Search Results (176)

We consider the problem of Bayesian parameter inference in the Merton structural credit risk model, where the posterior is induced by a jump-diffusion likelihood and the marginal evidence is not available in closed form. To approximate this posterior, we construct a variational family based on triangular sum-of-squares (SOS) polynomial flows, in which each component map is monotone by construction: its diagonal derivative is a positive definite quadratic form on a monomial basis, yielding a closed-form log-Jacobian and explicit gradients with respect to all flow parameters. The symmetric positive definite matrices parametrizing the flow are optimized by intrinsic Riemannian gradient ascent on the positive definite cone equipped with the affine-invariant metric, which preserves feasibility at every iterate without projection. We show that the rank-one Jacobian gradients produced by the SOS structure have unit norm in the affine-invariant metric, establishing a direct algebraic coupling between the transport family and the optimization geometry and implying a universal 1-Lipschitz bound for the log-Jacobian along geodesics. On the likelihood side, we derive exact score identities for all five structural parameters of the Merton model—drift, volatility, jump intensity, jump mean, and jump volatility—through both the Poisson log-normal mixture and the Fourier inversion representations. Strictly positive parameters are handled via exponential reparametrization, and the resulting gradients propagate end-to-end through the flow. We establish uniform truncation bounds on compact parameter sets for the infinite mixture and its associated score series, providing rigorous control over the finite approximations used in practice. The base distribution is chosen to be uniform on ${[0,1]}^5$, whose bounded support ensures uniform control of the monomial basis and stabilizes the polynomial calculus. These ingredients are assembled into a fully explicit modified ELBO with implementable gradients, combining Euclidean updates for vector parameters and intrinsic manifold updates for matrix parameters. Full article

In this work, a formal asymptotic framework based on infinite number expressions is employed to investigate structural relations associated with the Dirichlet representation of the Riemann zeta function. Within this framework, infinite number objects are interpreted through asymptotic representatives and serve as symbolic encodings of asymptotic behavior in the regime x → ∞. A divergent real series is constructed from the sum of entries of an n × n matrix in the asymptotic limit n → ∞ and analyzed in relation to the squared modulus of a Dirichlet-type series. When the common parameter coincides with the imaginary part of a nontrivial zero of the Riemann zeta function on the critical line, the framework yields a structured cancellation mechanism, leading to parameter-dependent decay or convergence toward the constant −γ/2. Additional formal asymptotic relations are derived linking nontrivial zeros, divergent expressions, and the Euler–Mascheroni constant. The theoretical analysis is accompanied by numerical computations in double-precision arithmetic, which serve as consistency checks of the predicted asymptotic behavior. The proposed approach provides a coherent representative asymptotic methodology for organizing and analyzing identities involving divergent expressions arising in analytic number theory. The resulting relations are interpreted within this representative framework and are intended as structural asymptotic identities rather than classical equalities of divergent series. Full article

In this paper, we study matrix transformations on spaces of generalized almost convergent double sequences with powers. Extending classical results of Lorentz, Maddox, and Nanda, we characterize several classes of infinite matrices that map between Maddox’s double sequence spaces and spaces of almost convergent (to zero) double sequences with powers. Our results generalize earlier characterizations for single sequence spaces obtained by the authors in previous work, providing a structured framework for studying summability and convergence in higher dimensions. Full article

Non-parametric information geometry has long faced an “intractability barrier”: in the infinite-dimensional setting, the Fisher–Rao metric is a weak Riemannian metric functional that lacks a bounded inverse, rendering classical optimization and estimation techniques computationally inaccessible. This paper resolves this barrier by building the statistical manifold on the Orlicz space $L_0^{\mathsf{\Phi }}\left(P_f\right)$ (the Pistone–Sempi manifold), which provides the necessary exponential integrability for score functions and a rigorous Fréchet differentiability for the Kullback–Leibler divergence. We introduce a novel Structural Decomposition of the Tangent Space ($T_{f}M=S\oplus S^{\perp }$), where the infinite-dimensional space is split into a finite-dimensional covariate subspace (S)—representing the observable system—and its orthogonal complement ($S^{\perp }$). Through this decomposition, we derive the Covariate Fisher Information Matrix (cFIM), denoted as $G_f$, which acts as the computable “Hilbertian slice” of the otherwise intractable metric functional. Key theoretical contributions include proving the Trace Theorem ($H_G\left(f\right)=\mathrm{Tr}\left(G_f\right)$) to identify G-entropy as a fundamental geometric invariant; demonstrating the Geometric Invariance of the Covariate Fisher Information Matrix (cFIM) as a covariant $(0,2)$-tensor under reparameterization; establishing the cFIM as the local Hessian of the KL-divergence; and characterizing the Efficiency Standard through a generalized Cramer–Rao Lower Bound for semi-parametric inference within the Orlicz manifold. Furthermore, we demonstrate that this framework provides a formal mathematical justification for the Manifold Hypothesis, as the structural decomposition naturally identifies the low-dimensional subspace where information is concentrated. By shifting the focus from the intractable global manifold to the tractable covariate geometry, this framework proves that statistical information is not a property of data alone, but an active geometric interaction between the environment (data), the system (covariate subspace), and the mechanism (Fisher–Rao connection). Full article

