A note on the modular representation on the $\mathbb Z/2$-homology groups of the fourth power of real projective space and its application

We write $BV_h$ for the classifying space of the elementary Abelian 2-group $V_h$ of rank $h,$ which is homotopy equivalent to the cartesian product of $h$ copies of $\mathbb RP^{\infty}.$ Its cohomology with $\mathbb Z/2$-coefficients can be identified with the graded unstable algebra $P^{\otimes h} = \mathbb Z/2[t_1, \ldots, t_h]= \bigoplus_{n\geq 0}P^{\otimes h}_n$ over the Steenrod ring $\mathcal A$, where grading is by the degree of the homogeneous terms $P^{\otimes h}_n$ of degree $n$ in $h$ generators with the degree of each $t_i$ being one. Let $GL_h$ be the usual general linear group of rank $h$ over $\mathbb Z/2.$ The algebra $P^{\otimes h}$ admits a left action of $\mathcal A$ as well as a right action of $GL_h.$ A central problem of homotopy theory is to determine the structure of the space of $GL_h$-coinvariants, $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ,$ where ${\rm Ann}_{\overline{\mathcal A}}H_n(BV_h; \mathbb Z/2) ={\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ denotes the space of primitive homology classes, considered as a representation of $GL_h$ for all $n.$ Solving this problem is very difficult and still unresolved for $h\geq 4.$ The aim of this Note is of studying the dimension of $\mathbb Z/2\otimes_{GL_h}{\rm Ann}_{\overline{\mathcal A}}[P^{\otimes h}_n]^{*}$ for the case $h = 4$ and the "generic" degrees $n$ of the form $k(2^{s} - 1) + r.2^{s},$ where $k,\, r,\, s$ are positive integers. Applying the results, we investigate the behaviour of the Singer cohomological "transfer" of rank $4$, which is a homomorphism from a certain subquotient of the divided power algebra $\Gamma(a_1^{(1)}, \ldots, a_4^{(1)})$ to mod-2 cohomology groups ${\rm Ext}_{\mathcal A}^{4, 4+n}(\mathbb Z/2, \mathbb Z/2)$ of the algebra $\mathcal A.$ Singer's algebraic transfer is one of the relatively efficient tools in determining the cohomology of the Steenrod algebra.

On the lambda algebra and Singer's cohomological transfer

Write $\mathbb A$ for the 2-primary Steenrod algebra, which is the algebra of stable natural endomorphisms of the mod 2 cohomology functor on topological spaces. Working at the prime 2, computing the cohomology of $\mathbb A$ is an important problem of Algebraic topology, because it is the initial page of the Adams spectral sequence converging to stable homotopy groups of the spheres. A relatively efficient tool to describe this cohomology is the Singer algebraic transfer of rank $n$ in \cite{Singer}, which passes from a certain subquotient of a divided power algebra to the cohomology of $\mathbb A.$ Singer predicted that this transfer is a monomorphism, but this remains open for $n\geq 4.$ This short note is to verify the conjecture in the ranks 4 and 5 and some generic degrees.

Beyond culture: Advancing the understanding of political and technological contexts in crisis communication

This study aims to theoretically advance the context-oriented tradition in crisis communication by highlighting the political and technological contexts for understanding organisational crises. Using China as a case, the study proposes a broader analytical framework that investigates the societal contexts’ impact on crisis communication from political and technological domains. The analytical framework includes, first, the examination of the authoritarian regime with a divided power structure as the political context in China. Assessing political ideology, political structure, and political history as political contexts allows for a more comprehensive understanding of the impact of political contexts on crisis communication temporally and structurally; second, the investigation of internet users’ voices in a government-regulated commercial space as the technological context in China. Online participation and internet language thereby emerge as prominent parts of the technological contexts for understanding crisis communication in China. The implications and directions of research are also discussed.

DIVIDED POWER ALGEBRAS OVER AN OPERAD

The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras. We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group and the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.

To the question of the newest trends in political processes in Latin America: prospects and challenges

The article deals with the major political problems of Latin America recently: the right drift, the conflict potential of political systems leading to the so called “divided power” (conflict between the Executive and Legislative branches of government), the function of the judicial power etc.

A Stronger Libertarian Paradigm and The Death of the New Deal Constitution

McCutcheon v. Federal Election Commission (FEC) and Citizens United v. FEC have transformed the playing field. Citizens United allows those who run corporations to leverage their influence by using the massive funds accumulated by these entities. McCutcheon empowers a few individuals to make contributions to multiple campaigns around the entire country. Now a few individuals can influence entire elections cycles—essentially purchase public policy. Just one year after Citizens United was decided, the share of total campaign giving by the top 0.01 percent of all campaign donors rose over 50 percent. Following McCutcheon, the number of $500,000-plus gifts increased from 14 to 134, and the number of $1 million plus gifts skyrocketed from eight to sixty-three. This strong libertarian paradigm has caused legislators to align with the interests of donors rather than constituents. This violates any number of constitutional pillars including divided power, federalism, the right to vote, and constituent representation.