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53 pages, 594 KiB

Open AccessArticle

Arnold’s Piecewise Linear Filtrations, Analogues of Stanley–Reisner Rings and Simplicial Newton Polyhedra

by Anatoly Kushnirenko

Mathematics 2022, 10(23), 4445; https://doi.org/10.3390/math10234445 - 24 Nov 2022

Viewed by 968

Abstract

In 1974, the author proved that the codimension of the ideal

(g_{1}, g_{2}, \dots, g_{d})

generated in the group algebra

K [Z^{d}]

over a field

K

of characteristic 0 by generic Laurent [...] Read more.

In 1974, the author proved that the codimension of the ideal

(g_{1}, g_{2}, \dots, g_{d})

generated in the group algebra

K [Z^{d}]

over a field

K

of characteristic 0 by generic Laurent polynomials having the same Newton polytope

Γ

is equal to

d! \times V o l u m e (Γ)

. Assuming that Newtons polytope is simplicial and super-convenient (that is, containing some neighborhood of the origin), the author strengthens the 1974 result by explicitly specifying the set

B^{s h}

of monomials of cardinality

d! \times V o l u m e (Γ)

, whose equivalence classes form a basis of the quotient algebra

K [Z^{d}] / (g_{1}, g_{2}, \dots, g_{d})

. The set

B^{s h}

is constructed inductively from any shelling

s h

of the polytope

Γ

. Using the

B^{s h}

structure, we prove that the associated graded

K

-algebra

g r^{Γ} (K [Z^{d}])

constructed from the Arnold–Newton filtration of

K

-algebra

K [Z^{d}]

has the Cohen–Macaulay property. This proof is a generalization of B. Kind and P. Kleinschmitt’s 1979 proof that Stanley–Reisner rings of simplicial complexes admitting shelling are Cohen–Macaulay. Finally, we prove that for generic Laurent polynomials

(f_{1}, f_{2}, \dots, f_{d})

with the same Newton polytope

Γ

, the set

B^{s h}

defines a monomial basis of the quotient algebra

K [Z^{d}] / (g_{1}, g_{2}, \dots, g_{d})

. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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7 pages, 249 KiB

Open AccessArticle

Automorphisms and Definability (of Reducts) for Upward Complete Structures

by Alexei Semenov and Sergei Soprunov

Mathematics 2022, 10(20), 3748; https://doi.org/10.3390/math10203748 - 12 Oct 2022

Cited by 1 | Viewed by 966

Abstract

The Svenonius theorem establishes the correspondence between definability of relations in a countable structure and automorphism groups of these relations in extensions of the structure. This may help in finding a description of the lattice constituted by all definability spaces (reducts) of the [...] Read more.

The Svenonius theorem establishes the correspondence between definability of relations in a countable structure and automorphism groups of these relations in extensions of the structure. This may help in finding a description of the lattice constituted by all definability spaces (reducts) of the original structure. Results on definability lattices were previously obtained only for ω-categorical structures with finite signature. In our work, we introduce the concept of an upward complete structure and define the upward completion of a structure. For upward complete structures, the Galois correspondence between definability lattice and the lattice of closed supergroups of the automorphism group of the structure is an anti-isomorphism. We describe the natural class of structures which have upward completion, we call them discretely homogeneous graphs, present the explicit construction of their completion and automorphism groups of completions. We establish the general localness property of discretely homogeneous graphs and present examples of completable structures and their completions. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

14 pages, 487 KiB

Open AccessArticle

On Strictly Positive Fragments of Modal Logics with Confluence

by Stanislav Kikot and Andrey Kudinov

Mathematics 2022, 10(19), 3701; https://doi.org/10.3390/math10193701 - 10 Oct 2022

Viewed by 1081

Abstract

We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as

K . 2

,

D . 2

,

D 4.2

and

S 4.2

with unimodal confluence [...] Read more.