In this paper, we discuss the geometric structure, i.e., Dirac structure, underlying port-Hamiltonian systems. The paper has a tutorial character, and thus it contains questions/exercises. We start with the general definition of a Dirac structure and show that on finite-dimensional spaces, there is a simple matrix characterization. By simple examples, we show that, even in the finite-dimensional case, a Dirac structure does not guarantee the existence of solutions for an associated ordinary differential or difference equation. For associated partial differential equations, i.e., on an infinite-dimensional Dirac structure, the existence problem becomes even more challenging. We show that the spaces have to be chosen with care, but when we have shown the existence of solutions, then the Dirac structure will give us the desired properties, such as conservation of energy. The Dirac structure also implies that the associated transfer function has nice properties. Full article

An infinite-horizon $H_{\infty }$ linear-quadratic control problem is considered. This problem has the following features: (i) the control cost in the cost functional has a positive small coefficient (small parameter), meaning that the control cost is much smaller than the state cost; (ii) the current cost of the fast state variable in the cost functional is a non-zero positive semi-definite quadratic form. These features require developing a significantly novel approach to asymptotic analysis of the matrix Riccati algebraic equation appearing in the solvability conditions of the considered $H_{\infty }$ problem. Using this solution, an asymptotic analysis of the $H_{\infty }$ problem is carried out. This analysis yields parameter-free solvability conditions for this problem and a simplified controller solving this problem. An example illustrating the theoretical results is presented. Full article

The electronic band structure of two-dimensional lithium is calculated using the Dirac equation. Lithium is modeled as a two-dimensional square lattice in which the two strongly bound inner electrons and the fixed nucleus are treated as a positively charged ion (+e), while the outer electron is assumed to be uniformly distributed within the cell. The electronic potential is obtained by considering Coulomb-type interactions between the charges inside the unit cell and those in the surrounding cells. A numerical method that divides the unit cell into small pieces is employed to calculate the potential and then the Fourier coefficients are obtained. The Bloch method is used to determine the energy bands, leading to an eigenvalue matrix equation (in momentum space) of infinite dimension, which is truncated and solved using standard matrix diagonalization techniques. Convergence is analyzed with respect to the key parameters influencing the calculation: the lattice period, the dimension of the eigenvalue matrix, the unit-cell partition used to compute the potential’s Fourier coefficients, and the number of neighboring cells that contribute to the electronic interaction. Full article

Frequency domain methods used in electromagnetic analyses, such as the Method of Auxiliary Sources (MAS) and the various Moment Methods (MoM), share many similarities but have notable differences in terms of numerical stability, accuracy, and computational cost. Computational cost differs from algorithmic complexity, which is easier to define. Consequently, it is rarely analyzed systematically in numerical studies. To this end, this work deals with the canonical problem of electromagnetic scattering from an externally excited circular cylinder of infinite conductivity and applies both MAS and MoM in order to assess their solutions and behaviors from the aforementioned perspectives. This problem is solved by meticulously applying MAS and two popular variants of MoM to achieve comparable stability and accuracy. Then, the methods are compared in terms of the associated computational cost, not only in solving the ensuing matrix equations, but also in computing the near and far fields at a large number of points. Full article

Circumferential guided wave detection technology can serve as an alternative method for detecting casing bond defects. Due to the presence of the cement cladding, the circumferential SH guided waves transmit shear waves into the cement cladding as they propagate in the cementing casing, which cause the circumferential SH guided waves to show attenuation characteristics. In this study, the cementing casing structure was considered as a steel substratum semi-infinite domain cemented cladding pipe structure, and the corresponding dispersion and attenuation characteristics of circumferential SH guided waves were numerically solved based on the state matrix and Legendre polynomial hybrid method. In addition, a finite element simulation model of cementing casing was established to explore the interaction between SH guided waves and bonding defects. The relationship between the amplitude of SH guided waves and the size of the bonding defects was established through the attenuation coefficient. Moreover, an experimental platform for cementing casing detection is constructed to detect bonding defects of different sizes and to achieve the acoustic analysis of cementing defects in cementing casing, which provides a research path for the non-destructive testing and evaluation of bonding defects in cementing casing. Full article

