Irreducible Ones Research Articles

A functorial approach to rank functions on triangulated categories

Abstract We study rank functions on a triangulated category 𝒞 via its abelianisation mod ⁡ C \operatorname{mod}\mathcal{C} . We prove that every rank function on 𝒞 can be interpreted as an additive function on mod ⁡ C \operatorname{mod}\mathcal{C} . As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category Mod ⁡ C \operatorname{Mod}\mathcal{C} . We study the connection between rank functions and functors from 𝒞 to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T c \mathcal{C}=\mathcal{T}^{c} for a compactly generated triangulated category 𝒯, this connection becomes particularly nice, providing a link between rank functions on 𝒞 and smashing localisations of 𝒯. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in 𝒯. Finally, if C = per ⁡ ( A ) \mathcal{C}=\operatorname{per}(A) for a differential graded algebra 𝐴, we classify homological epimorphisms A → B A\to B with per ⁡ ( B ) \operatorname{per}(B) locally finite via special rank functions which we call idempotent.

Exploring Hybrid H-bi-Ideals in Hemirings: Characterizations and Applications in Decision Making

The concept of the hybrid structure, as an extension of both soft sets and fuzzy sets, has gained significant attention in various mathematical and decision-making domains. In this paper, we delve into the realm of hemirings and investigate the properties of hybrid h-bi-ideals, including prime, strongly prime, semiprime, irreducible, and strongly irreducible ones. By employing these hybrid h-bi-ideals, we provide insightful characterizations of h-hemiregular and h-intra-hemiregular hemirings, offering a deeper understanding of their algebraic structures. Beyond theoretical implications, we demonstrate the practical value of hybrid structures and decision-making theory in handling real-world problems under imprecise environments. Using the proposed decision-making algorithm based on hybrid structures, we have successfully addressed a significant real-world problem, showcasing the efficacy of this approach in providing robust solutions.

Limit of Weierstrass measure on stable curves

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g \geqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne–Mumford compactification $\overline{\mathcal{M}}_g$ of the moduli $\mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $\mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.

Q4-irreducible even triangulations of the projective plane

We establish a new generating theorem of even triangulations of the projective plane using an operation called a Q4-reduction. That is, every even triangulation of the projective plane can be obtained from one of the Q4-irreducible even triangulations, which contains a special subgraph named a projective-triangular cupola (or simply PTC) as its subembedding, by a sequence of Q4-expansions. By the result, we show that the set of (P4,Q4)-irreducible even triangulations coincides with that of (P,Q)-irreducible ones on the projective plane as well as the spherical case. However, we show that the property does not hold in general for the other closed surfaces.

A Hamiltonian approach for obtaining irreducible projective representations and the k ⋅ p perturbation for anti-unitary symmetry groups

As is known, the irreducible projective representations (Reps) of anti-unitary groups contain three different situations, namely, the real, the complex and quaternionic types with torsion number 1, 2, 4 respectively. This subtlety increases the complexity in obtaining irreducible projective Reps of anti-unitary groups. In the present work, a physical approach is introduced to derive the condition of irreducibility for projective Reps of anti-unitary groups. Then a practical procedure is provided to reduce an arbitrary projective Rep into direct sum of irreducible ones. The central idea is to construct a Hermitian Hamiltonian matrix which commutes with the representation of every group element g ∈ G, such that each of its eigenspaces forms an irreducible representation space of the group G. Thus the Rep is completely reduced in the eigenspaces of the Hamiltonian. This approach is applied in the k ⋅ p effective theory at the high symmetry points (HSPs) of the Brillouin zone for quasi-particle excitations in magnetic materials. After giving the criterion to judge the power of single-particle dispersion around an HSP, we then provide a systematic procedure to construct the k ⋅ p effective model.

Scaling the existence state of hydrogen in metal binary oxides

Hydrogen in metal oxide is a classic but engaging topic in the scientific research and actual application; owing to the flexible property of hydrogen, however, a scaling relationship regarding the existence state of hydrogen remains in lack. Hereby, a systematic first-principles calculation is performed on the behavior of neutrally charged hydrogen atom in a broad variety of metal binary oxides. Three main incorporation sites for hydrogen (interstitial site, hydroxyl group and oxygen vacancy) and more than 100 metal oxides are considered. From the calculated defect energy, the relatively stable incorporation sites for hydrogen are identified, and the mechanism behind the stability is unravelled by the inherent redox and electronic structure properties of host oxides. Generally, hydrogen prefers to form hydroxyl group in the reducible oxides but occupy interstital site in the irreducible ones. More importantly, oxygen vacancy formation energy associated with the redox property of oxide is established as the pragmatic descriptor for scaling the existence state of incorporated hydrogen, and an intercept value of about 4.5 eV is determined for distinguishing H occupancy, coincidencent with the standard hydrogen electrode potential. The scaling rule is expected to be popularized to other oxides and solids associated with hydrogen behavior.

