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We determine the 2-fusion systems of J-component type in which the centralizer of some fully centralized involution has a maximal J-component that is the 2-fusion system of U 3 ( 3 ) .

We study a model where evaluation consists of multiple components of different nature and (cardinal) performances in all components are aggregated into a summary index between 0 and 1. We propose what we call the normalizer-based aggregation rules and characterize them by individual separability, monotonicity, anonymity, and component independence. Each member in this family is distinguished by three types of parameters: (i) a profile of weights that determines the relative importance of each component; (ii) a profile of “individual normalizers” that converts an agent's performance in each component into a raw score (for that component) in the normalized scale of [ 0 , 1 ] ; and (iii) a profile of “group normalizers” that adjusts a raw score for each component relative to all agents' performances. Given these parameters, the overall evaluation, or score, of an agent is obtained as a weighted average of his adjusted scores for all components produced by individual and group normalizers.

We prove that the triangular matrix algebra Λ = ( H 0 H H ) is an affine quasi-hereditary algebra if and only if H is an affine quasi-hereditary algebra. Moreover, the category of Δ-good Λ-modules, the global dimension and the characteristic tilting module of Λ are described by using the corresponding ones of H. In the appendix, we prove that certain centralizer algebra and quotient algebra of an affine quasi-hereditary algebra are affine quasi-hereditary.

We establish a surprising correspondence between groups definable in o-minimal structures and linear algebraic groups, in the nilpotent case. It turns out that in the o-minimal context, like for finite groups, nilpotency is equivalent to the normalizer property or to uniqueness of Sylow subgroups. As a consequence, we show algebraic decompositions of o-minimal nilpotent groups, and we prove that a nilpotent Lie group is definable in an o-minimal expansion of the reals if and only if it is a linear algebraic group., 12 pages

We prove that a finite index regular inclusion of$II_1$-factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of$II_1$-factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

A maximal subfield of a division ring is said to be self-invariant if it is its own normalizer. Subfields of this kind are important because they have a strong connection with Albert's conjecture on the cyclicity of division rings of prime index. We show that every maximal subfield of the Mal'cev-Neumann division ring, which is of infinite dimension, is self-invariant. We also apply the Mal'cev-Neumann structure to refute the conjecture that every noncentral subnormal subgroup of the multiplicative group of a division ring must contain a noncentral normal subgroup. Finally, among other things, we rely on self-invariant subfields to present a criterion for a division ring to have a finite-dimensional subdivision rings.

We construct orbifolds of holomorphic lattice vertex operator algebras for non-abelian finite automorphism groups G. To this end, we construct twisted modules for automorphisms g together with the projective representation of the centralizer of g on the twisted module. This allows us to extract the irreducible modules of the fixed-point VOA VG, and to compute their characters and modular transformation properties. We then construct holomorphic VOAs by adjoining such modules to VG. Applying these methods to extremal lattices in d=48 and d=72, we construct more than fifty new holomorphic VOAs of central charge 48 and 72, many of which have a very small number of light states., Journal of Algebra, 585, ISSN:0021-8693, ISSN:1090-266X

The norm N(G) of a group G is the intersection of the normalizer of every subgroup of G. In this paper, we generalize Ito’s theorem and prove that the p-length of a p-soluble group is not greater t...

Most researchers consider the action of projective general group on the cosets of its maximal subgroups leaving out non-maximal subgroups. In this paper, we consider the action of centralizer of an elliptic element which is a non maximal subgroup . In particular, we determine the subdegrees, rank and properties of the suborbital graphs of the action. We achieve this through the application of the action of a group by conjugation. We have proved that the rank is q and the subdegrees are and .

Let $$\Gamma $$ be a finite tree. Fix a base vertex $$x_0$$ of $$\Gamma $$ and let $$T=T^{(x_0)}$$ be the Terwilliger algebra of $$\Gamma $$ with respect to $$x_0$$ . Denote by H the group of automorphisms of $$\Gamma $$ that fix $$x_0$$ , and let $$S={\mathrm{End}}_H~(V)$$ be the centralizer algebra of H, where $$V={\mathbb {C}}X$$ is the standard module of T with X the underlying vertex set of $$\Gamma $$ . It is obvious that T is contained in S. We show how large the gap is between T and S by comparing irreducible representations of them; in particular we find precisely when $$T=S$$ holds.

