Machine learning algorithms are increasingly used to enhance agricultural productivity cost-effectively. A critical task in precision agriculture is locating a plant’s root collar. This is required for the site-specific fertilization of the plants. Though state-of-the-art machine learning models achieve stellar performance in object detection, they are often sensitive to noisy inputs and variation in environment settings. In this paper, we propose an innovative technique of smooth perturbations to improve the robustness of root collar detection tasks using the YOLOv5 neural network model. We train a YOLOv5 model on blueberry image data for root collar detection. A small amount noise is added as a smooth perturbation to the bounding box of dimensions 50× 50, and this perturbed image is fed for training. Furthermore, we introduce an additional test set that represents the out-of-distribution (O.O.D.) case by applying Gaussian blur on test images to simulate particle situation. We use three different image datasets to train our model, the (i) Estonian blueberry, (ii) Serbian blueberry image, and (iii) public dataset sourced from Roboflow datasets, of sample size 118, 2779, and 2993, respectively. We achieve an overall precision of 0.886 on perturbed blueberry images compared to 0.871 on original (unperturbed) images for the O.O.D. test set. Similarly, our smooth perturbation training has achieved an mAP of 0.828, which significantly increases against the result of normal training, which only reaches 0.794. The result proves that our proposed smooth perturbation is an effective method to increase the robustness and generalizability of the object detection task.

We propose a generalization of the concept of connection form by means of a graded q-differential algebra Ω_q, where q is a primitive Nth root of unity, and develop the concept of curvature N-form for this generalization of the connection form. The Bianchi identity for a curvature N-form is proved. We study an Ω_q-connection on module and prove that every projective module admits an Ω_q-connection. If the module is equipped with a Hermitian structure, we introduce a notion of an Ω_q-connection consistent with the Hermitian structure.

Motivated by a ternary generalization of the Pauli exclusion principle proposed by R. Kerner, we propose a notion of a ${\mathbb{Z}}_3$-skew-symmetric covariant $\mathrm{SO}\left(3\right)$-tensor of the third order, consider it as a 3-dimensional matrix, and study the geometry of the 10-dimensional complex space of these tensors. We split this 10-dimensional space into a direct sum of two 5-dimensional subspaces by means of a primitive third-order root of unity q, and in each subspace, there is an irreducible representation of the rotation group $\mathrm{SO}\left(3\right)\hookrightarrow \mathrm{SU}\left(5\right)$. We find two $\mathrm{SO}\left(3\right)$-invariants of ${\mathbb{Z}}_3$-skew-symmetric tensors: one is the canonical Hermitian metric in five-dimensional complex vector space and the other is a quadratic form denoted by $K(z,z)$. We study the invariant properties of $K(z,z)$ and find its stabilizer. Making use of these invariant properties, we define an $\mathrm{SO}\left(3\right)$-irreducible geometric structure on a five-dimensional complex Hermitian manifold. We study a connection on a five-dimensional complex Hermitian manifold with an $\mathrm{SO}\left(3\right)$-irreducible geometric structure and find its curvature and torsion.

We propose the notion of $(q,\sigma ,\tau )$ -differential graded algebra, which generalizes the notions of $(\sigma ,\tau )$ -differential graded algebra and q-differential graded algebra. We construct two examples of $(q,\sigma ,\tau )$ -differential graded algebra, where the first one is constructed by means of the generalized Clifford algebra with two generators (reduced quantum plane), where we use a $(\sigma ,\tau )$ -twisted graded q-commutator. In order to construct the second example, we introduce the notion of $(\sigma ,\tau )$ -pre-cosimplicial algebra.

We propose an approach to extend the concept of a Lie algebra to ternary structures based on $\omega$-symmetry, where $\omega$ is a primitive cube root of unity. We give a definition of a corresponding structure, called a ternary Lie algebra at cube roots of unity, or a ternary $\omega$-Lie algebra. A method for constructing ternary associative algebras has been developed. For ternary algebras, the notions of the ternary $\omega$-associator and the ternary $\omega$-commutator are introduced. It is shown that if a ternary algebra possesses the property of associativity of the first or second kind, then the ternary $\omega$-commutator on this algebra determines the structure of a ternary $\omega$-Lie algebra. Ternary algebras of cubic matrices with associative ternary multiplication of the second kind are considered. The structure of the 8-dimensional ternary $\omega$-Lie algebra of cubic matrices of the second order is studied, and all its subalgebras of dimensions 2 and 3 are determined.