Computing the equisingularity type of a pseudo-irreducible polynomial

Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over $$\mathbb {C}$$ , this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of a Weierstrass equation.

Symmetric Quantum Sets and L-Algebras

Abstract Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$, ${\mathbb{C}}$, or ${\mathbb{H}}$. Symmetric quantum sets are characterized as a class of $L$-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.

Reducible and irreducible approximation of complex symmetric operators

This paper aims to study reducible and irreducible approximation in the set $\textsl{CSO}$ of all complex symmetric operators on a separable, complex Hilbert space $\mathcal H$. When ${\rm dim} \mathcal H=\infty$, it is proved that both those reducible ones and those irreducible ones are norm dense in $\textsl{CSO}$. When ${\rm dim} \mathcal H<\infty$, irreducible complex symmetric operators constitute an open, dense subset of $\textsl{CSO}$.

Galois theory for general systems of polynomial equations

We prove that the monodromy group of a reduced irreducible square system of general polynomial equations equals the symmetric group. This is a natural first step towards the Galois theory of general systems of polynomial equations, because arbitrary systems split into reduced irreducible ones upon monomial changes of variables. In particular, our result proves the multivariate version of the Abel–Ruffini theorem: the classification of general systems of equations solvable by radicals reduces to the classification of lattice polytopes of mixed volume 4 (which we prove to be finite in every dimension). We also notice that the monodromy of every general system of equations is either symmetric or imprimitive. The proof is based on a new result of independent importance regarding dual defectiveness of systems of equations: the discriminant of a reduced irreducible square system of general polynomial equations is a hypersurface unless the system is linear up to a monomial change of variables.

Разложение граничных представлений на плоскости Лобачевского в сечениях линейных расслоений

Earlier we described canonical (labelled by λ ∈C) and accompanying boundary representations of the group G = SU(1,1) on the Lobachevsky plane D in sections of linear bundles and decomposed canonical representations into irreducible ones. Now we decompose representations acting on distributions concentrated at the boundary of D. In the generic case 2λ ∉N they are diagonalizable, in the exceptional case Jordan blocks appear.

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

This paper deals with properties of the algebraic variety defined as the set of zeros of a typical sequence of polynomials. We consider various types of nice varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero obstruction polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.

Hyperelliptic d-Tangential Covers and d× d-Matrix KdV Elliptic Solitons

Abstract More than $40$ years ago I. Krichever developed the Theory of (vector) Baker–Akhiezer functions and devised a criterion for a $d$-marked compact Riemann surface to provide $d\times d$-matrix solutions to the KdV equation. Later on he also found a criterion for a $d$-marked curve to provide $d\times d$-matrix solutions to the Kadomtsev-Petviashvili (KP) equation, doubly periodic with respect to $x$, the 1st KP flow. In particular, when both criteria apply, one should obtain $d\times d$-matrix KdV elliptic solitons. It seems, however, that the latter issue has been completely neglected until very recently (cf. [10] where the $d=2$ case is treated). In this article we fix a complex elliptic curve $X=\mathbb{C}/\Lambda$, corresponding to a lattice $\Lambda \subset \mathbb{C}$, and define so-called hyperelliptic $d$-tangential covers as $d$-marked covers of $X$ satisfying a geometric condition inside their Jacobians. They satisfy Krichever’s criteria and give rise, therefore, to families of $d\times d$-matrix KdV elliptic solitons. We also construct an anticanonical rational surface ${\mathcal S}$ naturally attached to $X$, with a Picard group of rank $10$. It turns out that the former covers of $X$ correspond to rational irreducible curves in suitable divisor classes of ${\mathcal S}$. We thus reduce their construction to proving that the associated Severi Varieties (of rational irreducible nodal curves) are not empty. The final key to the problem consists in finding rational reducible nodal curves in the latter divisor classes that can be deformed to irreducible ones, according to A. Tannenbaum’s criterion (see [5]). At last we deduce, for any $d\geq 2$, infinite families (of arbitrary high genus and degree) of hyperelliptic $d$-tangential covers, giving rise to $d\times d$-matrix KdV elliptic solitons.