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

Given an n × n matrix c over a unitary ring R, the centralizer of c in the full n × n matrix ring M n ( R ) is called a principal centralizer matrix ring, denoted by S n ( c , R ) . We investigate its structure and prove: (1) If c is an invertible matrix with a c-free point, or if R has no zero-divisors and c is a Jordan-similar matrix with all eigenvalues in the center of R, then M n ( R ) is a separable Frobenius extension of S n ( c , R ) in the sense of Kasch. (2) If R is an integral domain and c is a Jordan-similar matrix, then S n ( c , R ) is a cellular R-algebra in the sense of Graham and Lehrer. In particular, if R is an algebraically closed field and c is an arbitrary matrix in M n ( R ) , then S n ( c , R ) is always a cellular algebra, and the extension S n ( c , R ) ⊆ M n ( R ) is always a separable Frobenius extension.

In this paper, we prove that; Let M be a 2-torsion free semiprime which satisfies the condition for all and α, β . Consider that as an additive mapping such that holds for all and α , then T is a left and right centralizer.

In this paper, we prove that there is an equivariant basic Morita equivalence between an inertial block and its Brauer correspondent in the normalizer of a defect group of the block.

The centralizer of the image of the diagonal embedding of $U_q(\mathfrak{sl}_2)$ in the tensor product of three irreducible representations is examined in a Schur-Weyl duality spirit. The aim is to offer a description in terms of generators and relations. A conjecture in this respect is offered with the centralizers presented as quotients of the Askey-Wilson algebra. Support for the conjecture is provided by an examination of the representations of the quotients. The conjecture is also shown to be true in a number of cases thereby exhibiting in particular the Temperley-Lieb, Birman-Murakami-Wenzl and one-boundary Temperley-Lieb algebras as quotients of the Askey-Wilson algebra., 26 pages

We define skew matrix gamma ring and describe the constitution of Jordan left centralizers and derivations on skew matrix gamma ring on a -ring. We also show the properties of these concepts.

We identify the dimension of the centralizer of the symmetric group $\mathfrak{S}_d$ in the partition algebra $\mathcal{A}_d(\delta)$ and in the Brauer algebra $\mathcal{B}_d(\delta)$ with the number of multidigraphs with $d$ arrows and the number of disjoint union of directed cycles with $d$ arrows, respectively. Using Schur-Weyl duality as a fundamental theory, we conclude that each centralizer is related with the $G$-invariant space $P^d(M_n(\mathbf{k}))^G$ of degree $d$ homogeneous polynomials on $n \times n$ matrices, where $G$ is the orthogonal group and the group of permutation matrices, respectively. Our approach gives a uniform way to show that the dimensions of $P^d(M_n(\mathbf{k}))^G$ are stable for sufficiently large $n$., Comment: 20 pages, changes of wrong conditions, typos, and grammar. Brauer algebras. Journal of Algebra (2021)

Let G be a group. If for every proper normal subgroup N and element x of G with N〈x〉≠G, N〈x〉 is an FC-group, but G is not an FC-group, then we call G an NFC-group. In the present paper we consider the NFC-groups. We prove that every non-perfect NFC-group with non-trivial finite images is a minimal non-FC-group. Also we show that if G is a non-perfect NFC-group having no nontrivial proper subgroup of finite index, then G is a minimal non-FC-group under the condition “every Sylow p-subgroup is an FC-group for all primes p”. In the perfect case, we show that there exist locally nilpotent perfect NFC-p-groups which are not minimal non-FC-groups and also that McLain groups $M(\mathbb {Q},GF(p))$ for any prime p contain such groups. We give a characterization for torsion-free case. We also consider the p-groups such that the normalizer of every element of order p is an FC-subgroup.

We investigate the structure constants of the center H n of the group algebra Z [ S p n ( q ) ] over a finite field. The reflection length on the group G L 2 n ( q ) induces a filtration on the algebras H n . We prove that the structure constants of the associated graded algebra S n are independent of n. As tool in the proof we consider the embedding S p m ↪ S p n ( q ) and determine the behavior of the centralizers and intersection of centralizers under the embedding S p m ↪ S p n ( q ) . We determine explicit formulas for rate of the growth of the centralizer of an element U ∈ S p m ( q ) in terms of dimension of the fixed space of U.

