Search Results (578)

In this paper, the use of nilpotent Lie algebras as the basis for homomorphic encryption based on additive operations is explored. The $g$-setting is set up over $g{l}_n\left({\mathbb{Z}}_q\right)$) and the group $G=exp\left(g\right)$, and it is noted that the exponential and logarithm series are truncated by nilpotency in a natural way. From this, an additive symmetric conjugation scheme is constructed: given a message element $M$ and a central randomizer $U\in z\left(g\right),$ we encrypt $=Kexp\left(M+U\right)K^{-1}$ and decrypt to $M=log\left(K^{-1}CK\right)-U$. The scheme is additive in nature, with the security defined in the IND-CPA model. Integrity is ensured using an encrypt-then-MAC construction. These properties together provide both confidentiality and robustness while preserving the homomorphic functionality. The scheme realizes additive homomorphism through a truncated BCH-sum, so it is suitable for ciphertext summations. We implemented a prototype and took reproducible measurements (Python 3.11/NumPy) of the series $\{{10,10}^2,{10}^3,{10}^4,{10}^5\}$ over 10 iterations, reporting the medians and 95% confidence intervals. The graphs exhibit that the latency per operation remains constant at fixed values, and the total time scales approximately linearly with the batch size; we also report the throughput, peak memory usage, $\mid C\mid /\mid M\mid$ expansion rate, and achievable aggregation depth. The applications are federated reporting, IoT telemetry, and privacy-preserving aggregations in DBMS; the limitations include its additive nature (lacking general multiplicative homomorphism), IND-CPA (but not CCA), and side-channel resistance requirements. We place our approach in contrast to the standard FHE building blocks BFV/BGV/CKKS nd the emerging NIST PQC standards (FIPS 203/204/205), as a well-established security model with future engineering optimizations. Full article

This work establishes definitive conditions for the inheritance of categorical completeness and cocompleteness by categories of internal group objects. We prove that while the completeness of $\mathbf{Grp}\left(\mathcal{C}\right)$ follows unconditionally from the completeness of the base category $\mathcal{C}$, cocompleteness requires $\mathcal{C}$ to be regular, cocomplete, and admit a free group functor left adjoint to the forgetful functor. Explicit limit and colimit constructions are provided, with colimits realized via coequalizers of relations induced by group axioms over free group objects. Applications demonstrate cocompleteness in topological groups, ordered groups, and group sheaves, while Lie groups serve as counterexamples revealing necessary analytic constraints—particularly the impossibility of equipping free groups on non-discrete manifolds with smooth structures. Further results include the inheritance of regularity when the free group functor preserves finite products, the existence of internal hom-objects in locally Cartesian closed settings, monadicity for locally presentable $\mathcal{C}$, and homotopical extensions where model structures on $\mathbf{Grp}\left(\mathcal{M}\right)$ reflect those of $\mathcal{M}$. This framework unifies classical category theory with geometric obstruction theory, resolving fundamental questions on exactness transfer and enabling new constructions in homotopical algebra and internal representation theory. Full article

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that $di{m}_K\left(L\right)=\mid L/K\mid <\infty$, the transcendence degree $trd(L/K)<\infty$ and $trd(L/K)=\infty$ with $difftrd(L/K)<\infty$, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension $L=K\left(G\right)$; then, there is a Galois correspondence between the intermediate fields $K\subset K^{\prime }\subset L$ and the subgroups $e\subset G^{\prime }\subset G$, provided that $K^{\prime }$ is stable under a Lie algebra $\Delta$ of invariant derivations of $L/K$. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article

Just as Gauss’s interpretation of complex numbers as points in a number plane in the form of a suitably formulated axiom found its way into the vector representation of Fourier transforms, this is the case with Euler’s formula and Riemann’s Zeta function considered here. The description of the connection between variables through complex numbers as it is given in Euler’s formula and emphasized by Riemann is reflected here with great flexibility in the introduction of non-classically generalized complex numbers and the vector representation of the generalized Zeta function based on them. For describing such dependencies of two variables with the help of generalized complex numbers, we introduce manifolds underlying certain Lie groups as level sets of norms, antinorms or semi-antinorms. No undefined or “imaginary” quantities are used for this. In contrast to the approach of Hamilton and his numerous successors, the vector-valued vector product of non-classically generalized complex numbers is commutative, and the whole number system satisfies a weak distributivity property as considered by Hankel, but not the strong one. Full article

