Search Results (22)

The Krawtchouk and Hahn families are discrete orthogonal polynomials defined on the integer pixel grid, yet as polynomials, they are entire functions of a complex variable. Adopting this complex variable and complex-valued viewpoint, we develop an image enhancement framework that connects discrete orthogonal polynomial theory with geometric function theory. Two operators are introduced. The Krawtchouk operator exploits the binomial weight, for which the parameter p concentrates the basis around a selectable tonal level $x=pN$, producing a localized contrast enhancement steerable toward shadows, midtones, or highlights. The Hahn operator uses the two-parameter Hahn polynomials, the discrete analogue of the Jacobi family, in which $(\alpha ,\beta )$ give asymmetric control of dark and light bands. Each operator is the real restriction of a holomorphic near identity map $\mathfrak{F}\left(z\right)=z+{\sum }_k{c}_k{\tilde{\varphi }}_k\left(z\right)$, with the intensity entering the discrete basis through $x=NI$, realized as a monotone 256-entry lookup table. We prove a bounded deviation estimate $|F-\mathrm{id}|\le {\sum }_k\left|c_k\right|$ on the intensity segment $[0, 1]$ and a positive slope condition that, via the Noshiro–Warschawski criterion, is a univalence condition for the analytic transfer map, ruling out intensity order reversal and oscillatory folding. Because Hahn polynomials lose orthogonality at high order in naive arithmetic, we show that a three-term recurrence with log-gamma weights preserves orthonormality to within about ${10}^{-10}$ on the full 8-bit grid, where single precision computation fails, and that the same scheme remains at the double-precision roundoff level on 10-, 12-, and 16-bit grids ($N=1023,4095,65,535$). The operators cost $\mathcal{O}\left(mL\right)$ table construction plus one lookup per pixel (about 3 ms for a $1024\times 1024$ color image), and a histogram-based differential entropy criterion selects the focus parameters automatically. Experiments on imagery of fine art, wildlife, archaeology, and architecture show that the Krawtchouk operator yields stronger localized contrast compared to seven classical methods, while the Hahn operator attains higher PSNR/SSIM, both as deterministic fast slope-controlled transforms. A diffusion MRI example further demonstrates that matching the focus parameter to the tonal mass adapts the same operators to dark-dominated medical scan imagery. Full article

Machine learning practice often reduces a complex training or inference problem to a one-dimensional response curve, such as a validation-loss curve, calibration curve, robustness-budget profile, or checkpoint-interpolation path. This paper presents a functional-analytic formulation of proposed $(v,w)$–s–convex response-curve certification. The response curve is treated as an element of the Banach space of continuous functions under the supremum norm, while derivative-based certificates are handled in a Lipschitz and Sobolev-type norm when required. Generalized convexity is represented through a bounded structural operator, whose order condition defines a closed convex acceptance set. The violation score is measured by the positive part of the operator residual, and the Hermite–Hadamard, Fejér, and Ostrowski quantities are interpreted as bounded certificate functionals. The auxiliary profiles are constructed from validation-curve residuals through a split-calibrated procedure and then tested on held-out triples. The framework certifies only scalar response-curve summaries under explicit structural and empirical assumptions; it does not certify a full learning system, guarantee generalization, or replace dense sampling when the structural gate fails. Full article

The theory of analytic functions remains a fundamental area of geometric function theory, with particular emphasis on coefficient problems, differential subordinations, and determinant estimates. Motivated by recent developments in fractional calculus and quantum calculus, this paper introduces two new subclasses of normalized analytic functions by employing the subordination principle in combination with the q-derivative operator and the q-Sălăgeăn differential operator within the framework of quantum calculus. The inclusion of fractional and q-calculus techniques provides a more flexible and generalized approach to classical problems in complex analysis, enabling deeper structural insights into analytic function classes. Using the subordination framework, we derive coefficient relations for the proposed subclasses. Furthermore, we establish sharp upper bounds for the Fekete–Szegö functional $|a_3-\delta a_2^2|$ and for the second Hankel determinant $H_{2,2}\left(f\right)=a_2{a}_4-a_3^2$. The obtained results extend and unify several known works in the literature and demonstrate how the interaction between fractional calculus, quantum operators, and subordination theory can be effectively used in geometric function theory. Finally, the presented approach opens the door for further investigations involving higher-order Hankel determinants, other subclasses of analytic functions, and potential extensions involving special functions and fractional operators. Full article