The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations., 20 pages, 1 figure. The condition of finite generation in Thm 1.9 was replaced by countability. The proof of Thm 1.9 has been simplified. The notation used in 5 has been modified. Several minor corrections across the paper

A left Engel sink of an element g of a group G is a set $${\mathscr {E}}(g)$$ E ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$ [ . . . [ [ x , g ] , g ] , ⋯ , g ] belong to $${\mathscr {E}}(g)$$ E ( g ) . (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$ E ( g ) = { 1 } .) We prove that if a finite group G admits an automorphism $$\varphi $$ φ of prime order coprime to |G| such that for some positive integer m every element of the centralizer $$C_G(\varphi )$$ C G ( φ ) has a left Engel sink of cardinality at most m, then the index of the second Fitting subgroup $$F_2(G)$$ F 2 ( G ) is bounded in terms of m. A right Engel sink of an element g of a group G is a set $${\mathscr {R}}(g)$$ R ( g ) such that for every $$x\in G$$ x ∈ G all sufficiently long commutators $$[\ldots [[g,x],x],\dots ,x]$$ [ … [ [ g , x ] , x ] , ⋯ , x ] belong to $${\mathscr {R}}(g)$$ R ( g ) . (Thus, g is a right Engel element precisely when we can choose $${\mathscr {R}}(g)=\{ 1\}$$ R ( g ) = { 1 } .) We prove that if a finite group G admits an automorphism $$\varphi $$ φ of prime order coprime to |G| such that for some positive integer m every element of the centralizer $$C_G(\varphi )$$ C G ( φ ) has a right Engel sink of cardinality at most m, then the index of the Fitting subgroup $$F_1(G)$$ F 1 ( G ) is bounded in terms of m.

Let $\mathfrak {g}$ g be a complex simple Lie algebra. We consider subalgebras $\mathfrak {m}$ m which are Levi factors of parabolic subalgebras of $\mathfrak {g}$ g , or equivalently $\mathfrak {m}$ m is the centralizer of its center. We introduced the notion of admissible systems on finite order $\mathfrak {g}$ g -automorphisms 𝜃, and show that 𝜃 has admissible systems if and only if its fixed point set is a Levi factor. We then use the extended Dynkin diagrams to characterize such automorphisms, and look for automorphisms of minimal order.

Let $$A$$ be a complex $$n \times n$$ matrix. The set $$\mathcal{L}$$ of matrices $$X$$ satisfying the relation $${{X}^{T}}AX = A$$ is called the $$T$$ -congruence centralizer of $$A$$ . It is shown that the calculation of matrices from the nonlinear variety $$\mathcal{L}$$ can be reduced to solving a linear matrix equation.

Motion groups of links in the three sphere $\mathbb{S}^3$ are generalizations of the braid groups, which are motion groups of points in the disk $\mathbb{D}^2$. Representations of motion groups can be used to model statistics of extended objects such as closed strings in physics. Each $1$-extended $(3+1)$-topological quantum field theory (TQFT) will provide representations of motion groups, but it is difficult to compute such representations explicitly in general. In this paper, we compute representations of the motion groups of links in $\mathbb{S}^3$ with generalized axes from Dijkgraaf-Witten (DW) TQFTs inspired by dimension reduction. A succinct way to state our result is as a step toward a twisted generalization (Conjecture \ref{mainconjecture}) of a conjecture for DW theories of dimension reduction from $(3+1)$ to $(2+1)$: $\textrm{DW}^{3+1}_G \cong \oplus_{[g]\in [G]} \textrm{DW}^{2+1}_{C(g)}$, where the sum runs over all conjugacy classes $[g]\in [G]$ of $G$ and $C(g)$ the centralizer of any element $g\in [g]$. We prove a version of Conjecture \ref{mainconjecture} for the mapping class groups of closed manifolds and the case of torus links labeled by pure fluxes., Clarify the main conjecture as a twisted version of dimension reduction. To appear in Comm. Math Phys

This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

We define the degenerate two boundary affine Hecke-Clifford algebra H d , and show it admits a well-defined q ( n ) -linear action on the tensor space M ⊗ N ⊗ V ⊗ d , where V is the natural module for q ( n ) , and M , N are arbitrary modules for q ( n ) , the Lie superalgebra of Type Q. When M and N are irreducible highest weight modules parameterized by a staircase partition and a single row, respectively, this action factors through a quotient of H d . We then construct explicit modules for this quotient, H p , d , using combinatorial tools such as shifted tableaux and the Bratteli graph. These modules belong to a family of modules which we call calibrated. Using the relations in H p , d , we also classify a specific class of calibrated modules. The irreducible summands of M ⊗ N ⊗ V ⊗ d coincide with the combinatorial construction, and provide a weak version of the Schur-Weyl type duality.

