Search Results (371)

This study presents a comprehensive investigation of exact fractional wave solutions and bifurcation analysis for the (3 + 1)-dimensional extended shallow water wave (3D-eSWW) equation with $\beta$-derivative, which models nonlinear wave phenomena in fluid dynamics and coastal engineering. Leveraging the flexibility of the fractional derivative, the model provides a more generalized and adaptable framework for describing shallow water wave propagation. The Modified Extended Direct Algebraic Method (MEDAM) is systematically employed to derive a broad spectrum of novel exact analytical solutions. These include the following: dark solitary waves, singular solitons, singular periodic waves, periodic solutions expressed via trigonometric and Jacobi elliptic functions, polynomial solutions, hyperbolic wave patterns, combined dark–singular structures, combined hyperbolic–linear waves, and exponential-type wave profiles. Each solution family is presented with explicit parameter constraints that ensure both mathematical consistency and physical relevance, thereby offering a robust classification of wave regimes under diverse conditions. A thorough bifurcation analysis is conducted on the reduced dynamical system to examine parametric dependence and stability transitions. Critical bifurcation thresholds are identified, and distinct solution branches are mapped in the parameter space spanned by wave numbers, nonlinear coefficients, external forcing, and the fractional order $\beta$. The analysis reveals how solution dynamics undergo qualitative transitions—such as the emergence of solitary waves from periodic patterns or the appearance of singular structures—driven by the interplay of nonlinearity, dispersion, and fractional-order effects. These insights are crucial for understanding wave stability, predictability, and the onset of extreme events in shallow water contexts. Graphical representations of selected solutions validate the analytical results and illustrate the influence of $\beta$ on wave morphology, propagation, and stability. The simulations demonstrate that varying the fractional order can significantly alter wave profiles, highlighting the role of fractional calculus in capturing complex real-world behaviors. This work demonstrates the efficacy of the MEDAM technique in handling high-dimensional fractional nonlinear PDEs and provides a systematic framework for predicting and classifying wave regimes in real-world shallow water environments. The findings not only enrich the solution inventory of the 3D-eSWW equation but also advance the analytical toolkit for studying complex spatio-temporal dynamics in fractional mathematical physics and fluid mechanics. Ultimately, this research contributes to the development of more accurate models for coastal protection, tsunami forecasting, and marine engineering applications. Full article

We test whether $f\left(Q\right)$ symmetric teleparallel gravity theories can solve the Hubble tension consistently with DESI DR2 BAO. We consider three $f\left(Q\right)$ functional forms: logarithmic, exponential, and hyperbolic tangent. We extend these models by allowing a cosmological constant, and compare to phenomenological models with a flexible exponential, hyperbolic secant, and polynomial decay addition to the standard $\Lambda$CDM $H\left(z\right)$. We test these models against DESI DR2 BAO, CMB (Planck 2018 + SPT-3G + ACT DR6), local $H_0$, and Cosmic Chronometer data. The logarithmic and hyperbolic tangent $f\left(Q\right)$ models do not provide an adequate solution, but the exponential model does. Furthermore, it slightly reduces the $({\Omega }_m,H_0{r}_d)$ parameter space tension between CMB and BAO datasets to $2.56\sigma$, down from $2.65\sigma$ for $\Lambda$CDM. Although $\Lambda$CDM faces only $1.66\sigma$ tension in DESI data space, the $1\sigma$ higher tension in parameter space suggests a real anomaly. The models assisted by the cosmological constant perform slightly better still, at the cost of undermined theoretical motivation. They also perform poorly once local $H_0$ measurements are included. The phenomenological models fit all data reasonably well, yet the best-fitting models predict isotropically averaged BAO distances exceeding the DESI DR2 measurements at all redshifts. This highlights the difficulties of finding a theoretically motivated solution to the Hubble tension while remaining consistent with BAO data. Full article

