Advances in Applied Clifford Algebras最新文献

{"title":"Scaled-Difference and Angular Refinements of Eneström–Kakeya Bounds for Quaternionic Polynomials","authors":"Bilal Ahmad Rather","doi":"10.1007/s00006-026-01468-0","DOIUrl":"10.1007/s00006-026-01468-0","url":null,"abstract":"<div><p>We establish refined Eneström–Kakeya type bounds for the zeros of right quaternionic polynomials using a scaled-difference approach. The main result provides an optimizable upper bound in terms of weighted coefficient variation, while coefficient reversal gives corresponding lower bounds. The finite-defect formulas extend the classical monotonicity conditions, and angular restrictions give further estimates based on coefficient directions. We also develop orthogonal-frame and projection bounds, with Haar averaging providing the universal factor <span>(16/(3pi ))</span>. The numerical examples illustrate the effectiveness of scaling and compare the resulting bounds with the actual zero moduli.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"37 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149309895","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":"Smooth Geometry of Graded Skew Clifford Algebras","authors":"Andrés Rubiano","doi":"10.1007/s00006-026-01464-4","DOIUrl":"10.1007/s00006-026-01464-4","url":null,"abstract":"<div><p>We investigate the differential smoothness of graded skew Clifford algebras. Let <span>(C=C(mu ,M_1,ldots ,M_n))</span> be determined by a multiplicatively antisymmetric parameter matrix <span>(mu )</span> and a family of <span>(mu )</span>-symmetric matrices <span>(M_1,ldots ,M_n)</span>. Assuming that these matrices are linearly independent and satisfy an explicit compatibility condition relating their nonzero entries to the parameters <span>(mu _{ij})</span>, we construct a first-order differential calculus generated by <span>(dx_1,ldots ,dx_n)</span>. The associated twisting automorphisms commute, allowing the calculus to extend to a connected differential graded algebra of top degree <i>n</i>. We identify the volume form <span>(omega =dx_1wedge cdots wedge dx_n)</span> and prove integrability by exhibiting dual families of differential forms that implement the required Poincaré-type duality between differential and integral forms. Since the dimension of the calculus agrees with the Gelfand–Kirillov dimension of <i>C</i>, the algebra is differentially smooth. As a consequence, every graded Clifford algebra satisfying the corresponding independence condition is differentially smooth. Several examples illustrate how the criterion can be verified explicitly, while an additional example shows that it does not cover all graded skew Clifford algebras. These results provide a concrete family of noncommutative quadratic algebras equipped with integrable differential calculi and canonical volume forms.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149293023","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":"N-ary Anticommutators and Generalized Clifford Algebras in Finsler and Spectral Geometry","authors":"Carlos Castro Perelman","doi":"10.1007/s00006-026-01469-z","DOIUrl":"10.1007/s00006-026-01469-z","url":null,"abstract":"<div><p>It is shown that a careful study of the simplest family of generalized Clifford algebras (GCAs) associated with the <i>N</i>th root of unity in <i>d</i>-dimensions leads to the following generalized anti-commutator with <i>N</i> entries <span>( { e_{i_1}, e_{i_2}, e_{ i_3}, ldots , e_{ i_N} } = e_{i_1} e_{i_2} e_{ i_3} ldots e_{ i_N} + permutations = N! eta _{ i_1 i_2 ldots i_N } e )</span>, where <i>e</i> is the unit element and all the <i>N</i>! terms of the permutations appear with the same positive sign. The components of the rank-<i>N</i> metric are <span>( eta _{ i_1 i_2 ldots i_N } = 1, )</span> iff <span>( i_1 = i_2 = i_3 ldots = i_ N )</span>, and 0 otherwise. The range of the indices <span>( i_1, i_2, ldots i_N )</span> is <span>( 1,2, ldots , d )</span>. We proceed to explore the <i>N</i>th norm extensions of the quadratic norm and write down a generalized Finsler-like