Objectives: Sternal wound infections (SWIs) after cardiac surgery remain a major complication and represent a significant clinical challenge. This article aims to evaluate the effectiveness of closed-incision negative-pressure wound therapy (ciNPWT) in preventing postoperative wound complications in high-risk patients undergoing coronary bypass surgery via full median sternotomy. Methods: Data on all consecutive patients undergoing coronary artery bypass surgery at our facility between March 2021 and March 2023 were retrospectively collected. The ciNPWT group consisted of 71 patients. A control group receiving conventional wound dressings was selected by propensity matching. The primary outcome was postoperative sternal wound complication of any severity, as well as superficial and deep SWIs. The secondary outcomes were hospital stay length, in-hospital mortality, and need for perioperative wound revision. Results: The incidence of postoperative SWIs was significantly higher in the ciNPWT group than in the control group (18 [25.4%] vs. 7 [9.9%], p = 0.03). Of these 25 cases, 20 had received postoperative ciNPWT and 5 conventional wound dressings, which was statistically different (15 [21.1%] vs. 5 [7.0%], p = 0.03). ciNPWT was also significantly associated with positive bacterial cultures (13 [18.3%] vs. 4 [5.6%], p = 0.04) and perioperative wound revision (11 [15.5%] vs. 6 [8.5%], p = 0.05). Conclusions: In consecutive high-risk patients undergoing coronary bypass surgery, the use of prophylactic ciNPWT did not improve wound healing compared to conventional wound dressings, raising concerns about its effectiveness in high-risk patients. Our results do not support the routine use of ciNPWT in this setting. Its potential value may instead lie in carefully defined patient subgroups, underscoring the relevance of our findings for patient-tailored care strategies in cardiac surgery. Full article

This paper establishes a functorial algebraic isomorphism between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface $\Sigma$ of genus $g\ge 2$, for a complex reductive Lie group G, and the character variety ${\mathcal{M}}_{\mathcal{C}}^K({\Sigma }^{*},G)$ of representations of its fundamental group with relatively compact image. The dimension formula $dim{\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)=2(g-1){dim}_{\mathbb{C}}\left(G\right)+{\sum }_{i=1}^k{dim}_{\mathbb{R}}\left({\mathcal{C}}_i\right)$, where ${\mathcal{C}}_1,\dots ,{\mathcal{C}}_k$ are conjugacy classes in a maximal compact subgroup $K\subset G$, is derived for complex reductive Lie groups, and singularities are characterized as polystable bundles with non-trivial automorphism groups. As applications of the above geometric results to control theory, it is proved that topologically distinct polystable robotic navigation strategies around obstacles are classified by this character variety. The geometry of singular points in families of polystable control strategies is further investigated, revealing enhanced stability properties characterized by reduced tangent space dimensions arising from non-trivial automorphism groups. Full article

The Tibetan Kanjur has long been recognized as both a symbolic embodiment of the Buddhist canonical literature and as a ritual object, resulting in the production of various versions that differ in content, arrangement, and specific textual formulation. Since the late 1970s, the provenance, lineage affiliations, and historical development of these Kanjurs have attracted significant scholarly attention. In this paper, I present the findings of textual-critical research on the Tibetan translation of the Gaganagañjaparipṛcchā (Ggn), focusing particularly on two manuscript collections preserved at Nesar Monastery in Dolpo, namely the Nesar and Lang Kanjurs. Both Kanjurs, possibly dated as early as the thirteenth century, lie outside the two main lineages, Tshal pa and Them spangs ma, and demonstrate strong connections with Local or Independent Kanjurs, notably those of Phug brag and Namgyal. By undertaking a close comparison of selected passages from the Ggn across twenty-one canonical witnesses, this study finds that, for the Ggn: (a) the Nesar and Lang Kanjurs possess a group of unique textual variants which distinguish them from all other known Kanjur and Proto-Kanjur editions; (b) the Lang Kanjur appears to have been based chiefly on the Nesar Kanjur or an exemplar closely related to it; and (c) the compilers of the Lang Kanjur also relied on at least one other manuscript, which seems to have preserved readings of greater accuracy. These findings highlight the importance of the Nesar and Lang Kanjurs for textual-critical investigation and for understanding the transmission history of the Tibetan Buddhist canon. Ongoing research into these Kanjurs will yield crucial evidence for constructing a more nuanced and historically informed account of the formation, adaptation, and regional diffusion of the Tibetan Buddhist canon. Full article

This paper introduces weak variants of level convergence (L-convergence) and epigraph convergence (E-convergence) for nets of level functions on general topological spaces, extending the classical metric and real-valued frameworks to ordered codomains and generalized minima. We show that L-convergence implies E-convergence and that the two notions coincide when the limit function is level-continuous, mirroring the relationship between strong and weak variational convergence. In Hausdorff topological groups, we define robust level functions and prove that every level function can be approximated by robust ones via convolution-type operations, enabling perturbation-resilient modeling. These results both generalize and connect to $\Gamma$-convergence: they recover the classical metric, lower semicontinuous case, and extend the scope for optimization on Lie groups, fuzzy systems, and mechanics in non-Euclidean spaces. An explicit nonmetrizable example demonstrates the relevance of our theory beyond the reach of $\Gamma$-convergence. Full article

