Search Results (373)

A recent two-layer theory for long-runout turbidity currents explains sustained bypass by allowing a dense lower layer to exchange mass with a more dilute upper layer while avoiding rapid over-thickening. Here, a morphodynamic extension is developed that couples suspended load and bed exchange while treating the two-layer hydrodynamics as a prescribed background. A suspended-sediment balance with bed exchange and Exner’s equation are written on that background, the depositional state variable $B=E_s/\left(rC\right)$ is introduced, and an exact nonlinear evolution equation for B is derived within the prescribed-background setting. In the weak-exchange limit this equation reduces to an algebraic onset criterion, thereby identifying the regime in which the simpler threshold is valid. Applied to an Amazon-like local-normal-flow reconstruction, the model shows that finite exchange shifts depositional onset upstream relative to the weak-exchange estimate. Background-fidelity checks, grid-refinement tests and closure/inlet sensitivities are reported to delimit the quantitative use of the reduced application. The framework is therefore best interpreted as a coupled reduced theory for suspended load and bed exchange on a prescribed two-layer bypass background rather than a fully hydro-morphodynamic closure. Full article

Let X be a compact connected Riemann surface of genus $g\ge 2$ and let E be a holomorphic vector bundle of rank n over X. The compactness and connectedness of X imply that the characteristic polynomial of any holomorphic endomorphism $\phi \in H^0(X,\mathcal{E}nd\left(E\right))$ has constant coefficients, a fact we call the Principle of Spectral Constancy. As a consequence, the eigenvalues of $\phi$ are globally constant over X, the primary decomposition of E with respect to $\phi$ consists of globally defined holomorphic subbundles, and the Jordan decomposition $\phi ={\phi }_s+{\phi }_n$ into semisimple and nilpotent parts is globally well defined as a decomposition of sections of $\mathcal{E}nd\left(E\right)$. This paper provides a systematic analysis of Jordan normal forms for endomorphisms of holomorphic vector bundles over X, relating the Jordan type of $\phi$ to the stability properties of E. In particular, it is proved that endomorphisms of stable bundles are necessarily scalar, that the Jordan decomposition of an endomorphism of a polystable bundle is determined componentwise by the classical Jordan normal forms of matrices in the associated endomorphism algebra, and that finite-order endomorphisms are always semisimple. These results are applied to the study of fixed points of automorphisms of the moduli space ${\mathcal{B}}_X\left(\mathrm{SL}(n,\mathbb{C})\right)$ of rank n and trivial determinant polystable vector bundles over X. Specifically, a new result establishes that the commutative subalgebra of $H^0(X,\mathcal{E}nd\left(E\right))$ generated by the endomorphism associated with a fixed-point condition is semisimple, so nilpotent endomorphisms of E are precisely those incompatible with the fixed-point structure. Full article

This paper presents AeroelasticQ, a modular, high-performance aeroelastic simulation code for wind turbines, with particular emphasis on future applicability to multi-rotor configurations. The framework is organized into three core components: a flexible-blade structural solver, an airfoil-based aerodynamic solver, and a two-mesh aero-structural mapping module for transferring loads and kinematics between the aerodynamic and structural discretization. The implementation is written in C++17 using the Eigen linear algebra library (v5.0.0), and OpenMP (v5.1) is employed to enable rotor-level parallel execution for multi-rotor applications. The structural dynamics are formulated using Kane’s dynamic method combined with modal superposition, while the aerodynamic loads are computed using three-dimensional blade element momentum theory. The coupled and uncoupled modules are validated in the time domain against OpenFAST (v4.1.2) AeroDyn, ElastoDyn, and the coupled AeroDyn–ElastoDyn configuration using the NREL 5 MW reference wind turbine. The rotor-level aerodynamic validation gives mean absolute errors of 8.94 × 10⁻⁴, 2.82 × 10⁻⁴, and 2.71 × 10⁻⁵ for C_t, C_p, and C_q, respectively, while the coupled aeroelastic cases show close agreement in blade tip deflections, blade root loads, and aerodynamic power. A rigid three-rotor verification confirms the multi-rotor load-aggregation framework, with tower base thrust and overturning moment errors below 1.5% and 2% NRMSE, respectively, in both all rotors operating and one operating/two-parked configurations. In single-thread benchmarks, AeroelasticQ achieves speedups of 5.23×, 19.69×, and 3.65× in the aerodynamic-only, structural-only, and fully coupled modes, respectively. In the multi-rotor benchmark, the five-rotor case achieves a parallel speedup of 2.55× with a parallel efficiency of 51%. Full article

