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One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the “algebraic data” associated to the system. In technical terms, this translates to the problem of determining the existence of regular realizations of differential kernels via their possible prolongations. In this paper we effectively compute an improved upper bound for the number of prolongations needed to guarantee the existence of such realizations, which ultimately produces solutions to many types of systems of partial differential equations. This bound has several applications, including an improved upper bound for the order of characteristic sets of prime differential ideals. We obtain our upper bound by proving a new result on the growth of the Hilbert–Samuel function, which may be of independent interest. [ABSTRACT FROM AUTHOR]

A tensor T , in a given tensor space, is said to be h -identifiable if it admits a unique decomposition as a sum of h rank one tensors. A criterion for h -identifiability is called effective if it is satisfied in a dense, open subset of the set of rank h tensors. In this paper we give effective h -identifiability criteria for a large class of tensors. We then improve these criteria for some symmetric tensors. For instance, this allows us to give a complete set of effective identifiability criteria for ternary quintic polynomials. Finally, we implement our identifiability algorithms in Macaulay2. [ABSTRACT FROM AUTHOR]

Mumford showed that Schottky subgroups of PGL ( 2 , K ) give rise to certain curves, now called Mumford curves, over a non-archimedean field K . Such curves are foundational to subjects dealing with non-archimedean varieties, including Berkovich theory and tropical geometry. We develop and implement numerical algorithms for Mumford curves over the field of p -adic numbers. A crucial and difficult step is finding a good set of generators for a Schottky group, a problem solved in this paper. This result allows us to design and implement algorithms for tasks such as: approximating the period matrices of the Jacobians of Mumford curves; computing the Berkovich skeleta of their analytifications; and approximating points in canonical embeddings. We also discuss specific methods and future work for hyperelliptic Mumford curves. [ABSTRACT FROM AUTHOR]

Abstract: The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple™ package for rule-based programming, rewriting strategies and their combination with standard Maple™ code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation. [Copyright &y& Elsevier]

We correct a minor mistake in the paper of Hanrot and Zimmermann (2004). [ABSTRACT FROM AUTHOR]