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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 ) .

Several combinatorial actions of the affine Weyl group of type C ˜ n on triangulations, trees, words and permutations are compared. Addressing a question of David Vogan, we show that, modulo a natural involution, these permutation representations are multiplicity-free. The proof uses a general construction of Gelfand subgroups in the affine Weyl groups of types C ˜ n and B ˜ n .

In this note, we characterize $ b- $generalized derivations which are strong commutative preserving (SCP) on $ \mathscr{R} $. Moreover, we also discuss and characterize $ b- $generalized derivations involving certain $ \ast- $differential/functional identities on rings possessing involution.

Let ( R , ⁎ ) be a ring with involution and let A = M ( n , R ) be the matrix ring endowed with the ⁎-transpose involution. We study SL ⁎ ( 2 , A ) and the question of Bruhat generation over commutative and non-commutative local and adelic rings R. An important tool is the property of a ring being ⁎-Euclidean. In this regard, we introduce the notion of a ⁎-local ring R, prove that A is ⁎-Euclidean and explore reduction modulo the Jacobson radical for such rings. Globally, we provide an affirmative answer to the question whether a commutative adelic ring R leads towards the ring A being ⁎-Euclidean; while the non-commutative adelic quaternions are such that A is ⁎-Euclidean and SL ⁎ is generated by its Bruhat elements if and only if the characteristic is 2.

Burawoy (1999), Eyal (2001), Berdahl (2006), Cervinkova (2012) and others have proposed the concept of „postsocialism” as a way of describing the gap between socialism and capitalism. But, in a world marked by “rearrangements, reconfigurations and recombinations that yield new interweavings of the multiple social logics” (Stark, & Bruszt, 1998) what does postsocialism entail? This article aims to provide an overview of the different perspectives regarding the meaning of postsocialism, by bringing together and examining ideas represented in the theoretical literature. Starting from the idea of reframing the understanding of the past, this analysis explores postsocialism, as a process, from four perspectives: imitation, involution, emergence or zombie socialism. The critique serves as a starting point for future directions in postsocialism conceptualisations, in order to better understand the present based on the past.

We prove that there is a monadic adjunction between the category of bounded posets with involution and the category of orthomodular posets., Comment: arXiv admin note: text overlap with arXiv:1903.05399

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

This paper focuses on considering two inverse source problems (ISPs) for a multi-term time-fractional evolution equation with an involution term, interpolating the heat and wave equations. The frac...

Let [Formula: see text] be a field of characteristic zero and [Formula: see text] the algebra of [Formula: see text] matrices over [Formula: see text]. By the classical Amitsur–Levitzki theorem, it is well known that [Formula: see text] is the smallest degree of a standard polynomial identity of [Formula: see text]. A theorem due to Rowen shows that when the symplectic involution [Formula: see text] is considered, the standard polynomial of degree [Formula: see text] in symmetric variables is an identity of [Formula: see text]. This means that when only certain kinds of matrices are considered in the substitutions, the minimal degree of a standard identity may not remain being the same. In this paper, we present some results about the minimal degree of standard identities in skew or symmetric variables of odd degree of [Formula: see text] in the symplectic graded involution case. Along the way, we also present the minimal total degree of a double Capelli polynomial identity in symmetric variables of [Formula: see text] with transpose involution.

In this paper, we consider skew Lie product on an involutive ring and study several algebraic identities for it, which include generalized derivations of the ring. The results give information about the commutativity of the ring and a description of the generalized derivations.

A periodic group is called an OCn-group if the set of its element orders consists of all natural numbers from 1 to some natural n. W. Shi posed the question whether every OCn-group is locally finite. Until now, the case n = 8 remains open. Here we prove that if a group is generated by an involution and an element of order 3, and its element orders do not exceed 8, then it is finite. Thereby we obtain an affirmative answer to Shi’s question for n = 8 for (2, 3)-generated groups.

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

In 2020, Ferreyra et al. defined the weak core inverse of a complex matrix. We generalize it to a unitary ring with involution and investigate the relationship between the weak core inverse and the...

Let D be a division ring with involution ⋆ and S the set of all symmetric elements of D. Assume that the center F of D is uncountable and K is a division subring of D containing F. The main aim of this note is to show that S is right algebraic over K if and only if so is D. This result allows us to construct an example of division rings K ⊂ D such that D is right algebraic but not left algebraic over K.

We investigate the involutory matrices in the recently introduced group of almost-Riordan arrays. We prove that each such involution is a conjugate of a diagonal involution.