In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of |G| is a maximal subgroup.

Let R be a commutative ring. The lattice of subgroups of a Chevalley group G(Φ,R) containing the subgroup D(R) is studied, where D is a subfunctor of G(Φ, — ). Assuming that over any field F the normalizer of the group D(F) is “closed to be maximal,” it is proved that under some technical conditions the lattice is standard. A condition, on the normalizer of D(R) is studied in the case, where D(R) is the elementary subgroup of another Chevalley group G(Ψ,R) embedded into G(Φ,R).

This paper presents the task of normalizing Vietnamese transcribed texts in Speech-to-Text (STT) systems. The main purpose is to develop a text normalizer that automatically converts proper nouns a...

An $ (\alpha, \beta) $-Pythagorean fuzzy environment is an efficient tool for handling vagueness. In this paper, the notion of relative subgroup of a group is introduced. Using this concept, the $ (\alpha, \beta) $-Pythagorean fuzzy order of elements of groups in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups is defined and examined various algebraic properties of it. A relation between $ (\alpha, \beta) $-Pythagorean fuzzy order of an element of a group in $ (\alpha, \beta) $-Pythagorean fuzzy subgroups and order of the group is established. The extension principle for $ (\alpha, \beta) $-Pythagorean fuzzy sets is introduced. The concept of $ (\alpha, \beta) $-Pythagorean fuzzy normalizer and $ (\alpha, \beta) $-Pythagorean fuzzy centralizer of $ (\alpha, \beta) $-Pythagorean fuzzy subgroups are developed. Further, $ (\alpha, \beta) $-Pythagorean fuzzy quotient group of an $ (\alpha, \beta) $-Pythagorean fuzzy subgroup is defined. Finally, an $ (\alpha, \beta) $-Pythagorean fuzzy version of Lagrange's theorem is proved.

Preprint de J. Araújo, W. Bentz, and P.J. Cameron, “Primitive Permutation Groups and Strongly Factorizable Transformation Semigroups”, Journal of Algebra 565 (2021), 513-530. Let Ω be a finite set and T (Ω) be the full transformation monoid on Ω. The rank of a transformation t ∈ T (Ω) is the natural number |Ωt|. Given A ⊆ T (Ω), denote by 〈A〉 the semigroup generated by A. Let k be a fixed natural number such that 2 ≤ k ≤ |Ω|. In the first part of this paper we (almost) classify the permutation groups G on Ω such that for all rank k transformations t ∈ T (Ω), every element in St := 〈G, t〉 can be written as a product eg, where e2 = e ∈ St and g ∈ G. In the second part we prove, among other results, that if S ≤ T (Ω) and G is the normalizer of S in the symmetric group on Ω, then the semigroup SG is regular if and only if S is regular. (Recall that a semigroup S is regular if for all s ∈ S there exists s′ ∈ S such that s = ss′s.) The paper ends with a list of problems. The first author was partially supported by the Funda ̧c ̃ao para a Ciˆencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the projects UIDB/00297/2020 (Centro de Matemtica e Aplicaes), PTDC/MAT-PUR/31174/2017, UIDB/04621/2020 and UIDP/04621/2020. info:eu-repo/semantics/publishedVersion

Recent results on the Racah algebra $\mathcal{R}_n$ of rank $n - 2$ are reviewed. $\mathcal{R}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $\mathfrak{su}(1,1)$ in $\mathcal{U}(\mathfrak{su}(1,1))^{\otimes n}$. Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the $ (n-1)$ - sphere and a number of interesting realizations are provided., 18 pages, survey paper based on talk at the conference Representation Theory XVI in Dubrovnik, 2019. Version 2: some typos corrected and references updated. Accepted for publication in Contemporary Mathematics

We provide three new examples of twisted Hilbert spaces by considering properties that are "close" to Hilbert. We denote them \(Z(\mathcal J)\), \(Z(\mathcal S^2)\) and \(Z(\mathcal T_s^2)\). The first space is asymptotically Hilbertian but not weak Hilbert. On the opposite side, \(Z(\mathcal S^2)\) and \(Z(\mathcal T_s^2)\) are not asymptotically Hilbertian. Moreover, the space \(Z(\mathcal T_s^2)\) is a HAPpy space and the technique to prove it gives a "twisted" version of a theorem of Johnson and Szankowski (Ann. of Math. 176:1987-2001, 2012). This is, we can construct a nontrivial twisted Hilbert space such that the isomorphism constant from its \(n\)-dimensional subspaces to \(\ell_2^n\) grows to infinity as slowly as we wish when \(n\to \infty\).

