Comparison Maps for Relatively Free Resolutions

Dedicated to Murray Gerstenhaber and Jim Stasheff Kontsevich’s formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifoldMisL ∞ -quasi-isomorphic to its cohomology. The construction of theL ∞ -map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of\({\mathbb{R}}^{d}\)with standard volume form, we obtain anL ∞ -morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin–Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes. Mathematics Subject Classification 2010: Primary 53D55; Secondary 81T40, 81T45, 81T70Key wordsPoisson sigma modelBV quantizationEffectiveaction L ∞ algebrasCyclic homologyUnimodular Poisson structures

We compute the homology of the complex of formal operations on the Hochschild complex of differential graded commutative algebras as defined by Wahl and prove that these can be built as infinite sums of operations obtained from Loday's shuffle operations, Connes' boundary operator and the shuffle product.

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

We present an efficient algorithm to solve a class of two- and 2.5-dimensional (2-D and 2.5-D) Fredholm integrals of the first kind with a tensor product structure and nonnegativity constraint on the estimated parameters of interest in an optimization framework. A zeroth-order regularization functional is used to incorporate a priori information about the smoothness of the parameters into the problem formulation. We adapt the Butler-Reeds-Dawson (1981) algorithm to solve this optimization problem in three steps. In the first step, the data are compressed using singular value decomposition (SVD) of the kernels. The tensor-product structure of the kernel is exploited so that the compressed data is typically a thousand fold smaller than the original data. This size reduction is crucial for fast optimization. In the second step, the constrained optimization problem is transformed to an unconstrained optimization problem in the compressed data space. In the third step, a suboptimal value of the smoothing parameter is chosen by the BRD method. Steps 2 and 3 are iterated until convergence of the algorithm. We demonstrate the performance of the algorithm on simulated data.

This article describes a new algorithm to compute a free resolution of an ideal of (or module over) a commutative ring R combining the Koszul complex with Groebner basis methods. The algorithm computes the resolution of an ideal I via the resolutions of a sequence of subideals \({{(0)=I_0 \subset\ldots\subset I_r = I}}\) differing by one generator each time. The article discusses special orderings and criteria applicable to this algorithm and gives some timings based on an implementation within the computer algebra system Singular.

Using tilting modules we equip certain semisimple categories with a “reduced” tensor product structure. The fusion rules for this tensor product are determined via known character formulas for the involved modules.

Castelnuovo Mumford regularity with respect to multigraded ideals

A connection-like objects, termed hom-connections are defined in the realm of non-commutative geometry. The definition is based on the use of homomorphisms rather than tensor products. It is shown that hom-connections arise naturally from (strong) connections in non-commutative principal bundles. The induction procedure of hom-connections via a map of differential graded algebras or a differentiable bimodule is described. The curvature for a hom-connection is defined, and it is shown that flat hom-connections give rise to a chain complex.

Rewriting Systems and Hochschild-Mitchell Homology

We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let A A be a finitely generated algebra over a commutative ring K K with a (possibly infinite) Gröbner base G G on a free algebra F F , that is, A A is the quotient F / I ( G ) F/I(G) with the ideal I ( G ) I(G) of F F generated by G G . Given a Gröbner base H H for an A A -subbimodule L L of the free A A -bimodule A ⋅ X ⋅ A = A K ⊗ K ⋅ X ⊗ K A A \cdot X \cdot A = A_K \otimes K \cdot X \otimes _KA generated by a set X X , we have a morphism ∂ \partial of A A -bimodules from the free A A -bimodule A ⋅ H ⋅ A A \cdot H \cdot A generated by H H to A ⋅ X ⋅ A A \cdot X \cdot A sending the generator [ h ] [h] to the element h ∈ H h \in H . We construct a Gröbner base C C on F ⋅ H ⋅ F F \cdot H \cdot F for the A A -subbimodule Ker( ∂ \partial ) of A ⋅ H ⋅ A A \cdot H \cdot A , and with this C C we have the free A A -bimodule A ⋅ C ⋅ A A \cdot C \cdot A generated by C C and an exact sequence A ⋅ C ⋅ A → A ⋅ H ⋅ A → A ⋅ X ⋅ A A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A . Applying this construction inductively to the A A -bimodule A A itself, we have a free A A -bimodule resolution of A A .

In this paper we provide a classification theorem and a structure theorem for exact differential graded algebras, and we use the classification theorem to show that a differential graded algebra A is semisimple (as a differential graded algebra) precisely when the graded algebra Z(A) is semisimple (as a graded algebra) and A is an exact complex. We also relate exact differential graded algebras with a graded version of Hochschild cohomology.

Exact linear modeling using Ore algebras

In this article we study zeta functions enumerating subalgebras or ideals of Lie algebras over finite fields. We demonstrate a methodology for calculating such zeta functions in the most general setting, and develop this method to obtain explicit formulas for various new examples. We then discuss the implications of our data, in particular in relation to the general theory of zeta functions of Lie algebras over commutative rings.

Multivariate basis splines (B‐splines) and Non‐uniform rational B‐splines (NURBS) lack adaptivity due to their tensor product structure. Truncated hierarchical B‐splines (THB‐splines) provide a solution for this. THB‐splines organize the parameter space into a hierarchical structure, which enables efficient approximation and representation of functions with different levels of detail. The truncation mechanism ensures the partition of unity property of B‐splines and defines a more scattered set of basis functions without overlapping on the multi‐level spline space. Transferring these multi‐level splines into Bézier elements representation facilitates straightforward incorporation into existing finite element (FE) codes. By separating the multi‐level extraction of the THB‐splines from the standard Bézier extraction, a more general independent framework applicable to any sequence of nested spaces is created. The operators for the multi‐level structure of THB‐splines and the operators of Bézier extraction are constructed in a local approach. Adjusting the operators for the multi‐level structure from an element point of view and multiplying with the Bézier extraction operators of those elements, a direct map between Bézier elements and a hierarchical structure is obtained. The presented implementation involves the use of an open‐source Octave/MATLAB isogeometric analysis (IGA) code called GeoPDEs. A basic Poisson problem is presented to investigate the performance of multi‐level Bézier extraction compared to a standard THB‐spline approach.

Quantum-like modeling (QLM)—quantum theory applications outside of physics—are intensively developed with applications in biology, cognition, psychology, and decision-making. For cognition, QLM should be distinguished from quantum reductionist models in the spirit of Hameroff and Penrose, as well as Umezawa and Vitiello. QLM is not only concerned with just quantum physical processes in the brain but also with QL information processing by macroscopic neuronal structures. Although QLM of cognition and decision-making has seen some success, it suffers from a knowledge gap that exists between oscillatory neuronal network functioning in the brain and QL behavioral patterns. Recently, steps toward closing this gap have been taken using the generalized probability theory and prequantum classical statistical field theory (PCSFT)—a random field model beyond the complex Hilbert space formalism. PCSFT is used to move from the classical “oscillatory cognition” of the neuronal networks to QLM for decision-making. In this study, we addressed the most difficult problem within this construction: QLM for entanglement generation by classical networks, that is, “mental entanglement.” We started with the observational approach to entanglement based on operator algebras describing “local observables” and bringing into being the tensor product structure in the space of QL states. Moreover, we applied the standard states entanglement approach: entanglement generation by spatially separated networks in the brain. Finally, we discussed possible future experiments on “mental entanglement” detection using the EEG/MEG technique.