An example of a nonfinitely based involution monoid of order five has recently been discovered. We confirm that this example is, up to isomorphism, the unique smallest among all involution monoids.

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

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 note we show that every element of a simple Suzuki group $^2B_2(q)$ is a commutator of elements of coprime orders.

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.

Mammary development takes place during the growing and gestation periods in swine but it also continues after farrowing. In fact, a significant proportion of mammary accretion occurs during lactation and is stimulated by suckling. After piglets are weaned, there is involution of the mammary glands and the process of mammogenesis starts again during the next parity. Suckling of a teat for the first 12 to 14 h after farrowing is not sufficient to maintain lactation, and mammary involution accompanied by alterations in gene transcription will take place. The involution process is reversible within 1 day postpartum but is not reversible if a mammary gland is unsuckled for 3 days. Mammary glands that undergo involution early in lactation do not show further involution in the post-weaning period. The action of a teat being suckled does not only affect mammary development in the ongoing lactation but it also impacts mammogenesis in the following lactation. Indeed, when a mammary gland is not suckled in first parity it has a diminished development and lower milk yield in second parity. Furthermore, it was shown that suckling of a teat for only the first 2 days postpartum in primiparous sows is sufficient to ensure optimal mammary development and milk yield from that teat in the next lactation. The behavior of nursing piglets in early lactation is also affected by whether or not a teat was previously used. Such knowledge on lactation biology is essential in order to develop the best adapted management strategies for the currently used hyperprolific sow lines and to optimize growth rate of their piglets. This review gives an update on the role of suckling for mammary development in lactating sows and on how it can affect management strategies of primiparous sows.

In light of the emergent “history of capitalism” field in Euro-American history, this article reviews and critically situates how the category “capitalism” has been debated within the historiographies of China and South Asia. In discussions paralleling European historiography, Sinologists and Indologists explored whether the “prime mover” of capitalism was changes in production or in circulation. The example of South Asian studies shows how, from the 1960s through the 1980s, the dominant production-centered approach—drawing upon Marxist theory—produced stories of economic “failure” in Asia. The example of Chinese history since the 1990s points to the resurgence of a Smithian circulation-centered approach that challenges the Eurocentric story of failure. Each of these approaches emerged out of distinct eras of capital accumulation in the twentieth century: mid-century state-supported industrialization and late-century deregulated globalization. The tension between these approaches points towards a more integrative reinterpretation that sees the core dynamics of capitalism as a cyclical process of “capital accumulation,” one that integrates both production and circulation-centered approaches and also challenges Eurocentric histories of capitalism. This article's conclusion provides speculative thoughts on writing the histories of capitalism for China and South Asia today.

Let F be a field of characteristic two and G a finite abelian 2-group with an involutory automorphism η. If G = H × D with non-trivial subgroups H and D of G such that η inverts the elements of H (H without a direct factor of order 2) and fixes D element-wise, then the linear extension of η to the group algebra FG is called a nice involution. This determines the groups of unitary and symmetric normalized units of FG. We calculate the orders and the invariants of these subgroups.

Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\,>\,1$ is a positive integer. We characterize $a\,\in \,K$, the Lie algebra of skew elements in $R$, satisfying ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on $K$. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if $a,\,b\,\in \,R$ satisfy ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$, where either $n$ is even or $b\,=\,0$, then ${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some $\lambda \,\in \,C$.

For most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras, and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representation to its dual.

This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

Assume that $(G,+)$ is a commutative semigroup, $\unicode[STIX]{x1D70F}$ is an endomorphism of $G$ and an involution, $D$ is a nonempty subset of $G$ and $(H,+)$ is an abelian group uniquely divisible by two. We prove that if $D$ is ‘sufficiently large’, then each function $g:D\rightarrow H$ satisfying $g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$ for $x,y\in D$ with $x+y,x+\unicode[STIX]{x1D70F}(y)\in D$ can be extended to a unique solution $f:G\rightarrow H$ of the generalised Jensen functional equation $f(x+y)+f(x+\unicode[STIX]{x1D70F}(y))=2f(x)$.

We rectify two omissions in the list of generators and include a brief discussion of the localization theorem of Kosniowski and Stong.

