{"title":"Inverse Transmutation Operator for (alpha )-Hyperholomorphic Functions in Clifford Analysis","authors":"Doan Cong Dinh, Luong Thi Tuyet","doi":"10.1007/s00006-026-01462-6","DOIUrl":"10.1007/s00006-026-01462-6","url":null,"abstract":"<div><p>This paper revisits <span>(alpha )</span>-hyperholomorphic functions in Clifford analysis, that is, functions satisfying the equation <span>( Du - ualpha = 0, )</span> where <i>D</i> denotes the Dirac operator in <span>(mathbb {R}^n)</span> and <span>(alpha in mathbb {C}_{0,n})</span>. A transmutation operator for <span>(alpha )</span>-hyperholomorphic functions has previously been constructed, mapping each monogenic function in a star-like domain to an <span>(alpha )</span>-hyperholomorphic one. In the present work, we derive explicit integral kernels for the inverse of this operator.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149189663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Paley–Wiener-Type Theorems for the Quaternion Dunkl Transform","authors":"Shanshan Li, Minggang Fei","doi":"10.1007/s00006-026-01466-2","DOIUrl":"10.1007/s00006-026-01466-2","url":null,"abstract":"<div><p>The aim of this paper is to establish several versions of Paley–Wiener-type theorems for the two-sided quaternion Dunkl transform (QDT) in <span>(mathbb {R}^2)</span>. This transform combines the non-commutative structure of quaternions with the reflection symmetry and weighted analysis inherent in Dunkl theory.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149189335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Solution of Inverse Eigenvalue Problems for Quaternionic Matrices","authors":"Tanay Saha, Sandipan Dutta","doi":"10.1007/s00006-026-01467-1","DOIUrl":"10.1007/s00006-026-01467-1","url":null,"abstract":"<div><p>This paper explores inverse eigenvalue problems concerning quaternionic matrices. Initially, we establish a vital criterion determining whether specified sets of complex numbers and quaternionic vectors constitute eigenvalues and their corresponding eigenvectors for certain quaternionic matrices. Subsequently, we provide explicit constructions for the collection of such quaternionic matrices, if they indeed exist. Additionally, within this collection, we identify a specific matrix closest to a given quaternionic matrix. This process involves transforming the problem into a convex optimization challenge. Finally, we verify our findings through various numerical demonstrations.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149150626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On S-spectrum related to a regularity","authors":"Rachid Arzini, Ali Jaatit","doi":"10.1007/s00006-026-01465-3","DOIUrl":"10.1007/s00006-026-01465-3","url":null,"abstract":"<div><p>Let <span>( mathcal {L}(X) )</span> denote the algebra of bounded right linear operators on a two-sided quaternionic Banach space <span>( X )</span>. In this paper, we introduce the concept of regularity in <span>(mathcal {L}(X))</span> extending that given by [13, 14]. We also establish the spectral mapping theorem for the spherical spectrum defined by a regularity. Furthermore, certain common spectral properties of operator products are examined.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149132550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Explicit Formula for Inverse and Determinant in Geometric Algebras over Odd-dimensional Vector Spaces","authors":"Kamron Abdulkhaev, Dmitry Shirokov","doi":"10.1007/s00006-026-01461-7","DOIUrl":"10.1007/s00006-026-01461-7","url":null,"abstract":"<div><p>In this paper, we present explicit formulas for the inverse and determinant in geometric (Clifford) algebras over vector spaces of dimension <span>(n=7)</span>. The derivation of these formulas is made possible by generalizing the concept of conjugation to basis conjugation operations. We further develop a general method for constructing such formulas over odd-dimensional spaces from the known even-dimensional case. To validate computational utility of the results, we provide a numerical implementation of the formulas. The code implementation is available at the repository github.com/kamranuz/clifford_7d. These formulas extend previous results for lower dimensions and offer new insights for applications in mathematical physics and computational geometry.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149067342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Solving Higher Quaternion Matrix Equation (AXB+CYD=E) Based on Semi-tensor Product of Matrices with Application","authors":"Junhao Li, Ying Li, Jingyi Zhao, Ning Shao","doi":"10.1007/s00006-026-01460-8","DOIUrl":"10.1007/s00006-026-01460-8","url":null,"abstract":"<div><p>In this paper, we investigate the general solutions to higher quaternion matrix equation <span>(AXB + CYD = E)</span> based on the semi-tensor product of matrices. First, based on the semi-tensor product of matrices, we propose the left and right real element representations for quaternions and conduct a study on their properties. Then, we provide the real columns tacking form of <i>n</i>-quaternions and <i>n</i>-quaternion matrices. Next, we convert the higher quaternion matrix equation into a real linear system by means of the left and right real element representation of higher quaternion and give the equivalent condition for the existence of the general solutions to the higher quaternion matrix equation. Furthermore, we obtain the expressions of the general solutions to the higher quaternion matrix equation <span>(AXB + CYD = E)</span>. And then, we propose the algorithms and conduct numerical experiments to verify the validity of our method. Finally, we combine this method with group permutation and apply it to image encryption and decryption, achieving good results.