Introduction 2 1. Symmetric spectra 5 1.1. Simplicial sets 5 1.2. Symmetric spectra 6 1.3. Simplicial structure on Sp 8 1.4. Symmetric Ω-spectra 10 2. The smash product of symmetric spectra 11 2.1. Symmetric sequences 11 2.2. Symmetric spectra 14 2.3. The ordinary category of spectra 18 3. Stable homotopy theory of symmetric spectra 19 3.1. Stable equivalence 20 3.2. Model categories 26 3.3. Le...

given a graph $g$, let $g^sigma$ be an oriented graph of $g$ with the orientation $sigma$ and skew-adjacency matrix $s(g^sigma)$. then the spectrum of $s(g^sigma)$ consisting of all the eigenvalues of $s(g^sigma)$ is called the skew-spectrum of $g^sigma$, denoted by $sp(g^sigma)$. the skew energy of the oriented graph $g^sigma$, denoted by $mathcal{e}_s(g^sigma)$, is defined as the sum of the n...

A stable cutset in a graph is a stable set whose deletion disconnects the graph. It was conjectured by Caro and proved by Chen and Yu that any graph with n vertices and at most 2n − 4 edges contains a stable cutset. The bound is tight, as we will show that all graphs with n vertices and 2n− 3 edges without stable cutset arise recursively glueing together triangles and triangular prisms along an...

amonge other things we give sufficient and necessary conditions for the lau product of banachalgebras to be biflat or biprojective.

in this paper, we give a characterization of strongly jordan zero-product preserving maps on normed algebras as a generalization of  jordan zero-product preserving maps. in this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly jordan zero-product preserving maps are completely different. also, we prove that the direct p...

let g be a simple connected graph. the generalized polarity wiener index ofg is defined as the number of unordered pairs of vertices of g whosedistance is k. some formulas are obtained for computing the generalizedpolarity wiener index of the cartesian product and the tensor product ofgraphs in this article.

a detailed study of ostracoda changes against the stable isotopes of carbon and oxygen acrossthe cenomanian-turonian boundary mass extinction has been fully documented in one section, the south ofengland. this section comprises the uppermost cenomanian to lowermost turonian strata. the comparisonbetween the results obtained in this research and two rival models for the cenomanian-turonian bound...

Careful ligand design led to the detection of either an unprecedented, stable, 4-membered palladacycle or off-cycle catalyst degradation product.

Let X be a smooth projective variety. Using modified psi classes on the stack of genus zero stable maps to X, a new associative quantum product is constructed on the cohomology space of X. When X is a homogeneous variety, this structure encodes the characteristic numbers of rational curves in X, and specialises to the usual quantum product upon resetting the parameters corresponding to the modi...