We show that the proportion of permutations $g$ in $S_{\!n}$ or $A_{n}$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^{{\it\varepsilon}}\rceil$ is at least a constant multiple of ${\it\varepsilon}$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^{{\it\varepsilon}}\rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2\rfloor ^{{\it\varepsilon}}\rceil$ for a symplectic or orthogonal group.

In this paper, we characterize Jordan $*$-derivations of a 2-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the following. Let $\delta :\,R\,\to \,R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $. Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$-derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, char $R\,\ne \,2$, and let $\delta $ be a nonzero Jordan $*$-derivation of $R$. If $\delta $ is an elementary operator of $R$, then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.

An understanding of the mechanisms regulating milk yield in sows is crucial for producers to make the best management decisions during lactation. Suckling of mammary glands by piglets is one factor that is essential for development of these glands during lactation and for the maintenance of lactation in sows. The process of mammary development is not static as the majority of it takes place in the last third of gestation, continues during lactation, is followed by involution at weaning and starts over again in the next gestation. During involution, the mammary glands undergo a rapid and drastic regression in parenchymal tissue, and this can also occur during lactation if a gland is not suckled regularly. Indeed, the pattern of regression is similar for glands that involute at weaning or during lactation. Suckling during 12 to 14 h postpartum is insufficient to maintain lactation and the process of involution that occurs in early lactation is reversible within 1 day of farrowing but is irreversible if a gland is not used for 3 days. However, milk yield from a gland which is ‘rescued’ within the first 24 h remains lower throughout lactation. Suckling does not only affect milk yield in the ongoing lactation, but it also seems to affect that of the next lactation. Indeed, non-suckling of a mammary gland in first-parity sows decreased development and milk yield of that gland in second parity. Nursing behaviour of piglets in early lactation was also affected, where changes were indicative of piglets in second parity being hungrier when suckling glands that were not previously used. It is not known, however, if the same effects would be seen between the second and third lactation. Furthermore, the minimum suckling period required to ensure maximal milk yield from a gland in the next lactation is not known. This review provides an update on our current knowledge of the importance of suckling for mammary development and milk yield in swine.

The famous five halves theorem of Boardman states that, if T: Mm → Mm is a smooth involution defined on a non-bounding closed smooth m-dimensional manifold Mm (m > 1) and if

is the fixed-point set of T, where Fj denotes the union of those components of F having dimension j, then 2m ≤ 5n. If the dimension m is written as m = 5k − c, where k ≥ 1 and 0 ≤ c < 5, the theorem states that the dimension n of the fixed submanifold is at least β(m), where β(m) = 2k if c = 0, 1, 2 and β(m) = 2k − 1 if c = 3, 4. In this paper, we give, for each m > 1, the equivariant cobordism classification of involutions (Mm, T), for which the fixed submanifold F attains the minimal dimension β(m).

Let ${{M}^{m}}$ be an $m$-dimensional, closed and smooth manifold, equipped with a smooth involution $T\,:\,{{M}^{m}}\,\to \,{{M}^{m}}$ whose fixed point set has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$, where ${{F}^{n}}$ and ${{F}^{j}}$ are submanifolds with dimensions $n$ and $j$, ${{F}^{j}}$ is indecomposable and $n\,>\,j$. Write $n\,-\,j\,={{2}^{p}}q$, where $q\,\ge \,1$ is odd and $p\,\ge \,0$, and set $m(n\,-\,j)\,=\,2n\,+\,p\,-q\,+\,1$ if $p\,\le \,q\,+\,1$ and $m(n\,-\,j)\,=\,2n\,+\,{{2}^{p-q}}$ if $p\,\ge \,q$. In this paper we show that $m\,\le \,m(n-j)+2j+1$. Further, we show that this bound is almost best possible, by exhibiting examples $({{M}^{m(n-j)+2j}},\,T)$ where the fixed point set of $T$ has the form ${{F}^{n}}\,\bigcup \,{{F}^{j}}$ described above, for every $2\,\le \,j\,<\,n$ and $j$ not of the form ${{2}^{t}}\,-\,1$ (for $j\,=\,0$ and 2, it has been previously shown that $m(n\,-\,j)\,+\,2j$ is the best possible bound). The existence of these bounds is guaranteed by the famous 5/2-theorem of J. Boardman, which establishes that under the above hypotheses $m\,\le \,\frac{5}{2}\,n$.