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The recent survey published in Fuzzy Sets and Systems nicknamed ``Hitchhiker's Guide'' has raised the rating of quasi-copula problems in the dependence modeling community in spite of the lack of statistical interpretation of quasi-copulas. Some of the open problems listed there were solved, and some conjectured one way or the other. This paper concentrates on the Open Problem 5 of this list concerning bounds on the volume of a $d$--variate quasi-copula. We disprove a recent conjecture published in the same journal on the lower bound of this volume. We also give evidence that the problem is much more difficult than suspected and provide hints about its final solution., Comment: 17 pages, 1 figure, 4 tables

In this work we present a theoretical model for differentiable programming. We construct an algebraic language that encapsulates formal semantics of differentiable programs by way of Operational Calculus. The algebraic nature of Operational Calculus can alter the properties of the programs that are expressed within the language and transform them into their solutions. In our model programs are elements of programming spaces and viewed as maps from the virtual memory space to itself. Virtual memory space is an algebra of programs, an algebraic data structure one can calculate with. We define the operator of differentiation ($\partial$) on programming spaces and, using its powers, implement the general shift operator and the operator of program composition. We provide the formula for the expansion of a differentiable program into an infinite tensor series in terms of the powers of $\partial$. We express the operator of program composition in terms of the generalized shift operator and $\partial$, which implements a differentiable composition in the language. Such operators serve as abstractions over the tensor series algebra, as main actors in our language. We demonstrate our models usefulness in differentiable programming by using it to analyse iterators, deriving fractional iterations and their iterating velocities, and explicitly solve the special case of ReduceSum.

In this article we use algebro-geometric tools to describe the structure of a superintegrable system. We study degenerate Neumann system with potential matrix that has some eigenvalues of multiplicity greater than one. We show that the degenerate Neumann system is superintegrable if and only if its spectral curve is reducible and that its flow can be linearized on the generalized Jacobians of the spectral curve. We also show that the generalized Jacobians of the hypereliptic component of the spectral curves are models for the minimal invariant tori of the flow. Moreover the spectral invariants generate local actions that span the invariant tori, while the moment maps for the rotational symmetries provide additional first integrals of the system. Using our results we reproduce already known facts that the degenerate Neumann system is superintegrable, if its potencial matrix has eigenvalues of multiplicity greater or equal than $3$., Comment: submited to Communications in Mathematical Physics

In this paper we study the Neumann system, which describes the harmonic oscillator (of arbitrary dimension) constrained to the sphere. In particular we will consider the confluent case where two eigenvalues of the potential coincide, which implies that the system has S^{1} symmetry. We will prove complete algebraic integrability of confluent Neumann system and show that its flow can be linearized on the generalized Jacobian torus of some singular algebraic curve. The symplectic reduction of S^{1} action will be described and we will show that the general Rosochatius system is a symplectic quotient of the confluent Neumann system, where all the eigenvalues of the potential are double. This will give a new mechanical interpretation of the Rosochatius system., Comment: 17 pages, 1 figure