This study evaluates structure-preserving neural network architectures for learning holonomically constrained mechanical dynamics in Cartesian coordinates. In contrast to methods using reduced coordinates, the full ambient phase space ${\mathbb{R}}^{2n}$ is retained with explicit algebraic constraints $C_i\left(\mathbf{q}\right)=0$ to provide a test bed for constraint-aware learning. The Constraint-Aware Hamiltonian Neural Network (CA-HNN) is proposed, which augments the standard HNN with a dedicated multiplier network ${\lambda }_{\varphi }(\mathbf{q},\mathbf{p})$ for Lagrange multipliers and a composite loss function evaluated on predicted rollouts. The theoretical framework is grounded in the geometry of constrained Hamiltonian systems: the extended phase space ${\mathbb{R}}^{2n+m}$ carries a degenerate antisymmetric structure where an m-dimensional kernel encodes constraint directions, while the symplectic structure emerges on the $2(n-m)$-dimensional reduced manifold $\Sigma$. It is proven that the physical Hamiltonian is conserved on the constraint surface under augmented flow. Benchmarks on a pendulum ($C=x^2+y^2-l^2$), double pendulum, and bead on a parabola ($C=y-x^2$) demonstrate that CA-HNN reduces constraint violations $\parallel C\left(\mathbf{q}\right)\parallel$ by $5\times$ to $2400\times$ compared to standard HNNs. While the best energy conservation is achieved by PINNs, these findings clarify the roles of architectural inductive bias, constraint augmentation, and soft physics regularization. Full article

Dynamic Movement Primitives (DMPs) are widely used for learning and generalizing robot skills. However, standard quaternion DMPs, when modeling orientation trajectories, constrain only the final orientation and cannot freely specify the final angular velocity. This limitation restricts its application to dynamic tasks requiring precise boundary conditions, such as hitting or throwing. Although existing improved methods achieve velocity generalization to some extent, they often struggle to balance trajectory shape preservation with dynamic smoothness, frequently causing significant deviation from demonstrations or abrupt acceleration discontinuities. In this paper, we propose a novel robot skill generalization method that enables controllable final angular velocity for quaternion DMPs. Specifically, we construct a dynamic goal system driven by a quintic polynomial in Lie algebra space, analytically planning the target orientation’s evolution based on given multi-order boundary constraints. This mechanism not only achieves precise control over the final angular velocity but also inherently guarantees global C² continuous dynamics across primitive segments. Comparative simulations and real-world robot hitting experiments demonstrate that, compared to existing approaches, our proposed method effectively satisfies dynamic boundary constraints while exhibiting superior shape preservation, minimal trajectory deviation, and higher smoothness, thereby significantly improving skill generalization performance in complex dynamic tasks. Full article

The output voltage harmonics of electric vehicle (EV) charging converters directly affect grid power quality. This paper proposes a harmonic analysis method based on geometric algebra (GA), which employs a multivector representation of signals and least squares estimation to accurately extract fundamental, integer-order, and inter-harmonics. A coupling coefficient is defined to quantify the phase correlation between frequency components. Based on measured data, harmonic characteristics under four typical operating conditions are analyzed, and an adaptive PID controller is designed to dynamically adjust the virtual resistance for harmonic suppression. The results show that the GA method significantly reduces spectral leakage under non-integer-period sampling conditions, with amplitude estimation errors below ±2%. The total harmonic distortion (THD) decreases with increasing active power and increases with reactive power injection. The droop coefficient exhibits a non-monotonic effect, while reducing the proportional gain raises the THD. Adaptive control reduces the average THD by 14.0–28.5% with a total response time of less than 0.05 s. The coupling coefficient $C_{13}$ is strongly positively correlated with THD and negatively correlated with the maximum Lyapunov exponent computed using the Rosenstein small-data method (correlation coefficient −0.85), confirming the intrinsic relationship between coupling and stability. Compared with fast Fourier transform (FFT) and other methods, GA achieves higher accuracy under short data records and non-integer-period sampling. This paper provides a complete theoretical framework and engineering solution for harmonic suppression in charging converters. Full article

