allow one to find the exact distribution of the cardinalities of classes of similar mappings for the case of small numbers n and m. Practical relevance: the results obtained allow us to estimate the probability that with a random choice of mapping to enhance secrecy at the stage of secondary processing of the BB84 protocol, the situation will be neutralized when the adversary has access to k photons sent over a communication chan nel of his choice.

Learning regularized representations of categorically labelled surface EMG enables simultaneous and proportional myoelectric control

BackgroundProcessing the surface electromyogram (sEMG) to decode movement intent is a promising approach for natural control of upper extremity prostheses. To this end, this paper introduces and evaluates a new framework which allows for simultaneous and proportional myoelectric control over multiple degrees of freedom (DoFs) in real-time. The framework uses multitask neural networks and domain-informed regularization in order to automatically find nonlinear mappings from the forearm sEMG envelope to multivariate and continuous encodings of concurrent hand- and wrist kinematics, despite only requiring categorical movement instruction stimuli signals for calibration.MethodsForearm sEMG with 8 channels was collected from healthy human subjects (N = 20) and used to calibrate two myoelectric control interfaces, each with two output DoFs. The interfaces were built from (I) the proposed framework, termed Myoelectric Representation Learning (MRL), and, to allow for comparisons, from (II) a standard pattern recognition framework based on Linear Discriminant Analysis (LDA). The online performances of both interfaces were assessed with a Fitts’s law type test generating 5 quantitative performance metrics. The temporal stabilities of the interfaces were evaluated by conducting identical tests without recalibration 7 days after the initial experiment session.ResultsMetric-wise two-way repeated measures ANOVA with factors method (MRL vs LDA) and session (day 1 vs day 7) revealed a significant (p<0.05) advantage for MRL over LDA in 5 out of 5 performance metrics, with metric-wise effect sizes (Cohen’s d) separating MRL from LDA ranging from left|dright|=0.62 to left|dright|=1.13. No significant effect on any metric was detected for neither session nor interaction between method and session, indicating that none of the methods deteriorated significantly in control efficacy during one week of intermission.ConclusionsThe results suggest that MRL is able to successfully generate stable mappings from EMG to kinematics, thereby enabling myoelectric control with real-time performance superior to that of the current commercial standard for pattern recognition (as represented by LDA). It is thus postulated that the presented MRL approach can be of practical utility for muscle-computer interfaces.

Social cues modulate the representations underlying cross-situational learning

Because children hear language in environments that contain many things to talk about, learning the meaning of even the simplest word requires making inferences under uncertainty. A cross-situational statistical learner can aggregate across naming events to form stable word-referent mappings, but this approach neglects an important source of information that can reduce referential uncertainty: social cues from speakers (e.g., eye gaze). In four large-scale experiments with adults, we tested the effects of varying referential uncertainty in cross-situational word learning using social cues. Social cues shifted learners away from tracking multiple hypotheses and towards storing only a single hypothesis (Experiments 1 and 2). In addition, learners were sensitive to graded changes in the strength of a social cue, and when it became less reliable, they were more likely to store multiple hypotheses (Experiment 3). Finally, learners stored fewer word-referent mappings in the presence of a social cue even when given the opportunity to visually inspect the objects for the same amount of time (Experiment 4). Taken together, our data suggest that the representations underlying cross-situational word learning of concrete object labels are quite flexible: In conditions of greater uncertainty, learners store a broader range of information.

Construction and Researching Aircraft High Potential of Robust Stability Control System in the Form of Single-parameter Structurally Stable Mapping

The study presents a new approach to building control systems for objects with uncertain parameters in the form of single-parameter structurally stable mappings of catastrophe theory to synthesize highly efficient control system, which has an extremely wide field of robust stability.

Continuous -selection

The paper is concerned with properties of sets admitting a continuous selection from the set of nearly best approximations. Necessary and sufficient conditions are put forward for the existence of continuous additive and multiplicative -selections on closed sets. Sufficient conditions are given for the existence of continuous selections for stable set-valued mappings with not-necessarily-convex values. Bibliography: 8 titles.

Design of Control System in the Class of Two-Parametric Structurally Stable Mappings

In this paper, we proposed a method for design of control systems with a high potential of robust stability in a class of two-parametric structurally stable mappings. Research of robust stability is based on the geometric interpretation of the second Lyapunov method, and also definition of system stability in the state space.We propose a method for construct the control system, designed in two-parametric class of structurally stable mappings, which will be sustained indefinitely in a wide range of uncertain parameters of the control object. This work presents novelty theoretical fundamental results assisting in analyzing of the behavior of control systems, meaning of robust stability.

The Second Lyapunov Function Method in Construction of Control Systems with the Increased Potential of Robust Stability in the Class of Catastrophe “Hyperbolic Umbilic”

The current article introduces a new method of construction of control systems for objects with uncertain parameters in the form of three-parameter structurally stable mappings from the theory of catastrophe, allowing to synthesize the highly effective control systems, which demonstrate the extremely wide range of robust stability. The research of robust stability of control systems is based on a new approach to post-rhenium of the A.M Lyapunov’s functions. The method of creation of control system with the increased potential of robust stability is stated. It could be concluded that the application of the suggested approach may assure asymptotically stable invariant states for the system in both negative and positive regions of variations of uncertain parameters.

On the Origin of Utility, Weighting, and Discounting Functions: How They Get Their Shapes and How to Change Their Shapes

We present a theoretical account of the origin of the shapes of utility, probability weighting, and temporal discounting functions. In an experimental test of the theory, we systematically change the shape of revealed utility, weighting, and discounting functions by manipulating the distribution of monies, probabilities, and delays in the choices used to elicit them. The data demonstrate that there is no stable mapping between attribute values and their subjective equivalents. Expected and discounted utility theories, and also their descendants such as prospect theory and hyperbolic discounting theory, simply assert stable mappings to describe choice data and offer no account of the instability we find. We explain where the shape of the mapping comes from and, in describing the mechanism by which people choose, explain why the shape depends on the distribution of gains, losses, risks, and delays in the environment.Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2013.1853 .This paper was accepted by Yuval Rottenstreich, judgment and decision making.

Sections of stable harmonic convex functions

In this article, we consider the class SHC of normalized stable harmonic convex mappings f=h+g¯ in the unit disk and determine r such that every section of f is close-to-convex or convex in the disk ∣z∣<r. Also, we show that the convolution f∗f∈SHC whenever f∈SHC. In addition, we prove that the harmonic convolution f1∗f2 is stable harmonic close-to-convex whenever f1∈SHC and f2 is either a slanted half-plane mapping or an asymmetric vertical strip mapping or a mapping in the family PH0(α) which is contained in the family of harmonic close-to-convex functions in ∣z∣<1.

On the Largest Cartesian Closed Category of Stable Domains

Let SABC (resp., SABC˜) be the category of algebraic bounded complete domains with conditionally multiplicative mappings, that is, Scott-continuous mappings preserving meets of pairs of compatible elements (resp., stable mappings). Zhang showed that the category of dI-domains is the largest cartesian closed subcategory of ω-SABC and ω-SABC˜, with the exponential being the stable function space, where ω-SABC and ω-SABC˜ are full subcategories of SABC and SABC˜ respectively which contain countablly based algebraic bounded complete domains as objects. This paper shows that:i)The exponentials of any full subcategory of SABC or SABC˜ are exactly function spaces;ii)SDABC˜ the category of distributive algebraic bounded complete domains, is the largest cartesian closed subcategory of SABC˜; The compact elements of function spaces in the category SABC are also studied.