Clifford's geometric algebras originally arise from theoretical physics, the most developed mathematically of the empirical sciences, where they provide a natural and elegant description of phenomena, such as classical mechanics and electrodynamics. Unfortunately, however, they are less well known to students nowadays, since already at the end of the 19th century they were replaced by the hollow formalism of vector analysis. Decades later Pauli and Dirac had to reinvent geometric algebras to write their famous equations, but the breakthrough that led to their triumphant return to the epicenter of active research came this time from technology: robotics, navigation, computer vision, animated simulations, etc. In view of these trends, a valuable opportunity opens up in many engineering sciences to upgrade and improve their mathematical description, optimize their algorithms, and significantly simplify the academic curricula. In this paper we approach some of the classical problems in geodesy and photogrammetry from the perspective of geometric algebras, which have proven rather effective in other engineering disciplines, such as robotics and computer vision. More precisely, we tackle problems like self-calibration and 3D reconstruction, trilateration, triangulation, and synchronization of reference frames, using dual quaternions, the Projective geometric algebra (PGA) and the more general Conformal geometric algebra (CGA). As expected, this approach provides a simpler and more elegant mathematical description, more suitable for both pedagogical reasons and algorithm efficiency. The results presented here are also meant to be used as a framework for future research targeting more advanced problems in those areas.

Introduction: Foliations are geometric structures that decompose smooth manifolds into immersed submanifolds called leaves. While classical theory relies on differential-geometric methods and the Frobenius theorem, a purely algebraic characterization remains underexplored. This study develops an algebraic framework for foliations by translating their structure into the language of modules, derivations, and ring theory.

Methods: Building on the foundations of smooth manifolds and differential maps, three interconnected perspectives are developed. The classical approach examines integrable and involutive distributions via the Frobenius theorem. The Lie-theoretic viewpoint employs Lie algebras, Lie groups, and Hopf algebras to encode symmetries. The module-theoretic perspective uses the module of derivations Der(C-infinity(M, R)) to provide an algebraic description of tangent vector actions. Smooth Gelfand duality is employed as the categorical framework connecting geometric structures with their corresponding algebras of functions.

Results: The central result establishes that every foliation F on a manifold M determines a unique involutive projective submodule of Der(C-infinity(M, R)) that encodes the foliation’s structure. Furthermore, an ideal–leaf correspondence is established: each closed leaf L corresponds to an ideal I(L) of functions vanishing on L, preserved under the foliation action. The foliation algebra A_F, consisting of functions constant along leaves, is shown to encode the leaf space structure even when the geometric quotient M/F is singular.

Conclusions: This work demonstrates that foliations can be fully characterized in algebraic terms, establishing a dictionary between differential geometry and algebra. Tangent distributions correspond to submodules of derivations, Lie brackets to commutators, and integrability to involutivity. These results open new pathways for applying categorical methods to the study, classification, and reconstruction of foliations on smooth manifolds.

In this paper, we introduce and investigate new classes of sequence spaces defined by a Musielak–Orlicz function Π = (πk) over a linear space (V, q), where q is a seminorm. By combining the structural flexibility of Musielak–Orlicz functions with generalized averaging techniques inspired by the classical M(ϕ)-space, we define the sequence spaces ℓ1[Π, q], ℓ∞[Π, q], and M[Π, ϕ, q]. These spaces extend several well-known Orlicz-type and modulus-based sequence spaces and provide a unified framework for studying coordinate-wise variable modular growth conditions. The space M[Π, ϕ, q] is constructed as a natural generalization of the classical M(ϕ) sequence space by incorporating a Musielak–Orlicz modular structure. We investigate the fundamental algebraic and topological properties of these spaces, including linearity,solidity, symmetricity, monotonicity, and completeness. Suitable Luxemburg-type seminorms are introduced,and it is shown that these spaces become complete seminormed spaces whenever the underlying seminormed space (V, q) is complete. Furthermore, we establish several inclusion relations between the newly defined spaces under appropriate growth conditions on the sequence ϕ = (ϕs) and on Musielak–Orlicz functions satisfying the Δ2-condition. In particular, we prove that ℓ1[Π, q] ⊆ M[Π, ϕ, q] ⊆ ℓ∞[Π, q],and obtain additional inclusion results involving compositions and sums of Musielak–Orlicz functions. These results generalize earlier works on Orlicz sequence spaces and contribute to the further development of modular and generalized sequence space theory, opening new directions for the study of dual spaces, matrix transformations, and operator-theoretic properties in Musielak–Orlicz-type settings.