The paper presents the development of a correlation model for initial tensile elastic modulus for flexible polymers as a function of Shore hardness in OO and A scale based on measurement. Measured polymers are in groups of silicone rubber, nitrile butadiene rubber (NBR), thermoplastic polyurethane (TPU) and silicone. The model is composed of piecewise exponential functions with fixed coefficients chosen to minimize the S₂ error norm and absolute value of relative error at the measured data points. Every chosen section of the hardness scale has one exponential function correlating the hardness to tensile elastic modulus with the argument in the form of a polynomial up to the fourth degree. The coefficients for the polynomial arguments were determined by enforcing interpolation conditions in a chosen set of points in the logarithmic scale for the elastic modulus. The correlation model possesses C0 continuity. For each material, five specimens were used for hardness measurements and five for the elastic modulus testing. The correlation model gives a positive value for elastic modulus of 0 for hardness, and a “finite”, “reasonable” value of 100 for hardness and is monotonic. Tensile properties were evaluated using true stress and logarithmic (Hencky) strain, with iterative correction of the changing cross-sectional area to account for large strain. The maximum relative error achieved in the correlation model for the OO scale is 13.4%, while for the A scale it is 7%. The developed model provides a practical and rapid method for estimating the initial tensile elastic modulus from non-destructive hardness measurements and is particularly useful in industrial applications and in the development of material models for dental surgery simulations. Full article

The bichromatic triangle polynomial $P_G\left(k\right)$ counts vertex k-colorings in which every triangle uses exactly two colors. We develop a transfer matrix framework for three canonical families of 2-trees: book graphs $B_n$, 1-fans $F_n^1$, and triangulated ladders $T{L}_m$. In each case, $P_G\left(k\right)$ satisfies a second-order linear recurrence with an explicit closed form; for $T{L}_m$ this yields a Chebyshev representation, while for $F_n^1$ the binary specialization gives $P_{F_n^1}\left(2\right)=2F_{n+1}$. A spectral identity ${\alpha }^2=r_{+}$ links the dominant characteristic roots of the fan and ladder recurrences, implying identical exponential growth rates when indexed by vertex count, whereas book graphs grow strictly faster for $k\ge 4$. In fact, this correspondence is exact: for all $k\ge 2$, the triangulated ladder polynomial coincides with that of a suitably indexed 1-fan. Passing to line graphs, we interpret $P_{L\left(K_n\right)}\left(k\right)$ as counting edge colorings of $K_n$ that forbid both monochromatic and rainbow triangles, and we identify a sharp obstruction threshold at $n\ge 6$. Full article

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation. Full article

In this work, we construct a new class of Appell-type polynomials generated through extended truncated and truncated exponential kernels, and we analyze their core algebraic and operational features. In particular, we establish a suitable recurrence scheme and obtain the associated multiplicative and differential operators. By confirming the quasi-monomial structure, we further deduce the governing differential equation for the proposed family. In addition, we present both a series expansion and a determinant formulation, providing complementary representations that are useful for symbolic manipulation and computation. As special cases, we introduce and study subfamilies arising from this setting, namely, extended truncated exponential versions of the Bernoulli, Euler, and Genocchi polynomials, and discuss their structural identities and operational behavior. Overall, these developments broaden the theory of special polynomials and furnish tools relevant to problems in mathematical physics and differential equations. Full article

Visual object tracking from unmanned aerial vehicles (UAVs) demands both high accuracy and computational efficiency for real-time deployment on resource-constrained platforms. While state space models (SSMs) offer linear computational complexity, existing methods face critical deployment challenges. They rely on the HiPPO framework with complex discretization procedures and employ hardware-aware algorithms optimized for high-performance GPUs, which introduce deployment overhead and are difficult to transfer to edge platforms. Additionally, their fixed polynomial bases may cause information loss for tracking features with complex geometric structures. We propose LF-SSM, a lightweight HiPPO (High-order Polynomial Projection Operators)-free state space model that reformulates state evolution on Riemannian manifolds. The core contribution is the Geodesic State Module (GSM), which performs state updates through tangent space projection and exponential mapping on the unit sphere. This design eliminates complex discretization and specialized hardware kernels while providing adaptive local coordinate systems. Extensive experiments on UAV benchmarks demonstrate that LF-SSM achieves state-of-the-art performance while running at 69 frames per second (FPS) with only 18.5 M parameters, demonstrating superior efficiency for real-time edge deployment. Full article