Consider a unital ∗-algebra $\mathfrak{A}$ defined over the complex field $\mathbb{C}$. In this work, we establish that a mapping, referred to as a nonlinear mixed left bi-skew Jordan and right Jordan n-derivation, reduces to an additive ∗-derivation under certain conditions. As applications, we further investigate special classes of unital ∗-algebras, namely, prime ∗-algebras and factor von Neumann algebras using our main result. Full article

This paper is devoted to provide a proof of F. Morley’s theorem concerning triangles in the Euclidean plane $\mathcal{E}$ (see Theorem 1 in the Introduction section) phrased in terms of the geometric algebra $\mathcal{G}$ of $\mathcal{E}$ (called Wessel’s algebra). This algebra is studied in detail in Section 2, its uses in describing isometries of $\mathcal{E}$ in Section 3, its bearing on the geometry of Morley’s construction in Section 4, and the claimed proof in Section 5. Morley’s theorem can be extended by using all the trisectors (interior and exterior) of a triangle, and suitable intersections of them. These intersections form what we call Morley’s constellation and out of it 36 generalized Morley triangles can be formed. Among these triangles, 27 are equilateral and with sides parallel to the original Morley triangle (Appendix B). The 36 triangles are depicted in Appendix C. All graphics in this work have been created by the author. Full article

Human motivation is governed by a long-memory cognitive process in which the depth of temporal integration—how far into the past the system draws upon accumulated experience—is not fixed, but dynamically compressed under cognitive stress. Despite extensive empirical evidence that acute stress impairs working memory and narrows temporal integration in decision-making, no existing mathematical framework has formally coupled the memory depth of the governing operator to a physiologically grounded stress indicator. To address this gap, we propose a stress-adaptive variable-order fractional model for motivational intensity $M\left(t\right)$, in which the Caputo fractional order $\alpha \left(t\right)$ varies inversely with an aggregated stress indicator $\sigma \left(t\right)$ through the Hill-type coupling $\alpha \left(t\right)={\alpha }_{min}+({\alpha }_{max}-{\alpha }_{min})C/(C+\sigma \left(t\right))$, thereby encoding the empirically documented shift from deep integrative to shallow heuristic processing as cognitive load increases. Rather than deriving the model by algebraic manipulation of a differential equation, we formulate it directly as a causally consistent type-III Volterra integral equation, in which the memory kernel is evaluated at the history time s, ensuring that the weight assigned to each past state reflects the memory depth that was physiologically active when that state was experienced. Well-posedness is established rigorously via the Banach fixed-point theorem with explicit contraction constants, uniform boundedness and non-negativity of solutions are derived through the fractional Gronwall inequality, and numerical solutions are computed using an Adams–Bashforth–Moulton predictor–corrector scheme adapted to the variable-order kernel. Five numerical experiments demonstrate that stress-induced variation in $\alpha \left(t\right)$ produces qualitatively richer dynamics compared with the tested constant-order baselines: the proposed model achieves a steeper peak decline rate (0.48 versus 0.19–0.45), a larger burnout gap (3.15 versus 1.92–2.81), and faster recovery to ninety percent of peak motivation (4.2 versus 3.9–7.3 time units), while the empirically observed numerical convergence approaches $O\left(h^2\right)$ for sufficiently small step sizes. The framework offers a principled phenomenological substrate for memory-adaptive cognitive modelling, with direct implications for stress-aware intelligent tutoring systems that are capable of inferring $\alpha \left(t\right)$ in real time from biometric signals such as heart rate variability or galvanic skin response, and adjusting instructional complexity accordingly. Empirical calibration against learning-analytics and psychophysiological datasets, together with stochastic extensions for probabilistic burnout-risk prediction, are identified as immediate priorities for future research. Full article

