Introducing tox-Profiles of Chemical Reactions

In this Essay, we present a critical analysis of two common practices in modern chemistry—that is, of using speculations about the “greenness” and “nontoxicity” of developed synthesis procedures and of a priori labelling various compounds derived from natural sources as being environmentally safe. We note that every organic molecule that contains functional groups should be biologically active. Thus, analysis of the particular greenness and the potential environmental impact of a given chemical process should account for the biological activity of all its components in a measureable (rather than empirical) way. We highlight the necessity of clarifying discussions on biological activity and toxicity and propose possible ways of introducing tox-Profiles as a reliable overview of the overall toxicity of chemical reactions.     

Effect of Sulfation Route and Subsequent Oxidation on Derivatization Degree and Biocompatibility of Cellulose Sulfates

Sulfated cellulose (CS) represents an interesting biopolymer due to bioactivity comparable to heparin. However, use of CS for making surface coatings or hydrogels requires the presence of reactive groups for covalent reactions. Here, an approach is presented to oxidize cellulose sulfates for subsequent cross‐linking reactions with amino groups to form imine bonds. Cellulose is sulfated by direct sulfation or acetosulfation, followed by a M alaprade oxidation. The CS obtained is characterized by elemental analysis and ¹³C‐NMR spectroscopy. The resulting oxidized cellulose sulfates (oxCS) have different degrees of sulfation ranging from 0.79 to 1.13 and oxidation degrees from 0.18 to 0.34, but also different mass average molecular mass (M_W). Toxicity studies are carried out with mouse 3T3 fibroblasts exposed to aqueous solutions of oxCS. The results show that all oxCS are non‐toxic at lower concentrations (0.5 mg mL^?1), but with both increasing degree of oxidation and concentrations, toxic effects are observed particularly for acetosulfated and lesser for direct sulfated oxCS, which is related to a decrease in the M_W of the products. It is concluded that oxCS obtained by direct sulfation with M_W above 70 kDa may represent a biocompatible material for the applications suggested above.     

Existence and regularity for generalised harmonic maps associated to a nonlocal polyconvex energy of Skyrme type

We prove existence and regularity of critical points of arbitrary degree for a generalised harmonic map problem, in which there is an additional nonlocal polyconvex term in the energy, heuristically of the same order as the Dirichlet term. The proof of regularity hinges upon a special nonlinear structure in the Euler–Lagrange equation similar to that possessed by the harmonic map equation. The functional is of a type appearing in certain models of the quantum Hall effect describing nonlocal Skyrmions.     

Minoration conforme du spectre du laplacien de Hodge-de Rham

Let M ⁿ be an n-dimensional compact manifold, with n ≥ 3. For any conformal class C of riemannian metrics on M, we set , where μ _p,k (M,g) is the kth eigenvalue of the Hodge laplacian acting on coexact p-forms. We prove that . We also prove that if g is a smooth metric such that , and n = 0,2,3 mod 4, then there is a non-zero corresponding eigenform of degree with constant length. As a corollary, on a four-manifold with non vanishing Euler characteristic, there is no such smooth extremal metric.     

Some results on embeddings of algebras, after de Bruijn and McKenzie

In 1957, N.G. de Bruijn showed that the symmetric group Sym(Ω) on an infinite set Ω contains a free subgroup on 2^card(Ω) generators, and proved a more general statement, a sample consequence of which is that for any group A of cardinality card(Ω), the group Sym(Ω) contains a coproduct of 2^card(Ω) copies of A, not only in the variety of all groups, but in any variety of groups to which A belongs. His key lemma is here generalized to an arbitrary variety of algebras V, and formulated as a statement about functors Set V. From this one easily obtains analogs of the results stated above with “group” and Sym(Ω) replaced by “monoid” and the monoid Self(Ω) of endomaps of Ω, by “associative K-algebra” and the K-algebra End_K (V) of endomorphisms of a K-vector-space V with basis Ω, and by “lattice” and the lattice Equiv(Ω) of equivalence relations on Ω. It is also shown, extending another result from de Bruijn's 1957 paper, that each of Sym(Ω), Self(Ω) and End_K(V) contains a coproduct of 2^card(Ω) copies of itself.That paper also gave an example of a group of cardinality 2^card(Ω) that was not embeddable in Sym(Ω), and R. McKenzie subsequently established a large class of such examples. Those results are shown here to be instances of a general property of the lattice of solution sets in Sym(Ω) of sets of equations with constants in Sym(Ω). Again, similar results - this time of varying strengths - are obtained for Self(Ω), End_K(V), and Equiv(Ω), and also for the monoid Rel(Ω) of binary relations on Ω.Many open questions and areas for further investigation are noted.     

Lower bounds of some singular integrals and their applications

Let and . We are interested in the lower bounds of the integral:

Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms

In this paper we construct a lifting map from a vector space of generalized Jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the Petersson scalar products.     

Rigidity in one-dimensional tiling spaces

Suppose Tφ and Tθ are tiling spaces arising from primitive nonperiodic substitutions φ and θ. Suppose Fφ and Fθ denote the corresponding inflation and substitution maps on the respective tiling spaces. We prove that Tφ and Tθ are homeomorphic if and only if there exist positive integers m and n such that and are topologically conjugate.     

Spectral radius of graphs with given matching number

In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are K_n if n = 2β or 2β + 1; if 2β + 2 ? n < 3β + 2; or if n = 3β + 2; if n > 3β + 2, where is the empty graph on t vertices.     

On bicyclic graphs with maximal energy

Let λ₁,λ₂,…,λ_n be the eigenvalues of a graph G of order n. The energy of G is defined as E(G)=|λ₁|+|λ₂|+?+|λ_n|. Let be the graph obtained from two copies of C₆ joined by a path P_n-10, B_n be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles C_a and C_b (a,b?10 and a≡b≡2 (mod 4)) joined by an edge. In this paper, we show that is the graph with maximal energy in B_n, which gives a partial solution to Gutman’s conjecture in Gutman and Vidovi? (2001) [I. Gutman, D. Vidovi?, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41 (2001) 1002-1005].