The paper deals with the arbitrary finite-dimensional irreducible representations of simple complex Lie algebras g of types B2 and G2 and real
forms so(1,4), so(3,2) and G of these algebras. Involutive automorphisms
θ on Cartan subalgebras of these algebras are considered. Formulae for
characters value χ(θ) are obtained. This allows us to find the number of
linearly independent space-like and time-like vectors in the representation
space

Consider a simple complex Lie algebra g and an irreducible finite-dimensional
representation φ : g → sl(V). Denote by gr the real form of inner type for
algebra g. Then φ(gr) ⊆ su(p,q), where p − q = δ(gr) is the signature of
the invariant Hermitian form on V. It is possible to find δ(gr) using Weyl
character formula for the representation φ. In [1], F.I.Karpelevich
derived formulae for δ in the case of classical Lie algebras. In [2], Lie algebras
su(p,q) are considered convenient tables for δ in the case a small rank of
g is found. In [3, 4], formulae for |δ| in the case of gr = G,FI,FII,so(p,q)
were presented. Nevertheless, in applications, the exact tables including the
sign of δ are necessary. We
obtain in this paper the tables of δ in terms of the marks of the highest weight
of representation φ. In addition, a similar table for any representation of exceptional
Lie algebras of type G2 is derived. And it is not necessary to know the system
of all weights of the representation φ

References

\bibitem{1}F.I.Karpelevich:"Simple subalgebras of real Lie algebras", Trudy

Mosk.Mat.Obshch., Vol.4,(1955), pp.3-112.

\bibitem{2}J.Patera and R.T.Sharp:"Signatures of finite $\mathfrak{su}(p,q)$ representations",

J.Math.Phys., Vol.25,(1984),pp.2128-2131.MR0748387(85j:22042).

\bibitem{3}A.N.Rudy:"Signatures of finite representation of real, simple Lie algebras", J.Phys.A:Math.Gen., Vol.26,(1993),

pp.5873-5880.MR1252794(94i:17014).

\bibitem{4}A.N.Rudy:"Signatures of finite classical Lie algebra representations", J.Phys.A:Math.Gen., Vol.28,(1995),

pp.1641-1653.MR1338050(96e:17017).

Inspired by the Cayley–Hamilton theorem for linear operators, we introduce the notion of an algebraic polynomial automorphism. In this framework, a polynomial automorphism is called algebraic if it satisfies a nontrivial polynomial operator identity when regarded as an operator acting on the polynomial algebra; otherwise, it is called non-algebraic. This notion provides a new algebraic perspective on polynomial automorphisms, allowing them to be studied using ideas analogous to those appearing in the theory of algebraic linear operators and operator identities.
We investigate the algebraicity of several important classes of polynomial automorphisms. First, we prove that every elementary automorphism is algebraic. More generally, we show that any finite composition of commuting elementary automorphisms is algebraic as well. Moreover, we obtain explicit bounds for the degree of the corresponding minimal polynomial satisfied by these automorphisms. These results indicate that algebraicity naturally arises within a large and fundamental family of polynomial transformations.
On the other hand, algebraicity does not follow from tameness. To demonstrate this phenomenon, we construct a tame automorphism of k[x,y,z] which is non-algebraic. Finally, we study the classical Nagata automorphism together with a family of higher-dimensional Nagata-type automorphisms. We prove that they satisfy a cubic operator identity and hence are algebraic of degree three.

