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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]

We prove that equivalence of deterministic top-down tree-to-string transducers is decidable, thus solving a long-standing open problem in formal language theory. We also present efficient algorithms for subclasses: for linear transducers or total transducers with unary output alphabet (over a given top-down regular domain language), as well as for transducers with the single-use restriction. These results are obtained using techniques from multi-linear algebra. For our main result, we introduce polynomial transducers and prove that for these, validity of a polynomial invariant can be certified by means of an inductive invariant of polynomial ideals. This allows us to construct two semi-algorithms, one searching for a certificate of the invariant and one searching for a witness of its violation. Via a translation into polynomial transducers, we thus obtain that equivalence of general ydt transducers is decidable. In fact, our translation also shows that equivalence is decidable when the output is not in a free monoid but in a free group. [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]

Abstract: We introduce the notion of independent sequences with respect to a monomial order by using the least terms of polynomials vanishing at the sequence. Our main result shows that the Krull dimension of a Noetherian ring is equal to the supremum of the length of independent sequences. The proof has led to other notions of independent sequences, which have interesting applications. For example, we can show that is the maximum number of analytically independent elements in an arbitrary ideal J of a local ring R and that if are (not necessarily finitely generated) subalgebras of a finitely generated algebra over a Noetherian Jacobson ring. [Copyright &y& Elsevier]

The main purpose of this paper is to give a survey of algorithms in invariant theory, with emphasis on nonreductive groups and on recent developments. But the article has some novel elements: it contains a new algorithm for computing invariant rings, which works under the condition that the invariant field is the field of fractions of the invariant ring. We also prove that if n is the dimension of the invariant ring, then there exists a separating set of invariants whose size is at most 2 n + 1. [ABSTRACT FROM AUTHOR]

Abstract: Given a linear action of a group G on a K-vector space V, we consider the invariant ring , where is the dual space. We are particularly interested in the case where and G is the group of all upper unipotent matrices or the group of all upper triangular matrices in . In fact, we determine for and . The result is a complete intersection for all values of n and q. We present explicit lists of generating invariants and their relations. This makes an addition to the rather short list of “doubly parametrized” series of group actions whose invariant rings are known to have a uniform description. [Copyright &y& Elsevier]

Abstract: This paper studies separating subsets of an invariant ring or, more generally, of any set consisting of functions. We prove that a subset of a finitely generated algebra always contains a finite separating subset. We also show that a general version of Noether’s degree bound holds for separating invariants, independently of the characteristic. While the general finiteness result is non-constructive, the Noether bound provides an easy algorithm for computing separating invariants of finite groups. The paper also contains a conceptual investigation of the difference between separating and generating subsets. [Copyright &y& Elsevier]

Abstract: Let G be an affine algebraic group acting on an affine variety X. We present an algorithm for computing generators of the invariant ring in the case where G is reductive. Furthermore, we address the case where G is connected and unipotent, so the invariant ring need not be finitely generated. For this case, we develop an algorithm which computes in terms of a so-called colon-operation. From this, generators of can be obtained in finite time if it is finitely generated. Under the additional hypothesis that is factorial, we present an algorithm that finds a quasi-affine variety whose coordinate ring is . Along the way, we develop some techniques for dealing with nonfinitely generated algebras. In particular, we introduce the finite generation ideal. [Copyright &y& Elsevier]

Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X)G. Moreover, we give a constructive version of a theorem of Rosenlicht, which says that almost all orbits can be separated by rational invariants. More precisely, we give an algorithm for computing a nonempty open subset of X with a geometric quotient. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finite group acting on a polynomial ring A over the field K and let denote the corresponding ring of invariants. Let B be the subalgebra of generated by all homogeneous elements of degree less than or equal to the group order . Then in general B is not equal to if the characteristic of K divides . However we prove that the field of fractions coincides with the field of invariants . We also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for . We prove that there is always a nonzero transfer of degree , such that the localization can be generated by fractions of homogeneous invariants of degrees less than . If with finite-dimensional -module V, then c can be chosen in degree one and can be replaced by . Let denote the image of the classical Noether-homomorphism (see the definition in the paper). We prove that contains the transfer ideal and thus can be used to calculate generators for by standard elimination techniques using Gröbner-bases. This provides a new construction algorithm for . [Copyright &y& Elsevier]

In a previous paper we showed that, for any n ≥ m + 2, most sets of n points in ℝm are determined (up to rotations, reflections, translations and relabeling of the points) by the distribution of their pairwise distances. But there are some exceptional point configurations which are not reconstructible from the distribution of distances in the above sense. In this paper, we concentrate on the planar case m = 2 and present a reconstructibility test with running time O(n11). The cases of orientation preserving rigid motions (rotations and translations) and scalings are also discussed. [ABSTRACT FROM AUTHOR]

Consider the diagonal action of the projective group PGL3 on n copies of ℙ2. In addition, consider the action of the symmetric group Σ n by permuting the copies. In this paper we find a set of generators for the invariant field of the combined group Σ n×PGL3. As the main application, we obtain a reconstruction principle for point configurations in ℙ2 from their sub-configurations of five points. Finally, we address the question of how such reconstruction principles pass down to subgroups. [ABSTRACT FROM AUTHOR]

We study the cohomology modules Hi(G,R) of a p‐group G acting on a ring R of characteristic p, for i>0. In particular, we are interested in the Cohen–Macaulay property and the depth of Hi(G,R) regarded as an RG‐module. We first determine the support of Hi(G,R), which turns out to be independent of i. Then we study the Cohen–Macaulay property for H1(G,R). Further results are restricted to the special case that G is cyclic and R is the symmetric algebra of a vector space on which G acts. We determine the depth of Hi(G,R) for i odd and obtain results in certain cases for i even. Along the way, we determine the degrees in which the transfer map TrG R →RG has non‐zero image. [ABSTRACT FROM PUBLISHER]

One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations.In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples. [Copyright &y& Elsevier]

Let A=BG, where B is a Noetherian algebra over a field K of characteristic p≠0 and G is a finite group such that p divides &z.sfnc;G&z.sfnc;. We give estimates for the depth of A in terms of the codimension of the branch locus of the extension B/A. [Copyright &y& Elsevier]

This paper gives an algorithm for computing invariant rings of reductive groups in arbitrary characteristic. Previously, only algorithms for linearly reductive groups and for finite groups have been known. The key step is to find a separating set of invariants. [ABSTRACT FROM AUTHOR]

We propose an algorithm for computing the radical of a polynomial ideal in positive characteristic. The algorithm does not involve polynomial factorization. [Copyright &y& Elsevier]

We calculate explicit matrix generators of the seven-dimensional representations of the Ree groups in characteristic[SUP~]3. [ABSTRACT FROM AUTHOR]