Abstract:
Dendroidal sets have been introduced as a combinatorial model for homotopy coherent operads. We introduce the notion of fully Kan dendroidal sets and show that there is a model structure on the category of dendroidal sets with fibrant objects given by fully Kan dendroidal sets. Moreover we show that the resulting homotopy theory is equivalent to the homotopy theory of connective spectra.

Abstract:
We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from discrete rings to connective ring spectra. That is, if $\mathcal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the $(n+1)$-cube induced by the cyclotomic trace $$K(\mathcal A)\to TC(\mathcal A)$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.

Abstract:
In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical spectrum for the 1/4 Cantor measure always yields an alternate spectrum. We call this set an additive spectrum. The proof works by connecting the additive set to a spectrum formed by odd multiplicative scaling.

Abstract:
We define the notion of an additive model category, and we prove that any additive, stable, combinatorial model category has a natural enrichment over symmetric spectra based on simplicial abelian groups. As a consequence, every object in such a model category has a naturally associated endomorphism ring inside this spectra category. We establish the basic properties of this enrichment. We also develop some enriched model category theory. In particular, we have a notion of an adjoint pair of functors being a 'module' over another such pair. Such things are called "adjoint modules". We develop the general theory of these, and use them to prove a result about transporting enrichments over one symmetric monoidal model category to a Quillen equivalent one.

Abstract:
We construct a cofibrantly generated Quillen model structure on the category of small differential graded categories. ----- Nous construisons une structure de categorie de modeles de Quillen a engendrement cofibrant sur la categorie des petites categories differentielles graduees.

Abstract:
We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the idempotent and nilpotent commute the ring is called weak* nil clean. We characterize all $n\in \mathbb{N}$, for which $\mathbb{Z}_n$ is weak nil clean but not nil clean. We show that if $R$ is a weak* nil clean and $e$ is an idempotent in $R$, then the corner ring $eRe$ is also weak* nil clean. Also we discuss $S$-weak nil clean rings and their properties, where $S$ is a set of idempotents and show that if $S=\{0, 1\}$, then a $S$-weak nil clean ring contains a unique maximal ideal. Finally we show that weak* nil clean rings are exchange rings and strongly nil clean rings provided $2\in R$ is nilpotent in the later case. We have ended the paper with introduction of weak J-clean rings.

Abstract:
The concept of nil-symmetric rings has been introduced as a generalization of symmetric rings and a particular case of nil-semicommutative rings. A ring is called right (left) nil-symmetric if, for , where are nilpotent elements, implies . A ring is called nil-symmetric if it is both right and left nil-symmetric. It has been shown that the polynomial ring over a nil-symmetric ring may not be a right or a left nil-symmetric ring. Further, it is also proved that if is right (left) nil-symmetric, then the polynomial ring is a nil-Armendariz ring. 1. Introduction Throughout this paper, all rings are associative with unity. Given a ring , and denote the set of all nilpotent elements of and the polynomial ring over , respectively. A ring is called reduced if it has no nonzero nilpotent elements; is said to be Abelian if all idempotents of are central; is symmetric [1] if implies for all . An equivalent condition for a ring to be symmetric is that whenever product of any number of elements of the ring is zero, any permutation of the factors still gives the product zero [2]. is reversible [3] if implies for all ; is called semicommutative [4] if implies for all . In [5], Rege-Chhawchharia introduced the concept of an Armendariz ring. A ring is called Armendariz if whenever polynomials , satisfy , then for each . Liu-Zhao [6] and Antoine [7] further generalize the concept of an Armendariz ring by defining a weak-Armendariz and a nil-Armendariz ring, respectively. A ring is called weak-Armendariz if whenever polynomials ,？？ satisfy , then for each . A ring is called nil-Armendariz if whenever ,？？ satisfy , then for each . Mohammadi et al. [8] initiated the notion of a nil-semicommutative ring as a generalization of a semicommutative ring. A ring is nil-semicommutative if implies for all . In their paper it is shown that, in a nil-semicommutative ring , forms an ideal of . Getting motivated by their paper we introduce the concept of a right (left) nil-symmetric ring which is a generalization of symmetric rings and a particular case of nil-semicommutative rings. Thus all the results valid for nil-semicommutative rings are valid for right (left) nil-symmetric rings also. We also prove that if a ring is right (left) nil-symmetric and Armendariz, then is right (left) nil-symmetric. In the context, there are also several other generalizations of symmetric rings (see [9, 10]). 2. Right (Left) Nil-Symmetric Rings For a ring , and denote the full matrix ring and the upper triangular matrix ring over , respectively. We observe that if is a ring, then Definition 1. A ring

Abstract:
A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. For any commutative *-ring $R$, we prove that the algebraic extension $R[i]$ where $i^2=\mu i+\eta$ for some $\mu,\eta\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $\mu\eta$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are also obtained.