On the low-energy limit of the QED N-photon amplitudes: part 2

In recent work, Gies and Karbstein have discovered that the two-loop Euler–Heisenberg Lagrangians for scalar and spinor QED have non-vanishing reducible contributions in addition to the well-studied irreducible ones. This invalidates previous applications of those Lagrangians to the computation of the two-loop N-photon amplitudes in the low energy limit. Here we compute the corrections to those amplitudes due to the reducible contributions.

Canonical quantization of the covariant fields on de Sitter space–times

The properties of the covariant quantum fields on de Sitter space–times are investigated focusing on the isometry generators and Casimir operators in order to establish the equivalence among the covariant representations and the unitary irreducible ones of the de Sitter isometry group. For the Dirac quantum field, it is shown that the spinor covariant representation, transforming the Dirac field under de Sitter isometries, is equivalent with a direct sum of two unitary irreducible representations of the [Formula: see text] group, transforming alike the particle and antiparticle field operators in momentum representation. Their basis generators and Casimir operators are written down finding that the covariant representations are equivalent with unitary irreducible ones from the principal series whose canonical weights are determined by the fermion mass and spin.

Wintgen ideal submanifolds: reduction theorems and a coarse classification

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Mobius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Mobius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Mobius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms.

Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p>0, μ a dominant weight and W the Weyl group of G. Let G=G(q) be a finite classical group, where q is a p-power. For a weight μ of G the sum sμ of distinct weights w(μ) with w∈W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1T of T in sμ. The main case is where μ=(q−1)ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of sμ⋅St defined to be 0 at the p-singular elements of G. Let βμ denote the Brauer character of a representation of SLn(q) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters βμ⋅St for μ=(q−1)ω, and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1T in the restriction βμ|T for every maximal torus T of G.

Matrix semigroups with constant spectral radius

Multiplicative matrix semigroups with constant spectral radius (c.s.r.) are studied and applied to several problems of algebra, combinatorics, functional equations, and dynamical systems. We show that all such semigroups are characterized by means of irreducible ones. Each irreducible c.s.r. semigroup defines walks on Euclidean sphere, all its nonsingular elements are similar (in the same basis) to orthogonal. We classify all nonnegative c.s.r. semigroups and arbitrary low-dimensional semigroups. For higher dimensions, we describe five classes and leave an open problem on completeness of that list. The problem of algorithmic recognition of c.s.r. property is proved to be polynomially solvable for irreducible semigroups and undecidable for reducible ones.

Complex Semigroups for Oscillator Groups

An oscillator group $G$ is a semidirect product of a Heisenberg group with a one-parameter group. In this article we construct Olshanski semigroups for infinite-dimensional oscillator groups. These are complex involutive semigroups which have a polar decomposition. The main application will be for representations $\pi$ of $G$ which are semibounded, i.e., there exists a non-empty open subset $U$ of the Lie algebra $\mathfrak{g}$ such that the operators $id\pi(x)$ from the derived representation are uniformly bounded from above for $x\in U$. More precisely we show that every semibounded representation of an oscillator group $G$ extends to a non-degenerate holomorphic representation of such a semigroup and conversely each non-degenerate holomorphic representation of such a semigroup gives rise to a semibounded representation of $G$. The main application of this result is a classification of representations of the canonical commutation relations with a positive Hamiltonian, which will be obtained in a subsequent paper. Moreover it yields direct integral decomposition into irreducible ones and implies the existence of a dense subspace of analytic vectors for semibounded representations of $G$.

Irreducible Sobol’ sequences in prime power bases

Sobol’ sequences are a popular family of low-discrepancy sequences, in spite of requiring primitive polynomials instead of irreducible ones in later constructions by Niederreiter and Tezuka. We introduce a generalization of Sobol’ sequences that removes this shortcoming and that we believe has the potential of becoming useful for practical applications. Indeed, these sequences preserve two important properties of the original construction proposed by Sobol’: their generating matrices are non-singular upper triangular matrices, and they have an easy-to-implement column-by-column construction. We prove they form a subfamily of the wide family of generalized Niederreiter sequences, hence satisfying all known discrepancy bounds for this family. Further, their connections with Niederreiter sequences show these two families only have a small intersection (after reordering the rows of generating matrices of Niederreiter sequences in that intersection).