Search Results (209)

This paper establishes new Fejér and Hermite–Hadamard-type inequalities for functions of two variables whose mixed second-order partial derivatives satisfy coordinated GA-convexity or coordinated GA-quasi-convexity on a rectangle in the positive quadrant. Our main results are formulated for non-negative continuous weight functions that are not necessarily symmetric with respect to the geometric means of the interval endpoints, thereby extending the classical framework to genuinely asymmetric weights. However, to obtain explicit and sharp integral bounds in certain cases, we also employ a technical lemma that assumes a special symmetric setting where the weight function is symmetric on each coordinate with respect to $\sqrt{h_1{h}_2}$ and $\sqrt{k_1{k}_2}$. We clearly distinguish which theorems hold for general asymmetric weights and which depend on this symmetry condition. Our findings unify and extend numerous previously known results for both symmetric and non-symmetric weight functions. Full article

This paper develops a two-variable truncated-exponential-based Hahn–Appell polynomial family of order r in quantum q-calculus, generated by $\mathcal{A}\left(t\right){(1-\eta t^r)}^{-1}{e}_q^{wt}\left(\zeta t\right)$, where $\mathcal{A}\left(t\right)$ is a normalized determining function and $e_q^{wt}$ denotes the Hahn-type q-exponential. From this relation, we derive connections with the underlying Hahn–Appell basis, the finite expansion induced by the truncation factor, the Hahn lowering property, operational resolvents, step-r recurrences, determinant formulas, and Hahn-factorial representations. We also obtain origin values, normalized monicity, parameter-connection and parameter-differentiation identities, quasi-monomial operators, and the associated Hahn-grid difference equation. Limiting and special cases recover known q-truncated-exponential and classical truncated-exponential Appell constructions. For the unit-Appell specialization $\mathcal{A}\left(t\right)=1$, numerical computations for $q=1/2$, $r=2$, $\eta =1$, and $w=1$ provide explicit polynomials, zero tables, diagnostics, and plots that illustrate parameter sensitivity, Hahn-grid deformation, conjugate symmetry of non-real zeros, origin zeros in odd degrees, and the observed layered rightward spread of zero clouds. Full article

Discussion of consciousness and artificial intelligence has hitherto focused on the question of whether a large language model (LLM) exhibits signs of consciousness or understanding. This paper proposes to redirect attention elsewhere: not to the model itself, but to the orchestrating layer that governs the model—the backend, understood here as the collection of mechanisms (context management, retrieval, evaluation, planning, and tool-use control) that structure the model’s operation. We argue that the backend performs a function functionally analogous to the role of consciousness in the human brain: it stabilizes generative processes, directs attention, maintains context, and mitigates the entropic disintegration of thought. Consciousness fulfills this function through the phenomenal layer—qualia—which creates a persistent subjective “inner canvas”, used here as a metaphor for a more general multimodal phenomenal space. The backend fulfills it only algorithmically, without phenomenal quality. We further show that computation is an informationally conservative process in the sense of Shannon’s Data Processing Inequality (DPI), and therefore cannot increase Shannon information, even though it may yield novel or pragmatically useful recombinations of existing information. We conclude by proposing the hypothesis that consciousness constitutes a phenomenon orthogonal to computation—not an emergent property of complexity, but a qualitative leap into a different dimension. This hypothesis, which builds on the author’s prior work in this Special Issue and in Symmetry, is presented as a conceptual contribution rather than a formal theory, and may have implications for how future artificial intelligence research conceptualizes the limits of computational architectures. Full article