We consider Banach spaces ${\ell }_p\left(\mathcal{C}\right),$ $1\le p<\infty ,$ where the index set $\mathcal{C}$ is the classical Cantor set and study various groups of symmetries of ${\ell }_p\left(\mathcal{C}\right),$ associated with the binary representation of $\mathcal{C}.$ The main purpose of the paper is the investigation of polynomials on ${\ell }_p\left(\mathcal{C}\right)$ that are symmetric (i.e., invariant) with respect to the constructed groups $G.$ We are interested in finding systems of generators of algebras of G-symmetric polynomials for different groups G and we discuss possible applications of G-symmetric polynomials to highly composite physical systems. The generators are useful for descriptions of spectra of algebras of G-symmetric analytic functions on ${\ell }_p\left(\mathcal{C}\right),$ and for the construction of some nontrivial complex homomorphisms of these algebras. Finally, we establish the topological transitivity and hypercyclicity of some shift-like operators on ${\ell }_p\left(\mathcal{C}\right)$ and its subspaces, and translation operators on algebras of symmetric analytic functions on ${\ell }_p\left(\mathcal{C}\right).$ Full article

Within the framework of Somigliana’s displacement and traction identities, we propose an extended equivalent elastic model that enables a BEM that is singularity-free in the primary solution stage for two-dimensional elastostatics. The central idea is to shift the integration boundary from the physical contour $S_1$ to an auxiliary contour $S_2$, introducing a geometric separation that removes boundary-source singularities from the discrete system. When the separation between $S_1$ and $S_2$ is sufficiently large, all integrals in the assembled algebraic equations become regular, and post-processing can be performed in a unified manner using the same nonsingular expressions for both boundary and interior evaluation. We introduce a practical separation measure, the dimensionless parameter $\delta$, and verify that a moderate choice (e.g., $\delta \approx 0.5$) is effective through a rigid-body displacement test. Benchmark examples demonstrate that, at lower computational cost, the proposed method improves accuracy and convergence compared with the conventional direct BEM (displacement boundary integral equation (BIE) with free-term coefficient $c=1/2$) and compares favorably with the finite element method (FEM). In particular, thin structures can be treated directly without invoking plate or shell theories. Full article

Let $\epsilon >0$ and $\mathcal{TB}(\mathcal{X}\times \mathcal{X})$ be the Banach algebra of all $2\times 2$ bounded upper triangular operator matrices on a separable Hilbert space $\mathcal{X}\times \mathcal{X}$. In this paper, we first establish the spectrum equalities for special cases of upper triangular operator matrices—diagonal block operator matrix $M_0=\left({\displaystyle \genfrac{}{}{0pt}{}{A0}{0B}}\right)$. We obtain that ${\widehat{\Sigma }}_{bi,\epsilon }\left(M_0\right)={\Sigma }_{bi,\epsilon }\left(A\right)\cup {\Sigma }_{bi,\epsilon }\left(B\right)$, $i\in \{1,2,4\}$, where ${\Sigma }_{bi,\epsilon }(·)$ and ${\widehat{\Sigma }}_{bi,\epsilon }(·)$ denote the noncommutative pseudo-upper (resp. lower) semi-Browder essential spectrum, noncommutative pseudo-Browder essential spectrum, sub-noncommutative pseudo-upper (resp. lower) semi-Browder essential spectrum, and sub-noncommutative pseudo-Browder essential spectrum. Secondly, based on Cao and Bai’s works, we study the perturbation of the sub-noncommutative pseudo-Browder essential spectrum ${\widehat{\Sigma }}_{b4,\epsilon }(·)$ of a 2 × 2 bounded upper triangular operator matrix $M_C=\left({\displaystyle \genfrac{}{}{0pt}{}{AC}{0B}}\right)$ on a separable Hilbert space. We obtain that ${\displaystyle \bigcap _{C\in \mathcal{B}\left(\mathcal{X}\right)}{\widehat{\Sigma }}_{b4,\epsilon }\left(M_C\right)={\Sigma }_{b1,\epsilon }\left(A\right)\cup {\Sigma }_{b2,\epsilon }\left(B\right)\cup \Delta }$, where $\Delta =\{\lambda \in \mathbb{C}:$ there exist $P_i\in \mathcal{B}\left(\mathcal{X}\right)$ with $\parallel P_i\parallel <\epsilon ,i\in \{1,2\},$ such that $\alpha (A+P_1-\lambda I)+\alpha (B+P_2-\lambda I)\ne \beta (A+P_1-\lambda I)+\beta (B+P_2-\lambda I)\}$. Finally, we obtain ${\Sigma }_{bi,\epsilon }\left(A\right)\cup {\Sigma }_{bi,\epsilon }\left(B\right)={\widehat{\Sigma }}_{bi,\epsilon }\left(M_C\right)\cup W,i\in \{1,2,4\}$, where W is the union of certain holes in $({\Sigma }_{bi,\epsilon }\left(A\right)\cup {\Sigma }_{bi,\epsilon }\left(B\right))\backslash {\widehat{\Sigma }}_{bi,\epsilon }\left(M_C\right)$. Full article