In this paper, we construct complete sets of polynomials in bi-involution on nilpotent Lie algebras of dimension 7 in the list due to Gong. Thus we verify the generalized Mishchenko-Fomenko conjecture for all algebras in this list. Bibliography: 14 titles.

It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations.

The article reviews the history of the word “involution,” the empirical basis of the concept of “agricultural involution,” and the mechanisms operating behind that phenomenon. It then considers the very different empirical bases and mechanisms of “bureaucratic involution.” State and peasant might interact in a positive way that leads to development – as when the state in the Reform era gave peasants the power and right to respond to market stimuli and develop the “labor and capital dual intensifying” “new agriculture” that has led to genuine development, demonstrating how small peasants have been the true primary subjects of Chinese agriculture and the true key to genuine agricultural development. By contrast, if bureaucratic involution should force on peasants policies that run counter to realities, it can lead to malignant “ultra-involution.” Similar consequences can be seen in spheres with scarce opportunities relative to the number of people seeking them, once they are placed under the forces of bureaucratic involution, as in the “examinations-above-all-else education system” as well as in similar (public and private) enterprise management. That is why the word “involution” has recently triggered such widespread resonance among so very many people. What is needed is state-party policies that truly accord with the interests of the people and draw their active participation. That kind of combination is what can check tendencies toward ultra-involution.

When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ${}^{*}$-semigroups and commutative involution semigroups., Comment: 14 pages

We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals., 10 pages

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

In this paper, some characterizations of partial isometries, normal elements and strongly $EP$ elements are given by the construction of $EP$ elements. At the same time, the partial isometry elements are characterized by the existence of solutions of equations in rings in a given set, and also by the general form of solutions of given equations.

In this paper, the authors derive the existence criteria and the formulae of the weighted Moore-Penrose inverse, the e-core inverse and the f-dual core inverse in rings. Also, new characterizations between weighted Moore-Penrose inverses and one-sided inverses along an element are given.

In this paper the stability of the initial value problem for the third order partial delay differential equation with involution is investigated. The first order of accuracy absolute stable difference scheme for the solution of the differential problem is presented. Stability estimates for the solution of this difference scheme are proved. Numerical results are provided.

A time dependent source identification problem for parabolic equation with involution and Neumann condition is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. The stable difference scheme for the approximate solution of this problem and its stability estimates are presented. Numerical results are given.

In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlinear differential equation with damping term and involution is established.

Italy has experienced different kinds of involution of parliamentary democracy and the legislative output over the course of the past couple of decades. Following the ‘Berlusconi-era’ (1994–2011) t...

Let R be a ring with involution containing a nontrivial symmetric idempotent element e and δ: R → R be a generalized ∗-reverse derivable map. In this paper, our aim is to show that under some suitable restrictions imposed on R every generalized ∗-reverse derivable map of R is additive.

Suppose that G is a finite group and X is a G-conjugacy classes of involutions. The commuting involution graph $${\mathcal {C}}(G,X)$$ C ( G , X ) is the graph whose vertex set is X with $$x, y \in X$$ x , y ∈ X being joined if $$x \ne y$$ x ≠ y and $$xy = yx$$ x y = y x . Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.

Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gr\"obner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting., Comment: 36 pages, 2 figures; v2: reordered Sections 4 and 5, added references and examples; v3: some corrections, added exposition; v4: minor corrections; v5: fixed typo in equation (6.3), final version

For each k≥2, let Lk1 denote the monoid obtained by adjoining an identity element to the semigroup generated by two idempotents E and F subjected to the relation EFEF⋯=0, where the left side is a p...

Motivated by Dar and Ali’s [9] elegant work's, the goal of the paper is to study nth power *-property in rings with involution and its related problems. In fact, the main result of this paper gives...

In this paper, we investigate commutativity of rings with involution provided with derivations and generalized derivations satisfying some algebraic identities. Furthermore, we provide examples to ...

Let M be a -ring with involution . In this paper , we will introduce the concept of symmetric left(right) reverse *-4-centralizer of M . Then, we proved that the 4-additive mapping T:MxMxMxMM is a reverse *-4-centralizer of M under certain conditions .

We enumerate involutions avoiding any single pattern of length three according to the statistic ‘number of visible pairs’, namely, the number of non-adjacent columns in the bargraph representation ...