Graph Neural Networks built on the message-passing paradigm exchange information along a graph's edges, effectively learning from relational data while preserving structural symmetries. However, many real-world systems, such as social or biological networks, exhibit complex interactions better represented by higher-order topological domains. Geometric and Topological Deep Learning addresses this by leveraging such structures. Central to this field is the concept of lifting, which transforms data representations from basic graph forms to more expressive topologies before the application of graph models for learning.

In this work, we propose a structural lifting strategy based on Forman–Ricci curvature, an edge-based characteristic rooted in Riemannian geometry. Curvature reveals local and global graph properties, identifying network backbones, i.e., coarse, structure-preserving geometries connecting major communities. These backbones are most suitably represented as hyperedges, modeling information flow between distant clusters. Our approach assigns a curvature-based metric to each edge in an input graph and performs a structural lifting that maps this graph into a hypergraph representation, treating identified backbones as hyperedges and virtually shortening distances between relevant nodes for subsequent graph classification. Our approach thus offers a remedy to information distortion in message passing across long distances and graph bottlenecks, a phenomenon known as over-squashing.

We evaluate our method on multiple well-known graph learning benchmarks, including a large molecular property prediction dataset, e.g., NCI109 and OGBG-MolHIV. Using standard graph network models such as GCN and GAT, the lifted datasets consistently improve performance in 75% of test cases compared to non-lifted baselines. Notably, higher-order hypergraph models such as EDGNN and AllSetTransformer also benefit, showing improved performance in 80% of test cases when paired with our lifting step.

Large Language Models (LLMs) have demonstrated remarkable capabilities in modeling complex sequential data by learning long-range dependencies. Although originally developed for natural language processing, their autoregressive architecture is inherently domain-agnostic, making them suitable for applications well beyond text generation. This observation opens new opportunities for leveraging LLMs in engineering design and optimization. This work investigates a GPT-based framework, built on the GPT-2 architecture, for the topology optimization of planar truss structures. The classical design problem is reformulated as a sequential construction process, where design actions are governed by predefined grammar rules that ensure structural feasibility. These actions are encoded symbolically and mapped into text-like strings, allowing each truss configuration to be represented as a tokenized sequence. Using this formulation, a pretrained model is fine-tuned on a dataset of structurally meaningful designs. Structural performance is accounted for through a mechanically informed loss function, weighting training according to structural stiffness. This strategy effectively biases the model toward high-performing configurations while preserving diversity in the design space. The proposed approach is validated across six benchmark cases, achieving performance levels ranging from 82% to 100% of the corresponding global optima. The model also demonstrates the ability to generate novel, mechanically sound topologies. These results highlight the potential of LLM-based generative frameworks as complementary tools for exploring large and complex design spaces in structural optimization.

This paper proposes an enhanced framework for facial expression classification that integrates eigenvector-based feature extraction with an optimized Support Vector Machine (SVM) learning strategy. While Principal Component Analysis (PCA) and SVM are classical tools, the novelty of this work lies in a joint optimization scheme that couples adaptive eigenvector selection with kernel-driven margin optimization, leading to improved robustness and generalization under challenging imaging conditions. Unlike standard Eigenface approaches that retain components based solely on variance, the proposed method introduces a discriminative eigenvector selection criterion, ensuring that the retained subspace maximizes inter-class separability while minimizing intra-class variability. This is further reinforced by a systematic exploration and tuning of SVM kernels (linear, polynomial, and radial basis function), combined with cross-validated hyperparameter optimization to stabilize the decision boundary in noisy and high-dimensional settings. The framework is evaluated on the Extended Cohn–Kanade (CK+) dataset under controlled degradations, including noise perturbation and illumination variation. Comparative experiments against baseline models (standard PCA+SVM and raw-feature SVM) demonstrate that the proposed approach achieves consistent performance gains, reaching a classification accuracy above 97% while maintaining robustness across degraded scenarios. In addition, the analysis highlights the role of support vector sparsity control in improving generalization and reducing overfitting, providing further insight into the interaction between feature space structure and margin-based learning. These results suggest that embedding discriminative criteria within the eigen-decomposition stage, combined with kernel-aware optimization, constitutes an effective strategy for enhancing classical machine learning pipelines in real-world visual recognition tasks.