We study cocompleteness, co-wellpoweredness, and generators in the centralizer category of an object or morphism in a monoidal category, and the center or the weak center of a monoidal category. We explicitly give some answers for when colimits, cocompleteness, co-wellpoweredness, and generators in these monoidal categories can be inherited from their base monidal categories. Most importantly, we investigate cofree objects of comonoids in these monoidal categories.

Given a G-flow X, let $\mathrm{Aut}(G, X)$ , or simply $\mathrm{Aut}(X)$ , denote the group of homeomorphisms of X which commute with the G action. We show that for any pair of countable groups G and H with G infinite, there is a minimal, free, Cantor G-flow X so that H embeds into $\mathrm{Aut}(X)$ . This generalizes results of [2, 7].

An extension of X by Y is a short exact sequence of quasi Banach modules and homomorphisms 0⟶Y⟶Z⟶X⟶0. When properly organized all these extensions constitute a linear space denoted by ExtB(X,Y), where B is the underlying (Banach) algebra. In this paper we "compute" the spaces of extensions for the Schatten classes when they are regarded in its natural (left) module structure over B=B(H), the algebra of all operators on the ground Hilbert space. Our main results can be summarized as follows

Optimal classifications of Lie algebras of some well-known equations under their group of inner automorphism are re-considered. By writing vector fields of some known Lie algebras in the abstract format, we have proved that there exist explicit isomorphism between Lie algebras and sub-algebras which have already been classified. The isomorphism between Lie algebras is useful in the sense that the classifications of sub-algebras of dimension ≤4 have previously been carried out in literature. These already available classifications can be used to write classification of any Lie algebra of dimension ≤4. As an example, the explicit isomorphism between Lie algebra of variant Boussinesq system and sub-algebra A 3,5 1 / 2 ${A}_{3,5}^{1/2}$ is proved, and subsequently, optimal sub-algebras up to dimension four are obtained. Besides this, some other examples of Lie algebras are also considered for explicit isomorphism.

Let $V$ be a vertex operator subalgebra of $U$. Assume that $U$, $V$, and its commutant $V^c$ in $U$ are CFT-type, self-dual, and regular VOAs. Assume also that the double commutant $V^{cc}$ equals $V$. We prove that any intertwining operator of $V$ is a compression of intertwining operators of $U$., 15 pages, final revision, to appear in J. Algebra

In this paper, we investigate a suborbital graph for the normalizer of Γ0(N) ∈ PSL(2;R), where N will be of the form 24p2 such that p > 3 is a prime number. Then we give edge and circuit conditions on graphs arising from the non-transitive action of the normalizer.

We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$-groups of the continuous asymptotic centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra., 32 pages

A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (a GBS group). Every GBS group is the fundamental group π1(𝔸) of a suitable labeled graph 𝔸. We prove that if 𝔸 and 𝔹 are labeled trees, then the groups π1(𝔸) and π1(𝔹) are universally equivalent iff π1(𝔸) and π1(𝔹) are embeddable into each other. An algorithm for verifying universal equivalence is pointed out. Moreover, we specify simple conditions for checking this criterion in the case where the centralizer dimension is equal to 3.

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group such that $G=\mathbf{G}(\mathbb{R})^\circ$ is of Hermitian type. If $\Gamma \leq L$ is a torsion-free lattice of a finite connected covering of $\text{PU}(1,1)$, given a standard Borel probability $\Gamma$-space $(\Omega,\mu_\Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $\sigma:\Gamma \times \Omega \rightarrow G$. The Toledo remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $\mathbf{H}$ of a maximal cocycle $\sigma$ is reductive and the centralizer of $H=\mathbf{H}(\mathbb{R})^\circ$ is compact. If additionally $\sigma$ admits a boundary map, then $H$ is of tube type and $\sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. In the particular case $G=\text{PU}(n,1)$ maximality is sufficient to prove that $\sigma$ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles., Comment: 29 pages, more general definition of pullback added, explicit example of $G=\text{PU}(n,1)$. To appear on Geometriae Dedicata

This paper examines the relationship between the companion forms of regular matrix polynomials with singular leading coefficients. When two such polynomials have the same underlying finite and infinite Jordan structures, it is shown that their companion forms are connected by a strict equivalence transformation that can be parameterized using the commutant of the companion forms' common Weierstrass canonical form. The process developed herein for generating such parameterized transformations is applied to the useful class of diagonalizable quadratic polynomials.