This study explores the nonlinear dynamics and interdependencies among major commodity markets—Gold, Oil, Natural Gas, and Silver—by employing advanced chaos theory and information-theoretic tools. Using daily data from 2020 to 2024, we estimate key complexity measures including Lyapunov exponents, correlation dimension, Shannon and Rényi entropy, and mutual information. We also apply the stochastic SO(2) Lie group method to model dynamic correlations, and wavelet coherence analysis to detect time-frequency co-movements. Our findings reveal evidence of low-dimensional deterministic chaos and time-varying nonlinear relationships, especially among pairs like Gold–Silver and Oil–Gas. These results highlight the importance of using nontraditional approaches to uncover hidden structure and co-movement dynamics in commodity markets, providing useful insights for portfolio diversification and systemic risk assessment. Full article

We investigate left-invariant generalized cross-curvature solitons on simply connected three-dimensional Lorentzian Lie groups. Working with the assumption that the contravariant tensor $P_{ij}$ (defined from the Ricci tensor and scalar curvature) is invertible, we derive the algebraic soliton equations for left-invariant metrics and classify all left-invariant generalized cross-curvature solitons (for the generalized equation ${\mathcal{L}}_{X}g+\lambda g=2h+2\rho Rg$) on the standard 3D Lorentzian Lie algebra types (unimodular Types Ia, Ib, II, and III and non-unimodular Types IV.1, IV.2, and IV.3). For each Lie algebra type, we state the necessary and sufficient algebraic conditions on the structure constants, provide explicit formulas for the soliton vector fields X (when they exist), and compute the soliton parameter $\lambda$ in terms of the structure constants and the parameter $\rho$. Our results include several existence families, explicit nonexistence results (notably for Type Ib and Type IV.3), and consequences linking the existence of left-invariant solitons with local conformal flatness in certain cases. The classification yields new explicit homogeneous generalized cross-curvature solitons in the Lorentzian setting and clarifies how the parameter $\rho$ modifies the algebraic constraints. Examples and brief geometric remarks are provided. Full article

We present a topologically protected teleportation protocol based on projective parity measurements between spatially separated Majorana zero modes (MZMs), eliminating the need for dynamic braiding. Unlike conventional teleportation schemes, our method preserves logical information through nonlocal encoding and suppresses decoherence exponentially with Majorana separation. We provide a rigorous mathematical framework that includes six theorems and a lemma, proving fidelity bounds, no entropy increase under ideal QND parity measurement under quantum non-demolition (QND) measurements, and compliance with the no-cloning theorem. We demonstrate that all correction operations lie within the Clifford group, enabling efficient, fault-tolerant implementation. Furthermore, we outline a scalable architecture for multi-qubit teleportation and relate our framework to recent experimental advances in quantum-dot-based Kitaev chains and superconducting nanowire platforms. These results position Majorana-based teleportation as a thermodynamically stable and experimentally viable approach to scalable quantum information transfer. All operations discussed are Clifford-only; achieving universality requires non-Clifford resources and lies outside our scope. Full article

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any $q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\backslash \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\backslash \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\backslash \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

The Lorentz group ${\mathrm{Lor}}_{1,3}={\mathrm{SO}}_0(1,3)$ has two point fixgroups, namely $\mathrm{SO}\left(3\right)$ for time-like translations and ${\mathrm{SO}}_0(1,1)\times {\mathrm{I}\mathrm{R}}^2$ for light-like translations. However, for light-like translations, it is reasonable to consider a line fixgroup that leads to the Borel structure of the Lorentz group and provides appropriate helicities for massless particles. Therefore, whether a particle is massless or massive is not so much a physical question but rather a question of the underlying Lie group symmetry. Full article

In this paper, we present Lie symmetry analysis of a generalized (1+1)-dimensional porous medium equation characterized by parameters m and d. Through group classification, we examine how these parameters influence the Lie symmetry structure of the equation. Our analysis establishes conditions under which the equation admits either a three-dimensional or a five-dimensional Lie algebra. Using the obtained symmetry algebras, we construct optimal systems of one-dimensional subalgebras. Subsequently, we derive invariant solutions corresponding to each subalgebra, providing explicit formulas in relevant parameter regimes. These solutions deepen our understanding of the nonlinear diffusion processes modeled by porous medium equations and offer valuable benchmarks for analytical and numerical studies. Full article

The convective Cahn–Hilliard–Oono equation is analyzed under the conditions ${\mu }_1\ge 0$ and ${\mu }_3+{\mu }_4\le 0$. The Lie invariance criteria are examined through symmetry generators, leading to the identification of Lie algebra, where translation symmetries exist in both space and time variables. By employing Lie group methods, the equation is transformed into a system of highly nonlinear ordinary differential equations using appropriate similarity transformations. The extended direct algebraic method are utilized to derive various soliton solutions, including kink, anti-kink, singular soliton, bright, dark, periodic, mixed periodic, mixed trigonometric, trigonometric, peakon soliton, anti-peaked with decay, shock, mixed shock-singular, mixed singular, complex solitary shock, singular, and shock wave solutions. The characteristics of selected solutions are illustrated in 3D, 2D, and contour plots for specific wave number effects. Additionally, the model’s stability is examined. These results contribute to advancing research by deepening the understanding of nonlinear wave structures and broadening the scope of knowledge in the field. Full article