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23 pages, 363 KB

Open AccessArticle

BIBO Stability of Linear Control Systems on Lie Group Examples

by Víctor Ayala, María Luisa Torreblanca Todco and William Eduardo Valdivia Hanco

Mathematics 2026, 14(12), 2141; https://doi.org/10.3390/math14122141 - 15 Jun 2026

Viewed by 162

We develop a collection of nontrivial examples that illustrate and test recent stability results for linear control systems (

L C S

) on Lie groups. We treat the main structural classes: Abelian (

R^{n}

), nilpotent (Heisenberg), solvable non-nilpotent (rigid motions [...] Read more.

We develop a collection of nontrivial examples that illustrate and test recent stability results for linear control systems (

L C S

) on Lie groups. We treat the main structural classes: Abelian (

R^{n}

), nilpotent (Heisenberg), solvable non-nilpotent (rigid motions of the plane

S E (2)

), compact semisimple (

S O (3)

), noncompact semisimple (

S L (2, R)

via Iwasawa decomposition) and mixed/Levi-type groups. The examples are designed to (i) show the sharpness of geometric boundedness criteria, (ii) exhibit typical failure modes (exponential escape, polynomial central drift, noncompact neutrals), and (iii) demonstrate how the canonical quotient and suitable outputs recover

B I B O

stability. The executive framework (

I C S

existence/uniqueness, canonical quotient

G / Γ

,

B I B O

characterization, robustness and ISS-type bounds) is briefly recalled; the main part of the paper consists of detailed worked examples implementing the practical checklist for applying these theorems. Full article

(This article belongs to the Section E2: Control Theory and Mechanics)

71 pages, 727 KB

Open AccessArticle

Notes on Number Theory

by Miroslav Stoenchev, Slavi Georgiev and Venelin Todorov

Mathematics 2026, 14(4), 697; https://doi.org/10.3390/math14040697 - 16 Feb 2026

Viewed by 1074

Abstract

This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of

Q

and describe algebraic number fields through their realization as finite-dimensional

Q

-algebras (via [...] Read more.

This paper presents a set of survey-style notes linking core themes of pure algebra with central topics in algebraic and analytic number theory. We begin with finite extensions of

Q

and describe algebraic number fields through their realization as finite-dimensional

Q

-algebras (via multiplication operators and matrix representations), leading naturally to the arithmetic invariants—trace, norm, and discriminant—and to the ring of integers, ideals, Dedekind domains, and the ideal class group. We then develop the classical theory of cyclotomic fields, emphasizing their Galois structure and their role in abelian extensions of

Q

. Next, we discuss ramification in general extensions, including decomposition and inertia groups, the Frobenius element, and the Chebotarev density theorem. The exposition continues with a concise algebraic introduction to elliptic curves and their L-functions, and it places key conjectural links (including Birch and Swinnerton-Dyer) in context. Finally, a collection of examples highlights a common operational language between fractional calculus and number theory: Laplace and Mellin transforms turn convolution-type operators into multiplication, clarifying the appearance of

Γ

-factors, Dirichlet series, and zeta- and L-function structures in both settings. Full article

(This article belongs to the Special Issue Advanced Research in Pure and Applied Algebra, 2nd Edition)

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34 pages, 495 KB

Open AccessFeature PaperArticle

Rigidity and Toledo Invariant for Spin*(8)-Higgs Bundles

by Álvaro Antón-Sancho

Mathematics 2026, 14(2), 358; https://doi.org/10.3390/math14020358 - 21 Jan 2026

Viewed by 397

Abstract

In this paper, we study

{Spin}^{*} (8)

-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on

{SO}^{*} (8)

. The group

{Spin}^{*} (8)

is exceptional among classical real [...] Read more.