This study develops a Constraint-Driven Model of Intelligence to explain the emergence of structured meaning in complex systems, reconciling probability and cybernetics. It applies a conceptual–analytic procedure, conducted entirely through logical reasoning and theoretical analysis, without empirical measurement, data acquisition, experimental manipulation, or statistical testing, and is therefore methodologically separate from empirical artificial intelligence research. Phenomena such as model collapse are cited as theoretical instances for epistemic argumentation, without asserting empirical verification. Building on Émile Borel’s Infinite Monkey Theorem, which demonstrates the theoretical inevitability of order in unbounded stochastic processes, and Gregory Bateson’s principle of negative explanation, which defines structure as the result of systematically eliminated alternatives, the analysis formalizes how constraints break ergodicity and generate asymmetry. Shannon’s entropy quantifies the informational effects of constraints, while Simon’s bounded rationality and Turing’s algorithmic limits show how cognitive and computational boundaries produce tractable outcomes. Applied to modern AI, the model accounts for model collapse in recursive training, showing that the loss of asymmetric constraints produces low-entropy, repetitive outputs, demonstrating the epistemic necessity of constraint regulation. Comparing probabilistic and cybernetic accounts of emergence, the study shows that structured intelligence arises not from stochastic exploration alone, but from bounded, recursive, selective processes. This model is transdisciplinary, formalizing how constraints from socioeconomic pressures to subcultural circulation shape diversity, innovation, and functional asymmetry, establishing a generalizable cybernetic epistemology for the generation of structured intelligence and meaning across domains. By formalizing these concepts through set-theoretic derivations and integrative synthesis, this non-empirical model advances a cybernetic epistemology, separate from quantitative AI evaluations or experimental designs. Full article

The use of integral and differential operators in geometric function theory has continued to gain interest among researchers in the field of study in recent times. This is due to the wide range of its applications in science, technology and engineering. In this work, therefore, the authors defined and investigated a new subclass of analytic functions in the open unit disk using the $\mathfrak{q}$-Srivastava–Attiya convolution operator and the Jackson’s $\mathfrak{q}$-derivative, by means of the subordination. The authors used two well-known lemmas to determine a sharp upper-bound for the Fekete–Szeg$\ddot{o}$ functional in two different cases. In particular, the authors introduced a new generalized subclass of complex order univalent functions denoted by ${\mathcal{L}}_{\mathfrak{q},\mathfrak{b},\mathfrak{h}}^{\mathfrak{s}}\left(\tau ,,,\Phi \right)$ and derived the coefficient estimates $a_{\iota }(\iota =2,3)$ of the Taylor–Maclaurin series in this class, as well as the Fekete–Szeg$\ddot{o}$ inequality $\left|a_3,-,\aleph ,a_2^2\right|$ for functions in this class. The work generalizes many known results in the literature. Full article

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to $1+\mathrm{s}\mathrm{i}\mathrm{n}z$. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function $\sqrt{1+tanhz}$, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems. Full article

Volatility recovery is of paramount importance in contemporary finance. Volatility levels are heavily used in risk and portfolio management. We employ the Hull–White one- and two-factor models to describe the market condition. We computationally recover the volatility term structure as a piecewise-linear function of time. For every maturity, a cost functional, defined as the squared differences between theoretical and market prices, is minimized and the respective linear part is reconstructed. On the last time steps, before each maturity, the derivative price is decomposed in order to make the minimization problem analytically solvable. The procedure works fast since only scalar values are obtained on each minimization. However, the predictor–corrector nature of the algorithm allows for the precise recovery of very complex volatility functions. An implicit scheme is used to solve the PDEs on bounded domains. The computational simulations with artificial and real data show that the proposed algorithm is stable, accurate and efficient. Full article

The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics. Full article

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature. Full article

Geometric function theory, a subfield of complex analysis that examines the geometrical characteristics of analytic functions, has seen a sharp increase in research in recent years. In particular, by employing subordination notions, the contributions of different subclasses of analytic functions associated with innovative image domains are of significant interest and are extensively investigated. Since $\Re (1+sinh(z\left)\right)\ngtr 0,$ it implies that the class $S_{sinh}^{*}$ introduced in reference third by Kumar et al. is not a subclass of starlike functions. Now, we have introduced a parameter $\lambda$ with the restriction $0\le \lambda \le ln(1+\sqrt{2}),$ and by doing that, $\Re (1+sinh(\lambda z\left)\right)>0.$ The present research intends to provide a novel subclass of starlike functions in the open unit disk $\mathcal{U},$ denoted as $S_{sinh\lambda }^{*}$, and investigate its geometric nature. For this newly defined subclass, we obtain sharp upper bounds of the coefficients $a_n$ for $n=2,3,4,5.$ Then, we prove a lemma, in which the largest disk contained in the image domain of $q_0\left(z\right)=1+sinh\left(\lambda z\right)$ and the smallest disk containing $q_0\left(\mathcal{U}\right)$ are investigated. This lemma has a central role in proving our radius problems. We discuss radius problems of various known classes, including $S^{*}\left(\beta \right)$ and $\mathcal{K}\left(\beta \right)$ of starlike functions of order $\beta$ and convex functions of order $\beta$. Investigating $S_{sinh\lambda }^{*}$ radii for several geometrically known classes and some classes of functions defined as ratios of functions are also part of the present research. The methodology used for finding $S_{sinh\lambda }^{*}$ radii of different subclasses is the calculation of that value of the radius $r<1$ for which the image domain of any function belonging to a specified class is contained in the largest disk of this lemma. A new representation of functions in this class, but for a more restricted range of $\lambda$, is also obtained. Full article