Generalization of the ordinary hyperbolic functions called hyperbolic Ateb-functions is considered. They are the inverse of incomplete Beta-function. These functions are solutions of differential equations, which describe the aperiodic vibration motion. It is shown that hyperbolic Ateb-functions have different dependence levels on their parameters. Investigation of the domain of hyperbolic Ateb-functions is conducted. It is shown that the minimum value of the domain can be expressed in terms of the lemniscate constant. The formulas for derivatives of hyperbolic Ateb-functions are proved and the structure of the higher-order derivatives is obtained. Some other properties connected with symmetry are considered. Taylor expansions of Ateb-cosine and Ateb-sine are taken out. Based on the mathematical induction principle, the corresponding theorems are proven. Examples of Taylor expansions of Ateb-cosine and Ateb-sine for different parameters are presented. The comparison of Ateb-function calculation using Taylor expansion and numerical methods shows the advantage of the Taylor series approach. Full article

This study presents the development of a comprehensive framework wherein probabilistic and analytical techniques collaboratively produce identities pertinent to special functions. The fundamental premise is that ratios of random variables, particularly within the family of Gamma and Beta distributions, inherently generate integral representations and hypergeometric structures that can be harnessed to derive non-trivial summation formulas. A pivotal outcome of this investigation is that, in the context of independent and identically distributed random variables, symmetry plays a crucial role. A key property of ratios of i.i.d. random variables translates into straightforward probabilistic assertions that directly lead to precise analytic identities, thereby enabling the recovery of classical results such as Kummer’s and Watson’s summation theorems as consequences of distributional symmetry. When the assumption of identical distribution is relaxed, hypergeometric representations continue to exist; however, the absence of symmetry impedes closed-form evaluations of the same nature. This contrast underscores symmetry as the fundamental mechanism underlying exact summation formulas and elucidates why such identities are contingent upon specific parameter configurations. More generally, this methodology offers a probabilistic interpretation of hypergeometric identities and establishes a conceptual connection between probability theory and the theory of special functions. Furthermore, it suggests that more expansive constructions, based on non-identically distributed variables or iterated processes, could be fruitfully explored. In particular, examining ratios may lead to new identities or alternative derivations of classical results. Full article

This study uses a fractional operator technique to analyze a novel class of special polynomials. These polynomials are designated as fractional Gould–Hopper–Bell–Apostol-type polynomials. We first define the operational expression of the Apostol-type Gould–Hopper–Bell polynomials and then use a suitable fractional operator to generate a new fractional version of these polynomials. The accompanying generating function, series definition, and summation formulas are also derived. Furthermore, certain symmetry identities and monomiality results are investigated. The study also identifies specific members of this fractional family, such as fractional Gould–Hopper–Bell–Apostol–Bernoulli polynomials, fractional Gould–Hopper–Bell–Apostol–Euler polynomials, and fractional Gould–Hopper–Bell–Apostol–Genocchi polynomials, and finds similar results for each. The study makes use of Mathematica to display computational results, zero distributions, and graphical demonstrations for a specific case of the established class. Full article

This work investigates the solution and the stability of a generalized system of bi-quadratic–Drygas functional equations in non-Archimedean normed spaces with unknown coefficients. The presence of asymmetric coefficients and reflection terms induces nontrivial coupling effects and symmetry-breaking phenomena, while simultaneously capturing additive, quadratic, and mixed additive–quadratic behaviors. An examination of the coefficients shows that the trivial solution is the only one satisfying the system of equations in asymmetric parameter configurations. For symmetric configurations, by exploiting the ultrametric structure of non-Archimedean norms and applying an iterative method combined with symmetry-based decomposition into even and odd parts, we establish the existence and uniqueness of an exact solution approximating a given mapping. Several known stability results for bi-Drygas functional equations are recovered with improvement as special cases. Full article