ANDI of breast is a common phenomenon. They are more common than breast malignancy. In this study an attempt was made to study the spectrum of diseases that come under ANDI of breast, their clinical presentation, treatment and also to nd out the clinical and cytohistological correlation. Study group includes 152 cases of Upper Assam who attended the Surgery OPD of AMCH during the study period and also all the patients who were admitted in the department of surgery under the clinical diagnosis of ANDI of breast. Conrmation of diagnosis was made on the basis of history, clinical examination, pathological and radiological investigations. Pathological investigations include FNAC and HPE. Radiological investigations include Ultrasonography and mammography. The following observations were made in the present study: Ÿ The commonest disease encountered was broadenoma of breast (14.47%) followed by mastalgia (23.68%) brocystic disease (14.47%) breast cyst (9.87%). The less common are duct ectasia, scleroding adenosus and phyllodes tumour, 9.87% each. Ÿ Majority are present in the second, third and fourth decades of life. All broadenoma, brocystic disease and phyllodes tumour are present in the third decade. Youngest patient was of 11 years and oldest was of 47 years. mean age of presentation was 27.45 years and the standard deviation was 7.13. Ÿ 86.84% had regular menstrual history and 13.16% had irregular menstrual history. Majority were nulliparous (46.05%). The highest parity was 4. Ÿ 94.08% did not give history of OCP use and 98.68% did not give history of lactation. Ÿ The commonest presentation was pain in the breast (76.32%) followed by lump (71.05%). Ÿ Majority cases presented to the hospital within 3-6 months of their onset of symptoms. The minimum duration of symptom was 1 month and the maximum was 12 months. Ÿ Right breast was involved more commonly 961.84%) than the left (38.16%). Ÿ Upper outer quadrant was involved most commonly involved (34.21%) and the least commonly involved was the central quadrant (1.32%). Ÿ The majority of the breast lumps (85.05%) had size less than 5cm in their greatest dimension. 14.95% had size more than 5cm. Ÿ 63.82% of the lumps had rm consistency whereas 6.58% had soft consistency. There was no positive family history in majority of the patients (89.47%). Ÿ The correlation between clinical diagnosis and histopathological diagnosis had the following ndings: For broadenoma the sensitivity of FNAC is 84.9% and HPE is 86.54% .Both have positive predictive value of 100%.For breast cyst the sensitivity of FNAC is 93.3%.For phyllodes tumour the sensitivity of FNAC is 85.7% and HPE is 75%.. Ÿ In the management, 41.45% cases underwent excision followed by aspioration of cyst in 7.89%, microdochectomy in 7.24%, wide local excision in 3.29% and simple mastectomy in 1.97%. Among conservative treatment 14.47%b were given evening primrose oil, 12.50% were given topical analgesic and breast support and danazol was given in 5.26%. The cases were followed up at three monthly interval for a period of one year and there was no recurrence during this period.

The purpose of this paper is to study pair of left centralizers in prime rings with involution satisfying certain identities.

In this research work, we demonstrate the Hyers-Ulam stability for Cauchy-Jensen functional equation in multi-Banach algebras by the fixed point technique. In fact, we prove that for a function which is approximately Cauchy-Jensen in multi Banach algebra, there is a unique involution near it. Next, we show that under some conditions the involution is continuous, the multi-Banach algebra becomes multi-$ C^* $-algebra and the Banach algebra is self-adjoint.

This paper studies the concept of the -quasi-*-IFP (resp., -*-reflexive) *-rings, as a generalization of the quasi-*-IFP (resp., *-reflexive) *-rings and every quasi-*-IFP (resp., *-reflexive) *-ring is -quasi-*-IFP (resp., -*-reflexive). This paper also discusses the sufficient condition for the quasi-*-IFP (resp., *-reflexive) *-ring in order to be -quasi-*-IFP (resp., -*- reflexive). Finally, this study investigates the -quasi-*-IFP (resp., -*-reflexivity) by using some types of the polynomial rings.

We find a new refinement of Fine’s partition theorem on partitions into distinct parts with the minimum part odd. As a consequence, we obtain two companion partition identities. Both analytic and combinatorial proofs are provided.

In this paper, certain spectral properties related with the first order linear differential-operator expression with involution in the Hilbert space of vector-functions at finite interval have been examined. Firstly, the minimal and maximal operators which are generated by the first order linear differential-operator expression with involution in the Hilbert spaces of vector-functions has been described. Then, the deficiency indices of the minimal operator have been calculated. Moreover, the space of boundary values of the minimal operator have been constructed. Afterwards, by using the method of Calkin–Gorbachuk, the general form of all selfadjoint extensions of the minimal operator in terms of boundary values has been found. Later on, the structure of spectrum of these extensions has been investigated.