The New Polynomial Exponential Distribution (NPED), introduced by Beghriche et al. (2022), provides a flexible one-parameter family capable of representing diverse hazard shapes and heavy-tailed behavior. Regression frameworks based on the NPED, however, have not yet been established. This paper introduces two methodological extensions: (i) a generalized linear model (NPED-GLM) in which the distribution parameter depends on covariates, and (ii) a Poisson–NPED count regression model suitable for overdispersed and heavy-tailed count data. Likelihood-based inference, asymptotic properties, and simulation studies are developed to investigate the performance of the estimators. Applications to engineering failure-count data and insurance claim frequencies illustrate the advantages of the proposed models relative to classical Poisson, negative binomial, and Poisson–Lindley regressions. These developments substantially broaden the applicability of the NPED in actuarial science, reliability engineering, and applied statistics. Full article

This paper introduces a generalized class of Weyl-type, Witt-type, and non-associative algebras constructed over an exponential–polynomial (expolynomial) framework. For fixed scalars ${\iota }_1,\dots ,{\iota }_r\in \mathcal{A}$ and for fixed integers $\mathbf{p}=(p_1,\dots ,p_n)\in {\mathbb{N}}^n$, we define the $\mathbb{F}$-algebra $\mathbb{F}\left[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}\right],$ an expolynomial ring over a field $\mathbb{F}$ of characteristic zero, where $\mathcal{A}$ is an additive subgroup of $\mathbb{F}$ containing $\mathbb{Z}$. This formulation extends the classical Weyl algebra through the integer power parameter $\mathbf{p}$, which generates a family of non-isomorphic simple algebras. The corresponding Weyl-type algebra $A\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right),$ the Witt-type Lie algebra $W\left(\mathbb{F}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right),$ and their non-associative variants are examined in detail. The simplicity, grading, and automorphism structures of these algebras are established, and the dependence of these properties on the deformation parameter $\mathbf{p}$ is analyzed. All the constructed Weyl-type algebras, the corresponding Witt-type Lie algebras, and the non-associative algebras are shown to be simple under derivation structures. Many naturally occurring subalgebras, such as the integer-coefficient subalgebra $A\left(\mathbb{Z}[e^{\pm x^{\mathbf{p}}{e}^{\iota x}},e^{\mathcal{A}x},x^{\mathcal{A}}]\right)$, are also proven to be simple. Our analysis reveals that different choices of $\mathbf{p}$ result in non-isomorphic algebraic structures while retaining non-commutativity. The results obtained generalize several existing constructions of Weyl-type algebras and lay the theoretical foundation for further developments in transcendental and non-commutative algebraic frameworks. Full article

To address the lack of accurate strength evaluation methods of the TMS concrete, this study focused on establishing a multi-age correlation model between the RS and CS of the TMS concrete. Sixteen groups of the TMS concrete with differentiated mix proportions were designed, and XRF/XRD techniques were used to characterize the chemical and mineral compositions of the TMS. RS and CS tests were conducted on standard cubic specimens at 3 d, 7 d, and 28 d ages, and linear, quadratic polynomial, and exponential functions were adopted for fitting analysis. The optimal model for each age was screened using the coefficient of determination, F-test, Akaike information criterion, and Bayesian information criterion. To verify the model and eliminate size effect interference, a large-scale plate specimen was fabricated for tests. Results showed that the correlation between RS and CS of the TMS concrete varied with age. Linear function was optimal for 3 d, quadratic polynomial function for 7 d, and exponential function for 28 d. All models passed the F-test. The relative errors of the piecewise model in large-scale specimen verification were stably controlled within 5.0%, meeting engineering-allowable error requirements. Crucially, the validation confirmed that the size effect is negligible for TMS concrete components within the investigated mix proportion range, eliminating the need for size correction factors. Consequently, this model can be directly applied to the non-destructive strength testing of TMS concrete prepared with P.O 42.5 Portland cement at 3 d, 7 d, and 28 d ages without the need for parameter adjustment regarding component dimensions. Full article

Grammar induction runs into a serious problem due to the exponential growth of the number of possible derivation trees as sentence length increases, which makes unsupervised parsing both computationally demanding and highly indeterminate. This paper proposes a mathematics-based approach that alleviates this combinatorial complexity by introducing structural constraints based on Catalan and Fuss–Catalan numbers. By limiting the depth of the tree, the degree of branching and the form of derivation, the method significantly narrows the search space, while retaining the full generative power of context-free grammars. A filtering algorithm guided by Catalan structures is developed that incorporates these combinatorial constraints directly into the execution process, with formal analysis showing that the search complexity, under realistic assumptions about depth and richness, decreases from exponential to approximately polynomial. Experimental results on synthetic and natural-language datasets show that the Catalan-constrained model reduces candidate derivation trees by approximately 60%, improves F1 accuracy over unconstrained and depth-bounded baselines, and nearly halves average parsing time. Qualitative evaluation further indicates that the induced grammars exhibit more balanced and linguistically plausible structures. These findings demonstrate that Catalan-based structural constraints provide an elegant and effective mechanism for controlling ambiguity in grammar induction, bridging formal combinatorics with practical syntactic learning. Full article

