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In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…
A combinatorial polytope $P$ is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge…
Focus of this study is to explore some aspects of mathematical foundations for using complex manifolds as a model for space-time. More specifically, certain equations of motions have been derived as a Projective geodesic on a real manifold…
Let $C$ be an irreducible smooth complex projective curve of genus $g$, with $g_C \geqslant 2$. Let $E$ be a vector bundle on $C$ of rank $r$, with $r\geqslant 2$. Let $\mc Q:=\mc Q(E,\,d)$ be the Quot Scheme parameterizing torsion…
Quadratization for partial differential equations (PDEs) is a process that transforms a nonquadratic PDE into a quadratic form by introducing auxiliary variables. This symbolic transformation has been used in diverse fields to simplify the…
In this paper, we study a 2D tomography problem for point source models with random unknown view angles. Rather than recovering the projection angles, we reconstruct the model through a set of rotation-invariant features that are estimated…
Reciprocal transformations mix the role of the dependent and independent variables of (nonlinear partial) differential equations to achieve simpler versions or even linearized versions of them. These transformations help in the…
This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…
In this paper we present definitions of different four-dimensional (4D) geometric quantities (Clifford multivectors). New decompositions of the torque N and the angular momentum M (bivectors) into 1-vectors N_{s}, N_{t} and M_{s}, M_{t}…
We introduce a new approach for robotic manipulation tasks in human settings that necessitates understanding the 3D geometric connections between a pair of objects. Conventional end-to-end training approaches, which convert pixel…
Numerous 6D pose estimation methods have been proposed that employ end-to-end regression to directly estimate the target pose parameters. Since the visible features of objects are implicitly influenced by their poses, the network allows…
Geometric Invariant Theory (GIT) produces quotients of algebraic varieties by reductive groups. If the variety is projective, this quotient depends on a choice of polarisation; by work of Dolgachev-Hu and Thaddeus, it is known that two…
Most state-of-the-art deep geometric learning single-view reconstruction approaches rely on encoder-decoder architectures that output either shape parametrizations or implicit representations. However, these representations rarely preserve…
The purpose of the present paper is to show few examples of nonlinear PDEs (mostly with strong geometric features) for which there is a hidden convex structure. This is not only a matter of curiosity. Once the convex structure is…
Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper,…
Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not…
When flat or on a firm mechanical substrate, the atomic composition and atomistic structure of two-dimensional crystals dictate their chemical, electronic, optical, and mechanical properties. These properties change when the two-dimensional…
Projective geometry provides the preferred framework for most implementations of Euclidean space in graphics applications. Translations and rotations are both linear transformations in projective geometry, which helps when it comes to…
A difference set is said to have classical parameters if $ (v,k, \lambda) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian…
We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a…