arc-length based on the rank-<i>N</i> metric <span>( g_{mu _1 mu _2 ldots mu _N})</span>. We continue by constructing the different expressions of the Dirac operators associated with the (Generalized) Clifford Spaces corresponding to these GCA’s. Dirac operators are essential in the study of Spectral Geometry in Noncommutative Geometry after imposing the correspondence between the geodesic distance and the inverse of the Dirac operator (a fermion propagator). These generalized anti-commutators above are special types of an <i>N</i>-ary algebraic structure. We finalize by displaying the relation among GCAs and the algebras underlying the noncommutative fuzzy torus and discuss applications in condensed matter and quantum groups. We conclude with some remarks on <i>N</i>-ary algebras and their applications in Mathematics and Physics.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149287886","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":"The Doublet Tower of (text {Cl}(4,2)): Universal Nilpotency, Projector Identities, and the (mathfrak {sl}(8,mathbb {R})) Parabolic from Conformal Null Vectors","authors":"Steen H. Hansen","doi":"10.1007/s00006-026-01463-5","DOIUrl":"10.1007/s00006-026-01463-5","url":null,"abstract":"<div><p>The conformal geometric algebra <span>(text {Cl}(4,2)cong mathbb {R}(8))</span> carries a natural <span>(mathbb {Z}_2)</span>-grading defined by the dilatation bivector <span>(D=varvec{e}_{4}varvec{e}_{0})</span>, which satisfies <span>(D^2=1)</span> and splits the eight-dimensional spinor module into two four-dimensional eigenspaces <span>(S=S_+oplus S_-)</span>. We prove that the <span>(text {ad}_D)</span>-eigenvectors of <span>(text {Cl}(4,2))</span>, those elements with eigenvalue <span>(pm 1)</span> under the Lie bracket <span>([D,cdot ])</span>, are exhausted by 16 doublets of the form <span>(T^pm _J=n_bullet e_J)</span>, where <span>(n_bullet in {n_{infty },n_{text {o}}})</span> is a null vector and <span>(e_J)</span> is a product of Lorentz-sector basis vectors indexed by <span>(Jsubseteq {1,2,3,5})</span>. These doublets satisfy universal algebraic identities: <span>((T^pm _J)^2=0)</span> (nilpotency) and <span>(T^+_J,T^-_J=(-1)^{k(k+1)/2+1},eta _J 2Pi _+)</span> (single-projector proportionality), where <span>(k=|J|)</span>, <span>(eta _J=prod _{jin J}eta _{jj})</span> is the spectator metric, and <span>(Pi _+=tfrac{1}{2}(1+D))</span>. The contracted sums <span>(sum _{|J|=k}T^+_J T^-_J)</span> are shown to be controlled by the elementary symmetric polynomials <span>(S_k)</span> of the Lorentz signature values <span>((+,1,+,1,+,1,-,1))</span>, with the vanishing of <span>(S_2=0)</span> reflecting the (3, 1) signature of physical spacetime. The even-grade doublets (belonging to <span>(text {Cl}^+(4,2)cong text {Cl}(4,1))</span>) contribute <span>(8Pi _+)</span> to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra <span>(mathfrak {sl}(8,mathbb {R}))</span> in which the conformal algebra <span>(mathfrak {so}(4,2))</span> embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension <span>(text {Cl}(p+1,q+1))</span>; there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars, and the family extensions <span>(text {Cl}(4+2k,2))</span> yield an exact description of the tower-realized flavour couplings.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 5","pages":""},"PeriodicalIF":1.0,"publicationDate":"2026-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-026-01463-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149234781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inverse Transmutation Operator for (alpha )-Hyperholomorphic Functions in Clifford Analysis","authors":"Doan Cong Dinh,&nbsp;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,&nbsp;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,&nbsp;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,&nbsp;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,&nbsp;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,&nbsp;Ying Li,&nbsp;Jingyi Zhao,&nbsp;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}