In this paper, we study

{Spin}^{*} (8)

-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on

{SO}^{*} (8)

. The group

{Spin}^{*} (8)

is exceptional among classical real forms, as its complexification

Spin (8, C)

admits triality, an outer automorphism of order 3, but triality does not preserve the real form

{Spin}^{*} (8)

. We establish the Toledo bound

| τ | \leq 4 (g - 1)

for semistable

{Spin}^{*} (8)

-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the

{SO}^{*} (8)

moduli space with discrete fibers parametrized by spin structures, and has a dimension of

15 (g - 1)

, one less than expected. Using Morse theory, we establish connectedness of moduli spaces for

τ = 0

and maximal

| τ |

. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into

{Spin}^{*} (8)

. We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form. Full article

(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)

19 pages, 607 KB

Open AccessArticle

The Stability of Linear Control Systems on Low-Dimensional Lie Groups

by Víctor Ayala, William Eduardo Valdivia Hanco, Jhon Eddy Pariapaza Mamani and María Luisa Torreblanca Todco

Symmetry 2025, 17(10), 1766; https://doi.org/10.3390/sym17101766 - 20 Oct 2025

Viewed by 859

Abstract

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such [...] Read more.

This work investigates the stability analysis of linear control systems defined on Lie groups, with a particular focus on low-dimensional cases. Unlike their Euclidean counterparts, such systems evolve on manifolds with non-Euclidean geometry, where trajectories respect the group’s intrinsic symmetries. Stability notions, such as inner asymptotic, inner, and input–output (BIBO) stability, are studied. The qualitative behavior of solutions is shown to depend critically on the spectral decomposition of derivations associated with the drift, and on the algebraic structure of the underlying Lie algebra. We study two classes of examples in detail: Abelian and solvable two-dimensional Lie groups, and the three-dimensional nilpotent Heisenberg group. These settings, while mathematically tractable, retain essential features of non-commutativity, geometric non-linearity, and sub-Riemannian geometry, making them canonical models in control theory. The results highlight the interplay between algebraic properties, invariant submanifolds, and trajectory behavior, offering insights applicable to robotic motion planning, quantum control, and signal processing. Full article

(This article belongs to the Special Issue Symmetries in Dynamical Systems and Control Theory)

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22 pages, 388 KB

Open AccessArticle

Gauge-Invariant Slavnov–Taylor Decomposition for Trilinear Vertices

by Andrea Quadri

Universe 2025, 11(7), 228; https://doi.org/10.3390/universe11070228 - 11 Jul 2025

Viewed by 581

Abstract

We continue the analysis of the gauge-invariant decomposition of amplitudes in spontaneously broken massive gauge theories by performing the characterization of separately gauge-invariant subsectors for amplitudes involving trilinear interaction vertices for an Abelian theory with chiral fermions. We show that the use of [...] Read more.

We continue the analysis of the gauge-invariant decomposition of amplitudes in spontaneously broken massive gauge theories by performing the characterization of separately gauge-invariant subsectors for amplitudes involving trilinear interaction vertices for an Abelian theory with chiral fermions. We show that the use of Frohlich–Morchio–Strocchi gauge-invariant dynamical (i.e., propagating inside loops) fields yields a very powerful handle on the cancellations among unphysical degrees of freedom (the longitudinal mode of the massive gauge field, the Goldstone scalar and the ghosts). The resulting cancellations are encoded into separate Slavnov–Taylor invariant sectors for 1-PI amplitudes. The construction works to all orders in perturbation theory. This decomposition suggests a novel strategy for the determination of finite counter-terms required to restore the Slavnov–Taylor identities in chiral theories in the absence of an invariant regularization scheme. Full article

(This article belongs to the Section Field Theory)

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33 pages, 1380 KB

Open AccessArticle

Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

by Anthony Torres-Hernandez, Fernando Brambila-Paz and Rafael Ramirez-Melendez

Fractal Fract. 2024, 8(1), 16; https://doi.org/10.3390/fractalfract8010016 - 23 Dec 2023

Cited by 1 | Viewed by 8654

Abstract

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions [...] Read more.

