The injective chromatic number χi(G) [5] of a graph G is the minimum number of colors needed to color the vertices of G such that two vertices with a common neighbor are assigned distinct colors. In this paper we define injective chromatic sum and injective strength of a graph and obtain the injective chromatic sum of complete graph, paths, cycles, wheel graph and complete bipartite graph. We a...

An injective map from the vertex set of a graph G—its order may not be finite—to the set of all natural numbers is called an arithmetic (a geometric) labeling of G if the map from the edge set which assigns to each edge the sum (product) of the numbers assigned to its ends by the former map, is injective and the range of the latter map forms an arithmetic (a geometric) progression. A graph is c...

Let $$\mathfrak {M}(\Sigma )$$ be an open and connected subset of the space hyperbolic metrics on a closed orientable surface, {M}(M)$$ manifold dimension at least 3. We impose conditions M $${{\,\mathrm{\mathfrak {M}}\,}}(M)$$ , which are often satisfied when in have non-positive curvature. Under these conditions, data homotopy class maps from $$\Sigma $$ to allows us view )\times \mathfrak as...

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to nbp/2c for a p-th order tensor in Rnp . Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the firs...

In this paper certain injectivity conditions in terms of extensions of monomorphisms are considered. In particular, it is proved that a ring R is a quasi-Frobenius ring if and only if every monomorphism from any essential right ideal of RR into R (N) R can be extended to RR. Also, known results on pseudo-injective modules are extended. Dinh raised the question if a pseudo-injective CS module is...

The local multiplier C*-algebra Mloc(A) of any C*-algebra A can be ∗-isomorphicly embedded into the injective envelope I(A) of A in such a way that the canonical embeddings of A into both these C*-algebras are identified. If A is commutative then Mloc(A) ≡ I(A). The injective envelopes of A and Mloc(A) always coincide, and every higher order local multiplier C*-algebra of A is contained in the ...

Abstract In finite group theory, chief factors play an important and well-understood role in the structure theory. We here develop a theory of for Polish groups. development this we prove version Schreier refinement theorem. also trichotomy topologically characteristically simple The requires two independently interesting lines study. First consider injective, continuous homomorphisms with dens...

We introduce the injective tensor products and the projective tensor products of ordered vector spaces with Archimedean order unit and study their functorial properties. The local characterization of a nuclear space is given.