We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as

K . 2

,

D . 2

,

D 4.2

and

S 4.2

with unimodal confluence

⋄ □ p \to □ ⋄ p

as well as the products of modal logics in the set

\{K, D, T, D 4, S 4\}

, which contain bimodal confluence

⋄_{1} □_{2} p \to □_{2} ⋄_{1} p

. We show that the impact of the unimodal confluence axiom on the axiomatisation of strictly positive fragments is rather weak. In the presence of

⊤ \to ⋄ ⊤

, it simply disappears and does not contribute to the axiomatisation. Without

⊤ \to ⋄ ⊤

it gives rise to a weaker formula

⋄ ⊤ \to ⋄ ⋄ ⊤

. On the other hand, bimodal confluence gives rise to more complicated formulas such as

⋄_{1} p \land ⋄_{2}^{n} ⊤ \to ⋄_{1} (p \land ⋄_{2}^{n} ⊤)

(which are superfluous in a product if the corresponding factor contains

⊤ \to ⋄ ⊤

). We also show that bimodal confluence cannot be captured by any finite set of strictly positive implications. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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21 pages, 598 KiB

Open AccessArticle

On Dihedralized Gyrogroups and Their Cayley Graphs

by Rasimate Maungchang and Teerapong Suksumran

Mathematics 2022, 10(13), 2276; https://doi.org/10.3390/math10132276 - 29 Jun 2022

Cited by 3 | Viewed by 1099

Abstract

The method of constructing the generalized dihedral group as a semidirect product of an abelian group and the group

Z_{2}

of integers modulo 2 is extended to the case of gyrogroups. This leads to the study of a new class of gyrogroups, [...] Read more.

The method of constructing the generalized dihedral group as a semidirect product of an abelian group and the group

Z_{2}

of integers modulo 2 is extended to the case of gyrogroups. This leads to the study of a new class of gyrogroups, which includes generalized dihedral groups and dihedral groups as a special case. In this article, we show that any dihedralizable gyrogroup can be enlarged to a dihedralized gyrogroup. Then, we establish algebraic properties of dihedralized gyrogroups as well as combinatorial properties of their Cayley graphs. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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14 pages, 298 KiB

Open AccessArticle

On Undecidability of Finite Subsets Theory for Torsion Abelian Groups

by Sergey Mikhailovich Dudakov

Mathematics 2022, 10(3), 533; https://doi.org/10.3390/math10030533 - 08 Feb 2022

Viewed by 1375

Abstract

Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we [...] Read more.

Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted. In particular, this theory is undecidable. For example, the free monoid (the sets of all words with concatenation) has this property, the corresponding algebra of finite subsets is the theory of all finite languages with concatenation. Another example is an arbitrary Abelian group that is not a torsion group. But the method of proof significantly used an element of infinite order, hence, it can’t be immediately generalized to torsion groups. In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

11 pages, 3852 KiB

Open AccessArticle

Detection of Multi-Pixel Low Contrast Object on a Real Sea Surface

by Victor Golikov, Oleg Samovarov, Daria Chernomorets and Marco Rodriguez-Blanco

Mathematics 2022, 10(3), 392; https://doi.org/10.3390/math10030392 - 27 Jan 2022

Viewed by 1766

Abstract

Video images captured at long range often show low-contrast floating objects of interest on a sea surface. A comparative experimental study of the statistical characteristics of reflections from floating objects and from the agitated sea surface showed differences in the correlation and spectral [...] Read more.

Video images captured at long range often show low-contrast floating objects of interest on a sea surface. A comparative experimental study of the statistical characteristics of reflections from floating objects and from the agitated sea surface showed differences in the correlation and spectral characteristics of these reflections. The functioning of the recently proposed modified matched subspace detector (MMSD) is based on the separation of the observed data spectrum on two subspaces: relatively low and relatively high frequencies. In the literature, the MMSD performance has been evaluated in general and using only a sea model (i.e., additive Gaussian background clutter). This paper extends the performance evaluating methodology for low contrast object detection using only a real sea dataset. The methodology assumes an object of low contrast if the mean and variance of the object and the surrounding background are the same. The paper assumes that the energy spectrum of the object and the sea are different. The paper investigates a scenario in which an artificially created model of a floating object with specified statistical parameters is placed on the surface of a real sea image. The paper compares the efficiency of the classical matched subspace detector (MSD) and MMSD for detecting low-contrast objects on the sea surface. The article analyzes the dependence of the detection probability at a fixed false alarm probability on the difference between the statistical means and variances of a floating object and the surrounding sea. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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25 pages, 383 KiB