In this paper, we propose an analytic deformation acting on probability measures, designed to encompass and extend two fundamental operators in free probability: the $(a,b)$- and the $T_c$-deformations. This unified operator, indicated by ${\mathcal{X}}_{(a,b,c)}$, is introduced through a functional relation for the Cauchy–Stieltjes transform. We have ${\mathcal{X}}_{(a,b,0)}={\tilde{U}}^{(a,b)}$ and ${\mathcal{X}}_{(1,1,c)}=T_c$. We examine the structural properties of this transformation within the setting of Cauchy–Stieltjes kernel (CSK) families, with special emphasis on the behavior of the associated variance functions (VFs). An explicit formula for the VF corresponding to measure deformed by ${\mathcal{X}}_{(a,b,c)}$ is established. This result allows us to demonstrate a key invariance property: the free Meixner class of probability measures remains stable under the ${\mathcal{X}}_{(a,b,c)}$-transformation. Furthermore, a novel characterization of the semicircle law is obtained through the action of ${\mathcal{X}}_{(a,1,c)}$, highlighting the role of symmetry in the deformation and preservation of free-probabilistic distributions. Full article

Context/Objectives: Pectus excavatum (PE), the most common anterior chest wall deformity in children and adolescents, impacts posture and is frequently associated with axial deviations due to biomechanical alterations of the spine and the properties of the involved musculature. Methods: We assessed 35 patients with PE with a Haller index below 3.25, aged between 5 and 17 years, who completed a three months specialized physical exercise program after proper training and instruction by a specialist. All patients were assessed before starting the exercise program and at the end of the treatment. The assessment method used was myotonometry, employing the MyotonPRO device, targeting the trapezius muscle with all three fascicles and the pectoralis major muscle both on the left and the right side, measuring: frequency (Hz), stiffness (N/m), decrement, relaxation time (ms), and the ratio between relaxation time and deformation time (creep). Results: The analysis of myotonometric parameters reveals a pattern of selective adaptation, predominantly involving the left hemibody in most of the groups analyzed, without significant functional imbalances. This asymmetry may reflect either the functional predominance of the left hemibody during participants’ daily activities or increased activation induced by the exercise program; however, by the end of the intervention, bilateral stability was observed in most parameters. Conclusions: A three-month physical exercise program in children and adolescents with PE results in improvements in muscle properties, particularly in the pectoralis major and middle trapezius muscles bilaterally, and contributes to the restoration of functional symmetry, thereby supporting the effectiveness of the exercise program in optimizing neuromuscular control, tissue elasticity, and scapular stability. Full article

In this paper, we establish a new hybrid class of special polynomials, the Gould–Hopper–Fubini-based Appell polynomials. Using the monomiality principle, we derive their generating function and explore related properties and identities. We also investigate symmetry identities and obtain a determinant representation for these polynomials. Finally, we present and discuss results for several special cases of this family and support the derived results with computational studies and visual representations. Full article

Elastic wave scattering in polycrystalline materials has been a long-lasting topic in seismology and physical acoustics. Numerous analytical scattering models have been reported for polycrystals with random grain orientations. However, the elastic wave scattering in polycrystals with a preferred grain orientation (crystallographic texture) has not been well studied. This study develops a general ultrasonic scattering model that correlates the scattering coefficients and attenuation coefficients with orientation distribution coefficients (ODCs) for polycrystalline materials with a crystallographic texture. These models are valid for aggregates of triclinic grains with arbitrary texture symmetry. Since different terminologies for orientation distribution functions (ODFs) are adopted in quantitative texture analysis, the relations between different terminologies are also summarized in this study. Furthermore, for two special cases—hexagonal polycrystalline materials with a fiber texture and cubic polycrystalline materials with orthotropic texture symmetry—explicit expressions for the ultrasonic backscattering coefficient through ODCs are derived. The explicit relationship between ultrasonic backscattering and ODCs not only manifests how the individual texture coefficients impact ultrasonic scattering but also makes it possible to determine ODCs up to the eighth order experimentally from ultrasonic scattering measurements. This type of forward model also can be applied to the microstructure characterization of textured polycrystals. Full article

