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Manganese(I) carbonyl complexes bearing a MACHO‐type ligand (HN(CH2CH2PR2)2) readily react in their amido form with CO2 to generate 4‐membered {Mn−N−C−O} metallacycles. The stability of the adducts decreases with the steric demand of the R groups at phosphorous (R=isopropyl>adamantyl). The CO2‐adducts display generally a lower reactivity as compared to the parent amido complexes. These adducts can thus be interpretated as masked forms of the active amido catalysts and potentially play important roles as off‐loop species or branching points in catalytic transformations of carbon dioxide. [ABSTRACT FROM AUTHOR]

Assume that a ground‐based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and that a loudspeaker is either also mounted on the vehicle or placed at a fixed location in the room. We prove that the reconstruction is almost always possible if (1) no echoes are received from floors, ceilings, or sloping walls and the vehicle carries at least three noncollinear microphones, or if (2) walls of any inclination may occur, the loudspeaker is fixed in the room and there are four noncoplanar microphones. The difficulty lies in the echo‐matching problem: How to determine which echoes come from the same wall. We solve this by using a Cayley–Menger determinant. Our proofs use methods from computational commutative algebra. [ABSTRACT FROM AUTHOR]

The potential of Pd/Pt complexes for catalytic carboxylation of arenes with CO2 is investigated by means of computational chemistry. Recently we reported that the bis[(2‐methoxyphenyl)phosphino]‐benzenesulfonamido palladium complex 1 inserts CO2 reversibly in its Pd−C(aryl) bond generating carboxylato complex 2. In the present work we study how geometric and electronic factors of various ligands and substrates influence the overall activation barrier (energy span, ES) of a potential catalytic cycle for arene carboxylation comprising this elementary step. The tendency of the key intermediates to dimerize and thus deactivating the potential catalysts is examined as well as the role of the base, which inevitably is needed to stabilize the reaction product. We show that Pd and Pt complexes I(Pd)‐L16‐S1 and I(Pt)‐L16‐S1 do not dimerize, enable the computation of complete catalytic cycles, and show interestingly low ES values of 26.8 and 24.5 kcal/mol, respectively. [ABSTRACT FROM AUTHOR]

Consider an action of a connected solvable group G on an affine variety X. This paper presents an algorithm that constructs a semi-invariant f ∈ K [ X ] = : R and computes the invariant ring (R f) G together with a presentation. The morphism X f → Spec ((R f) G) obtained from the algorithm is a universal geometric quotient. In fact, it is even better than that: a so-called excellent quotient. If R is a polynomial ring, the algorithm requires no Gröbner basis computations. If R is a complete intersection, then so is (R f) G. [ABSTRACT FROM AUTHOR]

The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's [7] constructive characterization of the Krull dimension. While Lombardi's characterization uses the lexicographic monomial order, ours uses the graded (reverse) lexicographic order or, in fact, any graded rational monomial order. Apart from this, the paper contains some related results and some examples which readers may find illuminating. [ABSTRACT FROM AUTHOR]

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone. [ABSTRACT FROM AUTHOR]

This paper contains three new algorithms for computing invariant rings. The first two apply to invariants of a finite group acting on a finitely generated algebra over a Euclidean ring. This may be viewed as a first step in “computational arithmetic invariant theory.” As a special case, the algorithms can compute multiplicative invariant rings. The third algorithm computes the invariant ring of a reductive group acting on a vector space, and often performs better than the algorithms known to date. The main tool upon which two of the algorithms are built is a generalized version of an ideal that was already used by Derksen in his algorithm for computing invariants of linearly reductive groups. As a further application, these so-called extended Derksen ideals give rise to invariantization maps, which turn an arbitrary ring element into an invariant. For the most part, the algorithms of this paper have been implemented. [ABSTRACT FROM AUTHOR]

We determine the minimal number of separating invariants for the invariant ring of a matrix group G ≤ GL n (F q) over the finite field F q. We show that this minimal number can be obtained with invariants of degree at most | G | n (q − 1). In the non-modular case this construction can be improved to give invariants of degree at most n (q − 1). As examples we study separating invariants over the field F 2 for two important representations of the symmetric group. [ABSTRACT FROM AUTHOR]