Abstract. We consider and compare super Pluecker and super Ptolemy relations from the perspective of constructing super cluster algebras. We develop the super cluster structure of the super Grassmannians $Gr_{2|0}(n|1)$ for arbitrary $n$, which was indicated in our earlier work. We also analyze super Pluecker relations for ``weighted projective'' case of super Grassmannians and obtain a new simple form of the relations for $Gr_{r|1}(n|1)$. To this end, we establish properties of Berezinians of certain type matrices (which we call ``wrong''). For the super Ptolemy relation for the decorated super Teichmueller space of Penner-Zeitlin, we show how by a change of variables it can be transformed into the classical Ptolemy relation with the new even variables decoupled from odd variables.
Abstract. We propose a new kind of automatic architecture search algorithm. The algorithm alternates pruning connections and adding neurons, and it is not restricted to layered architectures only. Here architecture is an arbitrary oriented graph with some weights (along with some biases and an activation function), so there may be no layered structure in such a network. The algorithm minimizes the complexity of staying within a given error. We demonstrate our algorithm on the brightness prediction problem of the next point through the previous points on an image. Our second test problem is the approximation of the bivariate function defining the brightness of a black and white image. Our optimized networks significantly outperform the standard solution for neural network architectures in both cases.
Informal abstract. We completely solved the following problems: super Plücker embedding, coordinates, and relations. We proposed a super cluster structure for the case $Gr_{2|0}(n|1)$. For the Grassmannian $Gr_{r|0}(n|m)$ we built on the standard construction of the exterior powers using the quotient of the tensor algebra by anticommutativity relations. For the general case $Gr_{r|s}{n|m}$ we need Voronov-Zorich's super exterior powers $\Lambda^{r|s}(V)$; and for the target space, we have to use a weighted projective space of weights $1$ and $-1$, $P_{+1,-1}(\Lambda^{r|s}(V)\oplus \Lambda^{s|r}(\Pi V))$.
Abstract. We recall the notion of a differential operator over a smooth map (in linear and non-linear settings) and consider its versions such as formal $\hbar$-differential operators over a map. We study constructions and examples of such operators, which include pullbacks by thick morphisms and quantization of symplectic micromorphisms.
Abstract. We introduce a formal $\hbar$-differential operator $\Delta$ that generates higher Koszul brackets on the algebra of (pseudo)differential forms on a $P_{\infty}$-manifold. Such an operator was first mentioned by Khudaverdian and Voronov in arXiv:1808.10049. (This operator is an analogue of the Koszul--Brylinski boundary operator $\partial_P$ which defines Poisson homology for an ordinary Poisson structure.)
Here we introduce $\Delta=\Delta_P$ by a different method and establish its properties. We show that this BV type operator generating higher Koszul brackets can be included in a one-parameter family of BV type formal $\hbar$-differential operators, which can be understood as a quantization of the cotangent $L_{\infty}$-bialgebroid. We obtain symmetric description on both $\Pi TM$ and $\Pi T^*M$.
For the purpose of the above, we develop in detail a theory of formal $\hbar$-differential operators and also of operators acting on densities on dual vector bundles. In particular, we have a statement about operators that can be seen as a quantization of the Mackenzie-Xu canonical diffeomorphism. Another interesting feature is that we are able to introduce a grading, not a filtration, on our algebras of operators. When operators act on objects on vector bundles, we obtain a bi-grading.
Abstract. We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.
Informal abstract. We give a proof of Darboux's lond-standing conjecture that every Darboux transformation of arbitrary order of a 2D Schroedinger type operator can be factorized into Darboux transformations of order one. The proof is constructive.
Besides the main result, the paper contains an algebraic definition of a Darboux transformations for an arbitrary operator, which naturally implies additional properties that are usually had to be added artificially.
Informal Abstract.
We explore factorizations of differential operators on the algebra of densities (note that all densities are considered in one algebra together,
such was introduced by Khudaverdian and Voronov in connection with Batalin--Vilkovisky geometry).
Here we consider the case of the line, where unlike the familiar setting (where operators act on functions) there are obstructions for
factorization. We analyze these obstructions.
Then we have a surprising result: we consider "generalized Sturm-Liouville" operator, which was invented by Khudaverdian and Voronov recently as the generalization of classical Sturm-Liouville operator which (if we care about changes of variables) is defined on densities of weight $-\frac{1}{2}$ and map them into the space of densities of weight $\frac{3}{2}$. The generalized Sturm-Liouville operator acts on densities of all weights. We establish a criterion of its factorizability in terms of solution of the classical Sturm-Liouville equation (!!!). We also establish the possibility of an incomplete factorization.
Informal Abstract.
What is interesting here is that we consider Darboux transformations (DTs) for an operator of a very general form and there are not much of results for such!
We answered the questions: (1) When an operator has a DT of Wronskian type of order one? (2) What are possible "building blocks" i.e. Darboux transformations of order one for an operator? We also suggested a new construction (partially inspired by continued fractions) that allows to construct DTs of new previously not known type: "Continued DTs". (Other known types of DTs so far are: Laplace transformation, "Wronskian type", Type I, and Intertwining Type).
Abstract. We consider differential operators on a supermanifold of dimension $1|1$. We distinguish non-degenerate operators as those with an invertible top coefficient in the expansion in the superderivative $D$. They are remarkably similar with ordinary differential operators. We show that every non-degenerate operator can be written in terms of 'super Wronskians' (which are certain Berezinians). We apply this to Darboux transformations (DTs), proving that every DT of an arbitrary non-degenerate operator is the composition of elementary first order transformations. Hence every DT corresponds to an invariant subspace of the source operator and upon a choice of a basis in this subspace is expressed by a super-Wronskian formula. We consider also dressing transformations, i.e., the effect of a DT on the coefficients of an operator. We calculate them in examples and make some general statements.
Darboux transformations for factorizable operator has lot of structure. In particular, there is a correspondence between Darboux transformations of factorizable second-order operators and Darboux transformation of first order operators. However, the correspondence is not one-to-one, and lifting back process is not immediate.
Here we find a criterion for an arbitrary operator on the plane to have an invertible Darboux transformation.
We also study Wronkian-type formulas which allow to construct Darboux transformations for many different types of operators. These are non-invertible by construction. From time to time new works appear showing that Wronskian-type formulas generate Darboux transformations for some new type of operators. However, up until this work there has been no explicit example of an operator for which Wronkiant-type formulas do NOT work. We give an explicit example in this paper, and find sufficient conditions for Wronkians to work.
The conjecture was reduced to the solution of a complicated system of non-linear PDEs. Using moving frames method (of differential geometry, due to P.Olver et al.) I have found a generating set of joint differential invariants for the involved operators. Then after re-writing in terms of invariants, the initial system split into two much simpler systems, one of which was a system of linear PDEs.