This Editorial introduces the Symmetry Special Issue “Theory and Applications of Special Functions II” and summarizes the nine contributions collected therein. The papers span the analytic continuation of multivariate hypergeometric functions; stability theory for differential equations via integral transforms; numerical schemes for multi-space fractional partial differential equations based on nonstandard finite differences and orthogonal polynomials; applications of the Lambert W function to viscoelastic creep modeling; algebraic constructions of new Hermite-type polynomial families via the monomiality principle; higher-level generalizations of poly-Cauchy numbers; Bell-polynomial expansions for Laplace transforms of higher-order nested functions; and two complementary studies on the physical implementation and algebraic description of Gaussian quantum states. Beyond the contributions of the Special Issue, we highlight methodological connections—continued fractions and complex analysis, transform techniques, special polynomials, and combinatorial sequences—and emphasize the unifying role of symmetry across mathematical structures and applications. Full article

The current paper retrieves implicit quiescent soliton solutions to optical metamaterials with nonlinear chromatic dispersion with generalized temporal evolution. Seven forms of self-phase modulation structures, as proposed by Kudryashov with time, are taken up. The implemented integration algorithm is Lie symmetry. A few of the solutions are in quadratures, while others are in terms of special functions. We also characterize the parameters that constrain the existence of such solutions. Full article

This study is inspired by the rich symmetry and diverse applications of special polynomial families, with a particular focus on the q-Bernoulli polynomials, which have recently emerged as significant tools in bi-univalent function theory. These polynomials are distinguished by their mathematical versatility, analytical manageability, and strong potential for generalization, offering an elegant framework for advancing the study of such functions. In this paper, we introduce a novel subclass of bi-univalent functions defined through q-Bernoulli polynomials. We obtain coefficient estimates for functions in this class and investigate their implications for the Fekete–Szegö functional. Additionally, we present several new results to enrich the theoretical landscape of bi-univalent functions associated with q-Bernoulli polynomials. Full article

Background and Clinical Significance: Hemiplegia following a cerebrovascular accident (CVA) disrupts gait symmetry and efficiency, compromising functional independence. The integration of surface electromyography (sEMG) and inertial measurement units (IMU) enables quantitative assessment of muscle activation and segmental dynamics, providing objective data for therapeutic planning. Case presentation: A 57-year-old male with chronic right hemiplegia, eight years post-ischemic stroke of the left middle cerebral artery. The patient ambulated independently without assistive devices, exhibiting right lower-limb circumduction. Clinical assessment revealed the following scores: Barthel Index 85/100, Tinetti Performance-Oriented Mobility Assessment (POMA) 16/28, Timed Up and Go (TUG) test 13 s, and Modified Ashworth Scale (MAS) scores of 1 (upper limb) and 1+ (lower limb). Methods: Multichannel sEMG (Miotool 800^®, 8 channels) was recorded form the lumbar erectors, gluteus medius and maximus, vastus medialis, vastus intermedius, vastus lateralis, biceps femoris, tibialis anterior, medial gastrocnemius, and lateral gastrocnemius. Ag/AgCI electrodes were positioned according to SENIAM recommendations: sampling rate: 1000 Hz; band-pass filter: 20–500 Hz; notch filter: 60 Hz; normalization to %MVC. Simultaneously, IMU signals (Xsens DOT^®, 60 Hz) were collected from both ankles during slow, medium and fast walking (20 s each) and compared with a healthy control subject. Results: The patient exhibited reduced sEMG amplitude and increased peak irregularity on the affected side, particularly in the gluteus medius, tibialis anterior, and gastrocnemius, along with agonist desynchronication. IMU data revealed decreased range of motion and angular pattern irregularity, with inconsistent acceleration peaks in the right ankle compared to the control, confirming neuromuscular and kinematic asymmetry. Conclusions: The combined sEMG-IMU analysis identified deficits in selective motor control and propulsion on the affected hemibody, providing essential information to guide physiotherapeutic interventions targeting pelvic stability, dorsiflexion, and propulsive phase training, enabling objective follow-up beyond specialized laboratory settings. Full article