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through

Q R

decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval

[0, 1)

, among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown. Full article

(This article belongs to the Special Issue Symmetry and Solutions of Fractional Differential Equations with Their Developments)

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17 pages, 550 KB

Open AccessArticle

Spontaneous Emergence of a Causal Time Axis in Euclidean Space from a Gauged Rotational Symmetry Theory

by Michael Luke Walker

Symmetry 2024, 16(1), 4; https://doi.org/10.3390/sym16010004 - 19 Dec 2023

Viewed by 2479

Abstract

We demonstrate the emergence of an effective “time” axis in the ground state of a gauged rotational symmetry theory in four-dimensional Euclidean space. In so doing, we remove the necessity of Wick rotation to Lorentz spacetime, an arbitrary and sometimes ill-defined procedure, especially [...] Read more.

We demonstrate the emergence of an effective “time” axis in the ground state of a gauged rotational symmetry theory in four-dimensional Euclidean space. In so doing, we remove the necessity of Wick rotation to Lorentz spacetime, an arbitrary and sometimes ill-defined procedure, especially for gravity-related theories. We begin by adapting the Cho-Duan-Ge decomposition to the gauge theory of the four-dimensional rotational symmetry group

S O (4)

, where it identifies the maximal Abelian subgroup

S O (2) \otimes S O (2)

in a gauge covariant manner. We then find the one-loop effective theory to have a stable condensate of monopoles corresponding to the reduction of

S O (4)

symmetry to

S O (2) \otimes S O (2)

. The construction of the condensate ensures that the four-dimensional spatial direction of its field strength must coincide with that of this embedding, and that a magnetic potential must be worked against to divert a trajectory away from this direction. Indeed, movement along this direction represents minimal potential energy. We take it to be the time direction. The gauge-dependent nature of the condensate is such that different gauge choices may lead to different time axes and we show on very general grounds that these different coordinate systems must be relatable by transformations of Lorentz form. Full article

(This article belongs to the Special Issue Physics and Symmetry Section: Feature Papers 2023)

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14 pages, 295 KB

Open AccessArticle

A Note on Finite Dimensional Odd Contact Lie Superalgebra in Prime Characteristic

by Xiaoning Xu and Qiyuan Wang

Axioms 2023, 12(12), 1108; https://doi.org/10.3390/axioms12121108 - 8 Dec 2023

Cited by 1 | Viewed by 2502

Abstract

Over a field of characteristic

p > 3

, let

K O (n, n + 1; \underset{̲}{t})

denote the odd contact Lie superalgebra. In this paper, the super-biderivations of odd Contact Lie superalgebra [...] Read more.

Over a field of characteristic

p > 3

, let

K O (n, n + 1; \underset{̲}{t})

denote the odd contact Lie superalgebra. In this paper, the super-biderivations of odd Contact Lie superalgebra

K O (n, n + 1; \underset{̲}{t})

are studied. Let

T_{K O}

be a torus of

K O (n, n + 1; \underset{̲}{t})

, which is an abelian subalgebra of

K O (n, n + 1; \underset{̲}{t})

. By applying the weight space decomposition approach of

K O (n, n + 1; \underset{̲}{t})

with respect to

T_{K O}

, we show that all skew-symmetric super-biderivations of

K O (n, n + 1; \underset{̲}{t})

are inner super-biderivations. Full article

(This article belongs to the Section Algebra and Number Theory)

20 pages, 361 KB

Open AccessArticle

Sequences over Finite Fields Defined by OGS and BN-Pair Decompositions of PSL₂(q) Connected to Dickson and Chebyshev Polynomials

by Robert Shwartz and Hadas Yadayi

Mathematics 2023, 11(4), 965; https://doi.org/10.3390/math11040965 - 13 Feb 2023

Viewed by 1684

Abstract

The factorization of groups into a Zappa–Szép product, or more generally into a k-fold Zappa–Szép product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group and has recently been applied to public-key cryptography. [...] Read more.