We investigate the multifractal geometry of irregular sets arising from weighted averages of random variables, where the weights $\left(w_n\right)$ form a positive sequence with exponential growth. Our analysis applies in particular to sequences generated by linear recurrence relations of Fibonacci type, including higher-order generalizations such as the Tetranacci sequence $\left({\mathsf{T}}_n\right)$. Using a Cantor-type construction built from alternating free and forced blocks, we show that the associated exceptional sets may attain full Hausdorff and packing dimension, independently of the precise form of the recurrence. We further develop a probabilistic interpretation of $\left({\mathsf{T}}_n\right)$ through an appropriate Markov representation that encodes its combinatorial evolution and yields sharp asymptotic behavior. Finally, given $n+1$ consecutive terms of a Fibonacci-type sequence, one may construct a polynomial $P_n\left(x\right)$ of degree at most n via Lagrange interpolation; we show that this polynomial admits an implicit recursive representation consistent with the underlying recurrence. Full article

The existing methods used for estimating generator matrixes of BCH codes, which are based on Galois Field Fourier transforms, need to exhaustively test all the possible codeword lengths and corresponding primitive polynomials. With the increase of codeword length, the search space exponentially expands. Consequently, the computational complexity of the estimation scheme becomes very high. To overcome this limitation, a fast estimation method is proposed based on Gaussian elimination. Firstly, the encoded bit stream is reshaped into a matrix according to the assumed codeword length. Then, by using Gaussian elimination, the bit matrix is simplified as the upper triangle form. By testing the independent columns of the upper triangle matrix, the assumed codeword length is judged to be right or not. Simultaneously, by using an augmented matrix, the parity check matrix of a BCH code can be estimated from the simplification result in the procedure of Gaussian elimination. Furthermore, the generator matrix is estimated by using the orthogonality between the generator matrix and parity check matrix. To improve the performance of the proposed method in resisting bit errors, soft-decision data is adopted to evaluate the reliability of received bits, and reliable bits are selected to construct the matrix to be analyzed. Experimental results indicate that the proposed method can recognize BCH codes effectively. The robustness of our method is acceptable for application, and the computation required is much less than the existing methods. Full article

The aim of this paper is to introduce Leonardo finite operator polynomials and obtain some of their new properties. We first present the recurrence relation provided by Leonardo finite operator polynomials. Then, we give a Binet-like formula, generating function, exponential generating function, and a finite sum formula for Leonardo finite operator polynomials. We present a determinant representation for the nth term of Leonardo finite operator polynomials. Ultimately, by utilizing the generating function of the proposed polynomials, we establish generating relations for specific bilinear and bilateral polynomial families. This approach thus broadens the applicability of the finite operator framework to encompass a wider range of special functions. Full article

We develop a shared denominator Carathéodory–Fejér (CF) method for efficiently evaluating linear combinations of $\phi$-functions for matrices whose spectrum lies in the negative real axis, as required in exponential integrators for large stiff ODE systems. This entire family is approximated with a single set of poles (a common denominator). The shared pole set is obtained by assembling a stacked Hankel matrix from Chebyshev boundary data for all target functions and computing a single SVD; the zeros of the associated singular-vector polynomial, mapped via the standard CF slit transform, yield the poles. With the poles fixed, per-function residues and constants are recovered by a robust least squares fit on a suitable grid of the negative real axis. For any linear combination of resolvent operators applied to right-hand sides, the evaluation reduces to one shifted linear solve per pole with a single combined right-hand side, so the dominant cost matches that of computing a single $\phi$-function action. Numerical experiments indicate geometric convergence at a rate consistent withHalphen’s constant, and for highly stiff problems our algorithm outperforms existing Taylor and Krylov polynomial-based algorithms. Full article