Search Results (33)

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

This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of $\mathbb{Q}$ and describe algebraic number fields through their realization as finite-dimensional $\mathbb{Q}$-algebras (via multiplication operators and matrix representations), leading naturally to the arithmetic invariants—trace, norm, and discriminant—and to the ring of integers, ideals, Dedekind domains, and the ideal class group. We then develop the classical theory of cyclotomic fields, emphasizing their Galois structure and their role in abelian extensions of $\mathbb{Q}$. Next, we discuss ramification in general extensions, including decomposition and inertia groups, the Frobenius element, and the Chebotarev density theorem. The exposition continues with a concise algebraic introduction to elliptic curves and their L-functions, and it places key conjectural links (including Birch and Swinnerton-Dyer) in context. Finally, a collection of examples highlights a common operational language between fractional calculus and number theory: Laplace and Mellin transforms turn convolution-type operators into multiplication, clarifying the appearance of $\Gamma$-factors, Dirichlet series, and zeta- and L-function structures in both settings. Full article

Time evolution of open quantum systems is governed by the master equation. The master equation, which is a matrix formalism, is the time derivative of the density matrix, which contains the complete information on the state of a quantum system. Instead of implementing successive measurements on repeated identically prepared systems, which are inevitably imperfect and can only be performed a limited number of times, a state estimator can be designed to obtain the whole information about the state of a quantum system represented in a density matrix. Trace-one and positive semi-definite properties of the density matrix arising from physical constraints have to be preserved during state estimation in quantum systems. To address this challenge, we present a projection technique that incorporates Dykstra’s algorithm and physical constraints into state estimation. This technique, which is an iterative method, ensures convergence and includes a designed oracle that projects the estimated state into intersections of admissible closed convex sets. The oracle structure is constructed using Hilbert projection, which looks for the best approximation of the projected estimated state within a Hilbert space into a closed convex set. According to the Hilbert projection theorem, this proposed oracle guarantees the existence and uniqueness of the best approximation of the projected state. Full article

A context-aware system is a system that adapts its behaviours in response to changes in the system’s environment (i.e., context). Ensuring the correctness of such a system is difficult because the state of the environment changes frequently in an unpredictable manner according to the laws of physics. Hence, formal verification techniques like model-checking and theorem proving do not work in many cases. Runtime Verification (RV) is a lightweight formal verification technique that consists of checking at runtime whether the execution of the system violates the requirements of the system. The Calculus of Context-aware Ambients (CCA) is a process calculus for modelling context-aware systems and reasoning about their behaviours. This paper proposes an RV tool for CCA, called ccaRV. Given a model of a system in CCA and a property of the system written in LTL (Linear Temporal Logic), ccaRV verifies automatically at runtime if the execution of the system violates the property. We propose a semantic approach to RV, where the RV mechanism is defined at the semantics level and not as an add-on. A consequence of this is that there is no need for generating a monitor from the property specification nor for the instrumentation of a system during verification. We define a labelled reduction relation for CCA, where the labels are used to capture the execution traces at the semantics level. Then we extend LTL with spatial operators and context expressions in order to formulate properties about the system context. We use a case study of the MQTT (Message Queue Telemetry Transport) protocol to evaluate the proposed RV approach. The results show that the ccaRV tool is scalable and its decisions are accurate. Full article

