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In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive…
Accelerating the convergence of second-order optimization, particularly Newton-type methods, remains a pivotal challenge in algorithmic research. In this paper, we extend previous work on the \textbf{Quadratic Gradient (QG)} and rigorously…
The present work is devoted to the study of fractional powers of vector operators, with particular emphasis on the gradient operator with non-constant coefficients. Within the setting of Clifford algebra $\mathbb{R}_n$, this operator turns…
Regarding quaternions as normal matrices, we first characterize the $2\times 2$ matrix-valued functions, defined on subsets of quaternions, whose values are quaternions. Then we investigate the regularity of quaternionic-valued functions,…
Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided a new approach to…
This article improves the triangle inequality for complex numbers, using the Hermite-Hadamard inequality for convex functions. Then, applications of the obtained refinement are presented to include some operator inequalities. The operator…
A thorough discussion and development of the calculus of real-valued functions of complex-valued vectors is given using the framework of the Wirtinger Calculus. The presented material is suitable for exposition in an introductory Electrical…
Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing,…
In this article, we prove the following spectral theorem for right linear normal operators (need not to be bounded) in quaternionic Hilbert spaces: Let $T$ be an unbounded right quaternionic linear normal operator in a quaternionic Hilbert…
Ladder operators are useful, if not essential, in the analysis of some given physical system since they can be used to find easily eigenvalues and eigenvectors of its Hamiltonian. In this paper we extend our previous results on abstract…
In this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of $S$-spectrum. The proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and…
In this paper we provide a definition of fractional gradient operators, related to directional derivatives. We develop a fractional vector calculus, providing a probabilistic interpretation and mathematical tools to treat multidimensional…
Fractional integral operators connected with real-valued scalar functions of matrix argument are applied in problems of mathematics, statistics and natural sciences. In this article we start considering the case of a Gauss hypergeometric…
We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model…
Novel types of convolution operators for quaternion linear canonical transform (QLCT) are proposed. Type one and two are defined in the spatial and QLCT spectral domains, respectively. They are distinct in the quaternion space and are…
Due to the noncommutative nature of quaternions and octonions we introduce barred operators. This objects give the opportunity to manipulate appropriately the hypercomplex fields. The standard problems arising in the definitions of…
For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\varepsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} \Lambda_{\varepsilon}^{S}(T) := \sigma_S (T) \bigcup \left \{ q…
Gradient-based optimization is a key ingredient of variational quantum algorithms, with applications ranging from quantum machine learning to quantum chemistry and simulation. The parameter-shift rule provides a hardware-friendly method for…
We address the challenging problem of deep representation learning--the efficient adaption of a pre-trained deep network to different tasks. Specifically, we propose to explore gradient-based features. These features are gradients of the…
The theory of the operator $$G(x) = |\underline{x}|^2 \frac{\partial }{\partial x_0} + \underline{x} \sum_{j=1}^n x_j \frac{\partial }{\partial x_j} $$ is deeply associated with the slice monogenic function theory and has grown in recent…