In this paper, the uniform approximation of continuous functions on $[0,1]$ by rational functions with prescribed poles and bounded multiplicities is studied. A classical theorem of Fichera characterizes density in $C\left(\right[0,1\left]\right)$ through the divergence of a conformally invariant series involving the pole distribution. A modern reformulation of this result is developed and it is given an operator-theoretic interpretation in which the approximation property is equivalent to cyclicity and to the absence of nontrivial invariant subspaces in an associated Hardy-space model. In this framework, the classical Blaschke condition emerges as the fundamental obstruction to density, linking rational approximation to the structure of model spaces and non-selfadjoint operator algebras. The density criterion is interpreted in terms of symmetry: divergence corresponds to a balanced distribution of poles compatible with the conformal geometry of the slit domain, while convergence induces symmetry breaking and the emergence of invariant structures. Numerical models illustrate the sharpness of the criterion and provide a concrete manifestation of the Blaschke obstruction and cyclicity mechanism. This new approach places Fichera’s theorem within a broader operator-theoretic and spectral framework, connecting classical approximation theory with Hardy spaces, invariant subspace theory, and modern rational approximation methods. Full article

Let X be a compact Riemann surface of genus $g\ge 2$. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal $G_2\left(\mathbb{C}\right)$-bundle, where $G_2\left(\mathbb{C}\right)$ is the exceptional Lie group of automorphisms of the octonions. We prove that the natural inclusion $G_2\left(\mathbb{C}\right)\hookrightarrow \mathrm{SO}(7,\mathbb{C})$ induces a closed embedding of the moduli space ${\mathcal{M}}_{\mathrm{Oct}}\left(X\right)$ into the moduli space ${\mathcal{M}}_{\mathrm{SO}(7,\mathbb{C})}\left(X\right)$ of $\mathrm{SO}(7,\mathbb{C})$-bundles. We further analyze the normal bundle to this embedding, computing its rank as $7(g-1)$ and providing an explicit cohomological description of its fibers, which enables explicit computations of tangent spaces and provides a foundation for deformation theory. As applications of the embedding, we prove that the image is a closed irreducible subvariety not contained in the singular locus of the ambient space, and we derive the Whitney formula $c\left(T_{\mathrm{amb}}\right)=c\left(T\right)·c\left(N\right)$ relating the Chern classes of the tangent bundle of ${\mathcal{M}}_{\mathrm{Oct}}\left(X\right)$, the pullback of the ambient tangent bundle, and the normal bundle over the smooth locus. Full article

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by $\mathcal{C}l\left(H_V\right)$. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra $\mathcal{C}l\left(H_V\right)$ simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the “mother algebra”—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space $H_V$ are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with $\mathcal{C}l\left(H_V\right)$ are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived. Full article