We study compactness for the complex Green operator ${\mathsf{G}}_q$ associated with the Kohn Laplacian ${\square }_b$ on boundaries of pseudoconvex domains in Stein manifolds. Let $\Omega ⋐X$ be a bounded pseudoconvex domain in a Stein manifold X of complex dimension n with $C^1$ boundary. For $1\le q\le n-2$, we first prove a compactness theorem under weak potential-theoretic hypotheses: if $b\Omega$ satisfies weak $\left({\mathcal{P}}_q\right)$ and weak $\left({\mathcal{P}}_{n-1-q}\right)$, then ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$ are compact on ${\mathsf{L}}_{p,q}^2\left(b\Omega \right)$. This extends known $C^{\infty }$ results in ${\mathbb{C}}^n$ to the minimal regularity $C^1$ and to the Stein setting. On locally convexifiable $C^1$ boundaries, we obtain a full characterization: compactness of ${\mathsf{G}}_q$ is equivalent to simultaneous compactness of ${\mathsf{G}}_q$ and ${\mathsf{G}}_{n-1-q}$, to compactness of the $\overline{\partial }$-Neumann operators ${\mathsf{N}}_q$ and ${\mathsf{N}}_{n-1-q}$ in the interior, to weak $\left({\mathcal{P}}_q\right)$ and $\left({\mathcal{P}}_{n-1-q}\right)$, and to the absence of (germs of) complex varieties of dimensions q and $n-1-q$ on $b\Omega$. A key ingredient is an annulus compactness transfer on ${\Omega }_{+}={\Omega }_2\setminus \overline{{\Omega }_1}$, which yields compactness of ${\mathsf{N}}_q^{{\Omega }_{+}}$ from weak $\left(\mathcal{P}\right)$ near each boundary component and allows us to build compact ${\overline{\partial }}_b$-solution operators via jump formulas. Consequences include the following: compact canonical solution operators for ${\overline{\partial }}_b$, compact resolvent for ${\square }_b$ on the orthogonal complement of its harmonic space (hence discrete spectrum and finite-dimensional harmonic forms), equivalence between compactness and standard compactness estimates, closed range and ${\mathsf{L}}^2$ Hodge decompositions, trace-class heat flow, stability under $C^1$ boundary perturbations, vanishing essential norms, Sobolev mapping (and gains under subellipticity), and compactness of Bergman-type commutators when $q=1$. Full article

We introduce a quantum integrated-information measure $\Phi$ for multipartite states within the Relational Quantum Dynamics (RQD) framework. $\Phi \left(\rho \right)$ is defined as the minimum quantum Jensen–Shannon distance between an n-partite density operator $\rho$ and any product state over a bipartition of its subsystems. We prove that its square root induces a genuine metric on state space and that $\Phi$ is monotonic under all completely positive trace-preserving maps. Restricting the search to bipartitions yields a unique optimal split and a unique closest product state. From this geometric picture, we derive a canonical entanglement witness directly tied to $\Phi$ and construct an integration dendrogram that reveals the full hierarchical correlation structure of $\rho$. We further show that there always exists an “optimal observer”—a channel or basis—that preserves $\Phi$ better than any alternative. Finally, we propose a quantum Markov blanket theorem: the boundary of the optimal bipartition isolates subsystems most effectively. Our framework unites categorical enrichment, convex-geometric methods, and operational tools, forging a concrete bridge between integrated information theory and quantum information science. Full article

Biological processes often involve the attachment and detachment of extended molecules to substrates. Here, the classical dimer model is used to investigate these geometric effects on the free energy, which governs both the equilibrium state and the reaction dynamics. We present a simplified version of Fisher’s derivation of the partition function of a two-dimensional dimer model at filling factor $\nu =1$, which takes into account the blocking of two adjacent sites by each dimer. Physical consequences of the dimer geometry on the entropy that are not reflected in simpler theories are identified. Specifically, for dimers adsorbing on the DNA double helix, the dimer geometry gives a persistently nonzero entropy and there is a significant charge inversion as the force binding the particles to the lattice increases relative to the thermal energy, which is not true of the simple lattice gas model for the dimers, in which all the sites are independent. Full article

This is a review/research paper on anomalies applied in a bottom–up approach to standard model and gravity. It is divided into two parts. The first consists of a proper review of anomalies in quantum field theories. Anomalies are analyzed according to three different methods: a perturbative one based on a Feynman diagram, a non-perturbative one relying on the Schwinger–DeWitt approach, and, third, one hinging on the Atiyah–Singer family’s index theorem. The three methods are applied both to chiral gauge anomalies and trace anomalies. The fundamental distinction, which our presentation leads to, is between obstructive (O) and non-obstructive (NO) anomalies. The former is tied to the non-existence of fermion propagators, which fatally maim the corresponding theory. In the second part, we apply this analysis to the SM and a variety of its extensions, which are immersed in a gravitational background, and we find that they are all plagued by a residual chiral trace anomaly. To completely eliminate all kinds of dangerous anomalies in SM-like theories, we propose a somewhat unconventional scheme and exemplify it by means of an explicit model. The latter is a left–right symmetric model. We embed it in a Weyl geometry to render it a conformal invariant. We then deal with some of its quantum aspects, particularly its even (NO) trace anomalies and the means to preserve its conformal invariance at the quantum level. We briefly review renormalization and unitarity in the framework of similar models discussed in the existing literature. Finally, we present a possible (conjectural) application of the model to describe the junction between cosmology and quantum field theory. Full article