Brain tumor segmentation is performed using three-dimensional magnetic resonance images (MRIs) as a common practice. The accuracy and fast segmentation still remain a challenge. An algorithm is proposed using deep learning involving the UNet model with wavelet functions. In this algorithm, to enhance gradient flow and feature propagation, U-Net incorporates dense connections (residual block), which reduce the vanishing gradient issue and increase feature reuse. The encoder route blocks are extracted. Information is gradually abstracted, and the corresponding decoder path blocks reestablish the spatial resolution of the input images. The algorithm is validated on the data set MICCAI BratS2020 which includes a set of labelled brain tumor (2D MRI scans) dataset. A comparative study of UNet with Daubechies (db2,db4) and UNet with Haar tabulated for the Brats2020 dataset is included. The research design employed in this study combines residual blocks with UNet where the residual block is implemented using db2,db4 and Haar as activation function after batch normalization to improve training stability and accuracy. It was trained on 4308 2D MRI images, validated on 761 and tested on 895 MRI images.
Experimental results using db2 gave a validation accuracy of 0.99 with validation loss of 0.38 and a test accuracy of 0.98 with a test loss of 0.38 and CPU time taken was 9s 342 ms/steps.
Using db4 the validation accuracy was 0.99, with a validation loss of 0.4240 and test accuracy of 0.98 with test loss of 0.42 and CPU time noted was recorded as 7s 256ms/steps. For Haar wavelet a validation accuracy was 0.97, and loss wH 0.76 with test accuracy of 0.97 and loss of 0.75. Time taken was 6s 231 ms/step. It is noted that the deep learning strategy invoked along with db2 wavelet activation function shows remarkably improved accuracy and performance for segmentation as compared to its counterpart db4 and Haar wavelet family.

The existence of a mass gap in non-abelian Yang–Mills theory is a cornerstone prediction related to quark confinement, strongly supported by experimental observations and lattice simulations. The Clay Mathematics Institute designated its rigorous proof within continuum Quantum Field Theory (QFT) as a Millennium Prize Problem. Standard formulations rely on Osterwalder–Schrader (OS) axioms to ensure well-defined relativistic QFT possessing asymptotic freedom and the empirically verified weakening of interactions at high energies. This paper demonstrates a fundamental incompatibility between these established requirements. By analyzing the analytic structure of gauge-invariant two-point correlation functions via the Källén–Lehmann spectral representation (implied by OS axioms) constrained by the mass gap, and confronting it with the specific asymptotic behavior dictated by asymptotic freedom (derived from Renormalization Group analysis), a mathematical contradiction is rigorously derived. Specifically, the polynomial and logarithmic structure required by asymptotic freedom at a high momentum cannot be reconciled with the asymptotic behavior allowed by the spectral representation for a theory with a mass gap and satisfying OS axioms. This incompatibility strongly suggests that the premise of a fundamental spacetime continuum, underlying standard QFT formulations, is inconsistent with the observed physical reality of the mass gap and asymptotic freedom. This paper challenges this mainstream understanding: it argues that the very attempt to combine the standard axiomatic framework (OS axioms) with the physical requirements of a mass gap and asymptotic freedom in a continuum setting leads to a mathematical contradiction. The problem, therefore, might not be how to construct it, but whether it can be constructed at all under these combined assumptions in a continuum.