The factorization of groups into a Zappa–Szép product, or more generally into a k-fold Zappa–Szép product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group and has recently been applied to public-key cryptography. We provide a generalization of the k-fold Zappa–Szép product of cyclic groups, which we call

O G S

decomposition. It is easy to see that the existence of an

O G S

decomposition for all the composition factors of a non-abelian group G implies the existence of an

O G S

for G itself. Since the composition factors of a soluble group are cyclic groups, it has an

O G S

decomposition. Therefore, the question of the existence of an

O G S

decomposition is interesting for non-soluble groups. The Jordan–Hölder theorem motivates us to consider the existence of an

O G S

decomposition for finite simple groups. In 1993, Holt and Rowley showed that

P S L_{2} (q)

and

P S L_{3} (q)

can be expressed as a product of cyclic groups. In this paper, we consider an

O G S

decomposition of

P S L_{2} (q)

from a different point of view to that of Holt and Rowley. We look at its connection to the

B N

-

p a i r

decomposition of the group. This connection leads to sequences over

F_{q}

, which can be defined recursively, with very interesting properties, and are closely connected to Dickson and Chebyshev polynomials. Since every finite simple Lie-type group exhibits

B N

-

p a i r

decomposition, the ideas in this paper might be generalized to further simple Lie-type groups. Full article

20 pages, 335 KB

Open AccessArticle

Transposition Regular AG-Groupoids and Their Decomposition Theorems

by Yudan Du, Xiaohong Zhang and Xiaogang An

Mathematics 2022, 10(9), 1396; https://doi.org/10.3390/math10091396 - 22 Apr 2022

Cited by 3 | Viewed by 2127

Abstract

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be [...] Read more.

In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be decomposed into the union of disjoint Abelian subgroups; (2) L1/R1-transposition regular AG-groupoids are LR-transposition regular AG-groupoids, and an example is given to illustrate that not every LR-transposition regular AG-groupoid is an L1/R1-transposition regular AG-groupoid; (3) an AG-groupoid is an L1/R1-transposition regular AG-groupoid if it is an LR-transposition regular AG-groupoid satisfying a certain condition; (4) strong L2/R3-transposition regular AG-groupoids are equivalent to each other, and they are union of disjoint Abelian subgroups; (5) strong L3/R2-transposition regular AG-groupoids are equivalent to each other and they can be decomposed into union of disjoint AG subgroups. Their relations are discussed. Finally, we introduce various transposition regular AG-groupoid semigroups and discuss the relationships among them and the commutative Clifford semigroup as well as the Abelian group. Full article

(This article belongs to the Special Issue Algebra and Discrete Mathematics 2021)

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19 pages, 547 KB

Open AccessArticle

Two Types of Jets and Quark and Chromon Model in QCD

by Yongmin Cho

Universe 2019, 5(2), 62; https://doi.org/10.3390/universe5020062 - 14 Feb 2019

Cited by 3 | Viewed by 3440

Abstract

We discuss the importance of the color reflection symmetry of the Abelian decomposition in QCD. The Abelian decomposition breaks up the color gauge field to three parts, the neuron, chromon, and the topological monopole, gauge independently. Moreover, it refines the Feynman diagram in [...] Read more.

We discuss the importance of the color reflection symmetry of the Abelian decomposition in QCD. The Abelian decomposition breaks up the color gauge field to three parts, the neuron, chromon, and the topological monopole, gauge independently. Moreover, it refines the Feynman diagram in such a way that the conservation of color is explicit. This leads us to generalize the quark model to the quark and chromon model. We show how the Abelian decomposition reduces the non-Abelian color gauge symmetry to the simple discrete 24 element color reflection symmetry which assumes the role of the color gauge symmetry and plays the central role in the quark and chromon model. Full article

(This article belongs to the Special Issue Selected Papers from the 7th International Conference on New Frontiers in Physics (ICNFP 2018))

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