The potential for widespread applications of the geometric and mapping properties of functions of a complex variable has motivated this article. On the other hand, the basic or quantum (or q-) derivatives and the basic or quantum (or q-) integrals are extensively applied in many different areas of the mathematical, physical and engineering sciences. Here, in this article, we first apply the q-calculus in order to introduce the q-derivative operator ${\mathfrak{S}}_{\eta ,p,q}^{n,m}$. Secondly, by means of this q-derivative operator, we define an interesting subclass $\mathcal{T}{\aleph }_{\lambda ,p}^{n,m}(\eta ,\alpha ,\kappa )$ of the class of normalized analytic and multivalent (or p-valent) functions in the open unit disk $\mathbb{U}$. This p-valent analytic function class is associated with the class $\kappa$-$\mathcal{UCV}$ of $\kappa$-uniformly convex functions and the class $\kappa$-$\mathcal{UST}$ of $\kappa$-uniformly starlike functions in $\mathbb{U}$. For functions belonging to the normalized analytic and multivalent (or p-valent) function class $\mathcal{T}{\aleph }_{\lambda ,p}^{n,m}(\eta ,\alpha ,\kappa )$, we then investigate such properties as those involving (for example) the coefficient bounds, distortion results, convex linear combinations, and the radii of starlikeness, convexity and close-to-convexity. We also consider a number of corollaries and consequences of the main findings, which we derived herein. Full article

Based on the geometry of a radial function, a sequence of approximations for arcsine, arccosine and arctangent are detailed. The approximations for arcsine and arccosine are sharp at the points zero and one. Convergence of the approximations is proved and the convergence is significantly better than Taylor series approximations for arguments approaching one. The established approximations can be utilized as the basis for Newton-Raphson iteration and analytical approximations, of modest complexity, and with relative error bounds of the order of ${10}^{-16}$, and lower, can be defined. Applications of the approximations include: first, upper and lower bounded functions, of arbitrary accuracy, for arcsine, arccosine and arctangent. Second, approximations with significantly higher accuracy based on the upper or lower bounded approximations. Third, approximations for the square of arcsine with better convergence than well established series for this function. Fourth, approximations to arccosine and arcsine, to even order powers, with relative errors that are significantly lower than published approximations. Fifth, approximations for the inverse tangent integral function and several unknown integrals. Full article

The unit–power Burr X distribution (UPBXD), a bounded version of the power Burr X distribution, is presented. The UPBXD is produced through the inverse exponential transformation of the power Burr X distribution, which is also beneficial for modelling data on the unit interval. Comprehensive analysis of its key characteristics is performed, including shape analysis of the primary functions, analytical expression for moments, quantile function, incomplete moments, stochastic ordering, and stress–strength reliability. Rényi, Havrda and Charvat, and d-generalized entropies, which are measures of uncertainty, are also obtained. The model’s parameters are estimated using a Bayesian estimation approach via symmetric and asymmetric loss functions. The Bayesian credible intervals are constructed based on the marginal posterior distribution. Monte Carlo simulation research is intended to test the accuracy of various estimators based on certain measures, in accordance with the complex forms of Bayesian estimators. Finally, we show that the new distribution is more appropriate than certain other competing models, according to their application for COVID-19 in Saudi Arabia and the United Kingdom. Full article

In this study, a sensing device employing a gold-coated quartz tuning fork (QTF) modified with a self-assembled monolayer (SAM) of L-cysteine was evaluated for the sensitive detection of Cu²⁺ ions in aqueous solutions. Three copper (II) salts, CuSO₄, CuCl₂, and Cu(NO₃)₂, at four different concentrations (10⁻¹², 10⁻¹⁰, 10⁻⁸, and 10⁻⁶ M) in small (100 μL) water sample amounts were each used as analytes to investigate the influence of their counterions in the detection of the Cu²⁺ ions. It was found that, among the counterions, the sulfate anion had the largest effect upon the detection of Cu²⁺ in water, in the following order: SO₄²⁻ > Cl⁻ > NO₃⁻. The lower limit of detection of the Cu²⁺ ions detected was in the 10⁻¹² M range. The frequency shifts measured with the QTFs relative to deionized water were inversely proportional to the concentration/mass of the analytes. Density functional theory calculations were conducted to understand the effect of the counterions on the respective electronic interaction energies for the apparent host–guest binding of the analytes with L-cysteine and with gold surface-bound L-cysteine molecules. Gas phase (both with and uncorrected BSSE) and solution phase interaction energies (ΔIE) calculated at the B3LYP/LANL2DZ and ωB97XD levels of theory showed that the stability for the complexes were in the following order: [L-cysteine]⊃[CuSO₄] > [L-cysteine]⊃[CuCl₂] > [L-cysteine]⊃[Cu(NO₃)₂], which supports our experimental findings, as they were in the same order as the experimentally observed order for the copper salts tested: CuSO₄ > CuCl₂ > Cu(NO₃)₂. Full article