Open AccessArticle

Calculating Complete Lists of Belyi Pairs

by Nikolai M. Adrianov and George B. Shabat

Mathematics 2022, 10(2), 258; https://doi.org/10.3390/math10020258 - 15 Jan 2022

Cited by 1 | Viewed by 1470

Abstract

Belyi pairs constitute an important element of the program developed by Alexander Grothendieck in 1972–1984. This program related seemingly distant domains of mathematics; in the case of Belyi pairs, such domains are two-dimensional combinatorial topology and one-dimensional arithmetic geometry. The paper contains an [...] Read more.

Belyi pairs constitute an important element of the program developed by Alexander Grothendieck in 1972–1984. This program related seemingly distant domains of mathematics; in the case of Belyi pairs, such domains are two-dimensional combinatorial topology and one-dimensional arithmetic geometry. The paper contains an account of some computer-assisted calculations of Belyi pairs with fixed discrete invariants. We present three complete lists of polynomial-like Belyi pairs: (1) of genus 2 and (minimal possible) degree 5; (2) clean ones of genus 1 and degree 8; and (3) clean ones of genus 2 and degree 8. The explanation of some phenomena we encounter in these calculations will hopefully stimulate further development of the dessins d’enfants theory. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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12 pages, 285 KiB

Open AccessArticle

s-Sequences and Monomial Modules

by Gioia Failla and Paola Lea Staglianó

Mathematics 2021, 9(21), 2659; https://doi.org/10.3390/math9212659 - 21 Oct 2021

Viewed by 1114

Abstract

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over [...] Read more.

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

15 pages, 327 KiB

Open AccessArticle

Generalized Strongly Increasing Semigroups

by E. R. García Barroso, J. I. García-García and A. Vigneron-Tenorio

Mathematics 2021, 9(12), 1370; https://doi.org/10.3390/math9121370 - 13 Jun 2021

Cited by 1 | Viewed by 1489

Abstract

In this work, we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and other families of semigroups and we explicitly give their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a [...] Read more.

In this work, we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and other families of semigroups and we explicitly give their set of gaps. Moreover, an algorithm to obtain all the GSI-semigroups up to a given Frobenius number is provided and the realization of positive integers as Frobenius numbers of GSI-semigroups is studied. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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36 pages, 494 KiB

Open AccessReview

Polynomial Automorphisms, Deformation Quantization and Some Applications on Noncommutative Algebras

by Wenchao Zhang, Roman Yavich, Alexei Belov-Kanel, Farrokh Razavinia, Andrey Elishev and Jietai Yu

Mathematics 2022, 10(22), 4214; https://doi.org/10.3390/math10224214 - 11 Nov 2022

Viewed by 1158

Abstract

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich [...] Read more.

This paper surveys results concerning the quantization approach to the Jacobian Conjecture and related topics on noncommutative algebras. We start with a brief review of the paper and its motivations. The first section deals with the approximation by tame automorphisms and the Belov–Kontsevich Conjecture. The second section provides quantization proof of Bergman’s centralizer theorem which has not been revisited for almost 50 years and formulates several related centralizer problems. In the third section, we investigate a free algebra analogue of a classical theorem of Białynicki-Birula’s theorem and give a noncommutative version of this famous theorem. Additionally, we consider positive-root torus actions and obtain the linearity property analogous to the Białynicki-Birula theorem. In the last sections, we introduce Feigin’s homomorphisms and we see how they help us in proving our main and fundamental theorems on screening operators and in the construction of our lattice

W_{n}

-algebras associated with

{sl}_{n}

, which is by far the simplest known approach concerning constructing such algebras until now. Full article

(This article belongs to the Special Issue Combinatorial Algebra, Computation, and Logic)

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