Let $${\mathbb {F}}_{\Theta }=U/K_\Theta $$ be a partial flag manifold, where $$K_\Theta $$ is the centralizer of a torus in U. We study U-invariant almost Hermitian structures on $${\mathbb {F}}_{\Theta }$$ . The classification of these structures are naturally related with the system $$R_{\mathfrak {t}}$$ of $${\mathfrak {t}}$$ -roots associated to $${\mathbb {F}}_{\Theta }$$ . We introduced the notion of connectedness by triples with zero sum in a general subset of a vector space and proved that the set of $${\mathfrak {t}}$$ -roots satisfies this property. Using this result, the invariant $${\mathcal {G}}_1$$ structures on $${\mathbb {F}}_{\Theta }$$ are completely classified.

We determine the saturated 2-fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2-fusion system of a large group of Lie type over a field of odd order.

In this study, we investigate suborbital graphs Gu,n of the normalizer ΓB (N) of Γ0 (N) in PSL(2, ℝ) for N = 2α3β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of ΓB (N) on using the group (N) and then obtain some properties of the graphs arising from this action.

In this paper we mainly investigate the Coleman automorphisms and class-preserving automorphisms of finite AZ-groups and finite groups related to AZ-groups. For example, we first prove that $$Out_c(G)$$ of an AZ-group G must be a $$2'$$ -group and therefore the normalizer property holds for G. Then we find some classes of finite groups such that the intersection of their outer class-preserving automorphism groups and outer Coleman automorphism groups is $$2'$$ -groups, and therefore, the normalizer property holds for these kinds of finite groups. Finally, we show that the normalizer property holds for the wreath products of AZ-groups by rational permutation groups under some conditions.

We investigate a version of the Green correspondence for categories of complexes, including homotopy categories and derived categories. The correspondence is an equivalence between a category defined over a finite group G and the same for a subgroup H, often the normalizer of a p-subgroup of G. We present a basic formula for deciding when categories of modules or complexes have a Green correspondence and apply it to many examples. In several cases the equivalence is an equivalence of triangulated categories, and in special cases it is an equivalence of tensor triangulated categories.

Background: Circulating microRNAs (miRNAs) have proved to be promising biomarkers for early diagnosis and therapeutic monitoring in cancers. Particularly for hepatocellular carcinoma (HCC), detection of circulating miRNA biomarkers as a new diagnostic approach has been written into the latest Guidelines for Diagnosis and Treatment of Primary Liver Cancer in China (2019 edition). However, no general consensus on an ideal endogenous normalizer for circulating miRNAs quantification has been reached, so it will affect the accuracy of quantitative results. In this study, we aim to identify a stable endogenous normalizer for analyzing circulating miRNAs. Methods: Candidate miRNAs were selected by screening dataset GSE104310, as well as data statistics and analysis. Five commonly reference genes were chosen for further comparison and verification. Then, the expression levels of these genes in serum were analyzed by quantitative reverse transcription PCR (RT-qPCR) among four groups, including patients diagnosed with HCC, chronic hepatitis B (CHB), liver cirrhosis, and healthy subjects. Furthermore, the stability of target genes was evaluated using geNorm, NormFinder, comparative ΔCq programs, and validated by database. We also explored the availability of the miRNA combination, and compared the performance difference between combination and individuals, as well as the selectivity of miRNA references in the combinations. Results: 11 candidate miRNAs were obtained, and miR-4644 stood out among these miRNAs, and proved to be much more stable than other endogenous miRNAs. Further study showed that miR-4644 exhibited higher stability and expression abundance than other commonly miRNA reference controls. Finally, we discovered the combination of miR-4644 and miR-16 revealed high performance in stability when compared to miRNA individuals. Furthermore, the combination consisted of references with closer nature could give rise to amplification effects in stability. Conclusions: Our findings demonstrated that miR-4644 is an ideal endogenous normalizer for circulating microRNA quantification in hepatocellular carcinoma. Besides, combining miR-4644 with miR-16 into a whole as a reference control would greatly improve the accuracy of quantification.