Particular Kottler spacetimes are analytically investigated. The investigated spacetimes are spherically symmetric nonrotating spacetimes endowed with a Schwarzschild solid-angle element. SchwarzschildNairiai spacetimes, Schwarzschild spacetimes with a linear term, and Schwarzschild spacetimes with a linear term and a cosmological constant are studied. The infinite-redshift surfaces are analytically written. To this aim, the parameter spaces of the models are analytically investigated, and the conditions for which the analytical radii are reconducted to the physical horizons are used to set and to constrain the parameter spaces. The coordinate-singularity-avoiding coordinate extensions are newly written. Schwarzschild spacetimes with a linear term and a cosmological constant termare analytically studied, and the new singularity-avoiding coordinate extensions are detailed. The new roles of the linear term and of the cosmological constant term in characterizing the Schwarzschild radius are traced. The generalized Schwarzschild–deSitter case and generalized Schwarzschild–anti-deSitter case are characterized in a different manner. The weak field limit is newly recalled. The embeddings are newly provided. The quantum implementation is newly envisaged. The geometrical objects are newly calculated. As a result, for the Einstein field equations, the presence of quintessence is newly excluded. The Birkhoff theorem is newly proven to be obeyed. Full article

This study examines the concept of impulse controllability within singular hybrid coupled systems through the utilisation of decentralised proportional plus derivative (P-D) output feedback. By employing the Differential Mean Value Theorem, the nonlinear model can be converted into a linear parameter-varying large-scale system. Our analysis leads to the establishment of algebraic conditions that are both necessary and sufficient for the existence of a decentralised P-D output feedback controller that can guarantee impulse controllability in these complex systems. Moreover, we address the issue of admissibility within these systems by employing matrix trace inequalities. We present a novel sufficient condition for impulse controllability, which offers a new perspective on addressing this challenging problem. To validate our findings, we present numerical examples that demonstrate the effectiveness of the proposed methodologies in practice. Full article

The concept of symmetry under various transformations of quantities describing basic natural phenomena is one of the fundamental principles in the mathematical formulation of physical laws. Starting with Noether’s theorems, we highlight some well–known examples of global symmetries and symmetry breaking on the particle level, such as the separation of strong and electroweak interactions and the Higgs mechanism, which gives mass to leptons and quarks. The relation between symmetry energy and charge symmetry breaking at both the nuclear level (under the interchange of protons and neutrons) and the particle level (under the interchange of u and d quarks) forms the main subject of this work. We trace the concept of symmetry energy from its introduction in the simple semi-empirical mass formula and liquid drop models to the most sophisticated non-relativistic, relativistic, and ab initio models. Methods used to extract symmetry energy attributes, utilizing the most significant combined terrestrial and astrophysical data and theoretical predictions, are reviewed. This includes properties of finite nuclei, heavy-ion collisions, neutron stars, gravitational waves, and parity–violating electron scattering experiments such as CREX and PREX, for which selected examples are provided. Finally, future approaches to investigation of the symmetry energy and its properties are discussed. Full article

Vehicular ad-hoc networks (VANETs) can significantly improve the level of urban traffic management. However, the sender unlinkability has become an intricate issue in the field of VANETs’ encryption. As the sender signcrypts a message, the receiver has to use the sender’s identity or public key to decrypt it. Consequently, the sender can be traced using the same identity or public key, which poses some security risks to the sender. To address this issue, we present an unlinkable and revocable signcryption scheme (URSCS), where an efficient and powerful signcryption mechanism is adopted for communication. The sender constructs a polynomial to generate a unique session key for each communication, which is then transmitted to a group of receivers, enabling the same secret message to be sent to multiple receivers. Each time a secret message is sent, a new key pair is generated, and an anonymization mechanism is introduced to conceal the true identity of the vehicle, thus preventing malicious attackers from tracing the sender through the public key or the real identity. With the introduction of the identification public key, this scheme supports either multiple receivers or a single receiver, where the receiver can be either road side units (RSUs) or vehicles. Additionally, a complete revocation mechanism is constructed with extremely low communication overhead, utilizing the Chinese remainder theorem (CRT). Formal and informal security analyses demonstrate that our URSCS scheme meets the expected security and privacy requirements of VANETs. The performance analysis shows that our URSCS scheme outperforms other represented schemes. Full article