The mechanical integrity of advanced porous materials and perforated structures at the nanoscale is critically governed by the interaction of surface effects and stress concentration around pore architectures. This paper investigates the residual stress field induced by surface tension around two arbitrarily shaped nano-perforations within an infinite elastic matrix, a configuration highly relevant to nanoporous metals and functional composites. By leveraging the complex variable method and conformal mapping techniques, the physical domains of the perforations (approximated as triangular and square shapes, paired with an elliptical perforation) are transformed into unit circles. This transformation allows for the derivation of semi-analytical solutions for the complex potentials and the subsequent stress field. Systematic numerical case studies reveal that a reduced inter-perforation distance dramatically intensifies the hoop stress concentration at the adjacent vertices, identifying these sites as potential initiation points for mechanical failure. Conversely, an increase in the size of one perforation can effectively shield its neighbor and reduce the overall stress level. These findings provide quantitative, physics-based guidelines for the microstructural design of nanoporous materials. By consciously tailoring the spatial distribution, size, and shape of perforations, the mechanical reliability of nanomaterials can be rationally optimized for applications in nanoscale systems. Full article

Let $A_{ex}\left(\mathcal{G}\right)$ be the extended adjacency matrix of G. The eigenvalues of $A_{ex}\left(\mathcal{G}\right)$ are called extended adjacency eigenvalues of $\mathcal{G}$. The sum of the absolute values of eigenvalues of the $A_{ex}$-matrix is called the extended adjacency energy ${\mathcal{E}}_{ex}\left(\mathcal{G}\right)$ of $\mathcal{G}$. In this paper, we obtain the $A_{ex}$-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the $A_{ex}$-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the $A_{ex}$-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the $A_{ex}$-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the $A_{ex}$-eigenvalues of the power graph of finite abelian groups. Full article

In this study, we analyze a multi-species mutualistic Lotka–Volterra model with Lévy jumps and regime-switching. A defining feature of the work lies in modeling the random environment through state-dependent switching in an infinite countable state space. Our main objective is to establish the sufficient conditions of the extinction and stochastic permanence of the model. First, we analyze the existence and uniqueness of the model’s solution, followed by an examination of the solution’s stochastic ultimate boundedness. Moreover, the challenges arising from state-dependent switching are addressed using the stochastic comparison method. Due to the presence of the jump component, more complex conditions are required to achieve a finite partition of the countably infinite space. Furthermore, the M-matrix theory is also used to obtain the stochastic permanence property. Finally, two specific examples are provided to illustrate the conclusions in this paper. Full article

The matrix transfer function (MTF) is fundamental to the analysis and control of multivariable descriptor systems, especially under zero initial conditions. Its importance lies in its direct relation to input–output behavior and its natural use in frequency-domain methods. Unlike classical approaches that obtain MTF through companion linearizations or indirect Weierstrass–Kronecker reductions, our method derives a closed-form MFD directly from the descriptor pencil $\left(\lambda \mathit{E}-\mathit{A}\right)$, avoiding linearizations and preserving descriptor structure. This yields (i) an explicit parameterization of state feedback gains via finite/infinite Jordan pairs, (ii) a normalization law that removes impulsive modes by design, and (iii) improved reproducibility through block-polynomial operations suited to large-scale MIMO systems. The framework further extends eigenstructure assignment to descriptor models, combining clarity of analysis with practical control design. These results establish a systematic basis for scalable methods in MIMO descriptor systems. Full article

Usually, when the computation of limiting distributions of (in)homogeneous (in)finite continuous-time Markov chains (CTMC) has to be performed numerically, the algorithm has to be told when to stop the computation. Such an instruction can be constructed based on available ergodicity bounds. One of the analytical methods to obtain ergodicity bounds for CTMCs is the logarithmic norm method. It can be applied to any CTMC; however, since the method requires a guessing step (search for proper Lyapunov functions), which may not be successful, the obtained bounds are not always meaningful. Moreover, the guessing step in the method cannot be eliminated or automated and has to be performed in each new use-case, i.e., for each new structure of the infinitesimal matrix. However, the simplicity of the method makes attempts to expand its scope tempting. In this paper, such an attempt is made. We present a new technique that allows one to apply, in one unified way, the logarithmic norm method to QBDs with countable state spaces. The technique involves the preprocessing of the infinitesimal matrix of the QBD, finding bounding for its blocks, and then merging them into the single explicit upper bound. The applicability of the technique is demonstrated through a series of examples. Full article

Wigner’s little groups are the subgroups of the Poincaré group whose transformations leave the four-momentum of a relativistic particle invariant. The little group for a massive particle is SO(3)-like, whereas for a massless particle, it is E(2)-like. Multiple approaches to group contractions are discussed. It is shown that the Lie algebra of the E(2)-like little group for massless particles can be obtained from the SO(3) and from the SO(2, 1) group by boosting to the infinite-momentum limit. It is also shown that it is possible to obtain the generators of the E(2)-like and cylindrical groups from those of SO(3) as well as from those of SO(2, 1) by using the squeeze transformation. The contraction of the Lorentz group SO(3, 2) to the Poincaré group is revisited. As physical examples, two applications are chosen from classical optics. The first shows the contraction of a light ray from a spherical transparent surface to a straight line. The second shows that the focusing of the image in a camera can be formulated by the implementation of the focal condition to the [ABCD] matrix of paraxial optics, which can be regarded as a limiting procedure. Full article