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148986107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Geometric Algebra-Inspired Spatio-temporal Learning Model for Sea Level Variability Reconstruction Under Incomplete Observations","authors":"Dongshuang Li, Yadi Huang, Chengqiang Huang, Wen Luo, Zhaoyuan Yu, Linwang Yuan","doi":"10.1007/s00006-026-01457-3","DOIUrl":"10.1007/s00006-026-01457-3","url":null,"abstract":"<div><p>Reconstructing complete sea level variability (SLV) from sparse observations is a central problem in ocean science. Accurate reconstruction critically depends on modeling cross-variable dependencies, as SLV is governed by coupled ocean processes and the associated state variables are intrinsically interdependent. Most deep learning approaches process such variables as independent channels, which may limit their ability to capture coordinated multivariate interactions under data sparsity. To address this issue, we develop a geometric algebra (GA)-inspired spatio-temporal learning model in which temperature, salinity, and horizontal velocities are encoded as coefficients associated with the basis blades of a multivector. The core module, a GA-based convolutional gated recurrent unit (GAConvGRU), replaces conventional channel-wise convolution with algebraically structured convolution within a recurrent architecture. The framework is further complemented by temporal embedding, dual-path convolution, and linear mapping to support multi-scale spatio-temporal modeling. Experiments across three coastal regions with varying levels of missing observations validate the proposed model. Ablation results show that GAConvGRU yields increasingly pronounced performance gains as data sparsity grows. Comparative evaluations against state-of-the-art methods demonstrate consistently competitive or superior reconstruction accuracy, with advantages more evident in high-missing-rate scenarios. Overall, the results indicate that GA-based multivariate interaction modeling enhances robustness in spatio-temporal learning under incomplete observations and provides an effective approach for SLV reconstruction and other geophysical applications involving strongly interdependent variables.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148927405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dendrification of Hom-Malcev Algebras","authors":"F. Harrathi, S. Mabrouk, O. Ncib, S. Silvestrov","doi":"10.1007/s00006-026-01450-w","DOIUrl":"10.1007/s00006-026-01450-w","url":null,"abstract":"<div><p>The main goal of this work is to introduce a notion of Hom-M-dendriform algebras which can be seen as the dendriform version of Hom-Malcev algebras. In fact they are the algebraic structures behind the <span>(mathcal {O})</span>-operator of Hom-pre-Malcev algebras. They also fit into a larger framework as Hom-Malcev algebraic analogues of Hom-L-dendriform algebras. Furthermore, we show a connections between Hom-M-dendriform algebras and Hom-alternative quadri-algebras.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148782321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Another 6D Multivector Inverse Algorithm","authors":"David Arthur Jones","doi":"10.1007/s00006-026-01455-5","DOIUrl":"10.1007/s00006-026-01455-5","url":null,"abstract":"<div><p>A simple algorithm for the inverse of a non-singular 6D multivector is presented. It is indifferent to signature (including degeneracy). A simpler algorithm is unlikely. A 4D algorithm of the same form is also presented. The formulas here are unlike the grade-involution-based inverse algorithms from which they were derived; instead, they rescale the scalar component. This algorithm requires two fewer multivector multiplications than its antecedent. A general expression for an annihilating polynomial of any 6D self-reverse multivector is given, with coefficients formed solely from the scalar parts of powers of the multivector. Opportunities for optimization are analyzed, and a strategy is presented that can reduce the number of real multiplications required to form the inverse to fewer than are required to multiply two arbitrary multivectors. Python code is presented, with additional code and documentation available in a public repository (see Ref. [7]).</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148728730","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}