This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko–Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements. Full article

The axiomatic structure of the $\kappa$-statistcal theory is proven. In addition to the first three standard Khinchin–Shannon axioms of continuity, maximality, and expansibility, two further axioms are identified, namely the self-duality axiom and the scaling axiom. It is shown that both the $\kappa$-entropy and its special limiting case, the classical Boltzmann–Gibbs–Shannon entropy, follow unambiguously from the above new set of five axioms. It has been emphasized that the statistical theory that can be built from $\kappa$-entropy has a validity that goes beyond physics and can be used to treat physical, natural, or artificial complex systems. The physical origin of the self-duality and scaling axioms has been investigated and traced back to the first principles of relativistic physics, i.e., the Galileo relativity principle and the Einstein principle of the constancy of the speed of light. It has been shown that the $\kappa$-formalism, which emerges from the $\kappa$-entropy, can treat both simple (few-body) and complex (statistical) systems in a unified way. Relativistic statistical mechanics based on $\kappa$-entropy is shown that preserves the main features of classical statistical mechanics (kinetic theory, molecular chaos hypothesis, maximum entropy principle, thermodynamic stability, H-theorem, and Lesche stability). The answers that the $\kappa$-statistical theory gives to the more-than-a-century-old open problems of relativistic physics, such as how thermodynamic quantities like temperature and entropy vary with the speed of the reference frame, have been emphasized. Full article

The anomalous dimension ${\gamma }_m=1$ in the infrared region near the conformal edge in the broken phase of the large $N_f$ QCD has been shown by the ladder Schwinger–Dyson equation and also by the lattice simulation for $N_f=8$ and for $N_c=3$. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, ${\gamma }_m=1$, by comparing the pion matrix element of the trace of the energy-momentum tensor $⟨\pi \left(p_2\right)|(1+{\gamma }_m)·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=⟨\pi \left(p_2\right)|{\theta }_{\mu }^{\mu }|\pi \left(p_1\right)⟩=2M_{\pi }^2$ (up to trace anomaly) with the estimate of $⟨\pi \left(p_2\right)|2·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=2M_{\pi }^2$ through the Feynman–Hellmann theorem combined with an assumption $M_{\pi }^2\sim m_f$ characteristic of the broken phase. We show that this is not justified by the explicit evaluation of each matrix element based on the dilaton chiral perturbation theory (dChPT): $⟨\pi \left(p_2\right)|2·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=2M_{\pi }^2+[(1-{\gamma }_m)M_{\pi }^2·2/(1+{\gamma }_m)]=2M_{\pi }^2·2/(1+{\gamma }_m)\ne 2M_{\pi }^2$ in contradiction with his estimate, which is compared with $⟨\pi \left(p_2\right)|(1+{\gamma }_m)·{\sum }_{i=1}^{N_f}{m}_f{\overline{\psi }}_i{\psi }_i|\pi \left(p_1\right)⟩=(1+{\gamma }_m)M_{\pi }^2+\left[(1-{\gamma }_m)M_{\pi }^2\right]=2M_{\pi }^2$ (both up to trace anomaly), where the terms in $\left[\right]$ are from the $\sigma$ (pseudo-dilaton) pole contribution. Thus, there is no constraint on ${\gamma }_m$ when the $\sigma$ pole contribution is treated consistently for both. We further show that the Feynman–Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the $\sigma$ pole contribution is absent, and with $M_{\pi }^2\sim m_f^{2/(1+{\gamma }_m)}$ instead of $M_{\pi }^2\sim m_f$, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of $f_0\left(500\right)$ to $\pi \pi$ for $N_f=2$. It is shown to be consistent with the reality, when $f_0\left(500\right)$ is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance. Full article