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Let f (x , y) = 1 + ∑ p = 1 m + n = p ∞ a m , n x m y n be a formal power series. We convert f(x, y) into the formal product ∏ p = 1 m + n = p ∞ (1 + g m , n x m y n) , namely the power product expansion in two independent variables. By developing new machinery involving the majorizing infinite product, we provide estimates on the domain of absolute convergence of the infinite product via the Taylor series coefficients of f(x, y). This machinery introduces a myriad of "mixed expansions", uncovers various algebraic connections between the (a m , n) and the (g m , n) , and leads to the identification of the domain of absolute convergence of the power product as the Cartesian product of polydiscs associated with a quadratic equation. This makes it possible to use the truncated power product expansions ∏ p = 1 m + n = p P (1 + g m , n x m y n) as approximations to the analytic function f(x, y). We derive an asymptotic formula for the g m , n , with mfixed, associated with the majorizing infinite product. We also discuss various combinatorial interpretations provided by these power product expansions and derive an additional theorem which shows that with g m , n ≥ 0 , the polydisc of absolute convergence for the power product is identical to that of its Taylor series. [ABSTRACT FROM AUTHOR]

The work is devoted to the holomorphy of a formal homogeneous series whose restrictions to analytic curves passing through 0 are holomorphic. In content, the main result of the paper is a variation of the well-known Forelli theorem on the holomorphic continuation of formal series along with a set of complex lines. [ABSTRACT FROM AUTHOR]

Let f (x , y) = 1 + ∑ p = 1 m + n = p ∞ a m , n x m y n be a formal power series. We convert f(x, y) into the formal product ∏ p = 1 m + n = p ∞ (1 - h m , n x m y n) - 1 , namely the inverse power product expansion in two independent variables. By developing new machinery involving the majorizing inverse power product, we provide estimates on the domain of absolute convergence of the inverse power product via the Taylor series coefficients of f(x, y). This machinery introduces a myriad of "mixed expansions," uncovers various algebraic connections between (a m , n) and (h m , n) , and leads to the identification of the domain of absolute convergence of the inverse power product as a Cartesian product of polydiscs. This makes it possible to use the truncated inverse power product expansions ∏ p = 1 m + n = p P (1 - h m , n x m y n) - 1 as approximations to the analytic function f(x, y). We derive an asymptotic formula for the h m , n , with mfixed, associated with a majorizing inverse power product. We also discuss various combinatorial interpretations provided by these inverse power product expansions. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature are re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models. [ABSTRACT FROM AUTHOR]

In this note we are concerned with the question: When are the polynomial solutions to a system of linear constant coefficient PDEs dense in the set of all its solutions? [ABSTRACT FROM AUTHOR]

In this paper, an eco-economic model with harvesting on biological population is established, which takes the form of a differential-algebra system. The impact of the economic profit from harvesting upon the dynamics of the model is studied. By using a suitable parameterization for the differential-algebra system, we derive an equivalent parameterized system which gives the stability results for the positive equilibrium point of our model. Moreover, based on the parameterized system as well as the approaches of normal form and formal series, the conditions on the Hopf bifurcation and the stability of center are obtained. Several numerical simulations for demonstrating the theoretical results are also presented. Lastly, according to the dynamical analysis, we provide a threshold value for the economic profit, which can maintain the sustainable development of our eco-economic system. [ABSTRACT FROM AUTHOR]

We study a model problem describing vibrational resonance by means of a high-order averaging technique based on so-called word series. With the technique applied here, the tasks of constructing the averaged system and the associated change of variables are divided into two parts. It is first necessary to build recursively a set of so-called word basis functions and, after that, all the required manipulations involve only scalar coefficients that are computed by means of simple recursions. As distinct from the situation with other approaches, with word-series, high-order averaged systems may be derived without having to compute the associated change of variables. In the system considered here, the construction of high-order averaged systems makes it possible to obtain very precise approximations to the true dynamics.

Using the techniques of the modern umbral calculus, we derive several combinatorial identities involving s-Appell polynomials. In particular, we obtain identities for classical polynomials, such as the Hermite, Laguerre, Bernoulli, Euler, N orlund, hypergeometric Bernoulli, and Legendre polynomials. Moreover, we obtain a generalization of Carlitz's identity for Bernoulli numbers and polynomials to arbitrary symmetric s-Appell polynomials. [ABSTRACT FROM AUTHOR]

A set is called recurrent if its minimal automaton is strongly connected and birecurrent if it is recurrent as well as reversal. We prove a series of results concerning birecurrent sets. It is already known that any birecurrent set is completely reducible (that is, such that the minimal representation of its characteristic series is completely reducible). The main result of this paper characterizes completely reducible sets as linear combinations of birecurrent sets [ABSTRACT FROM AUTHOR]

Over the last two decades, a great deal of work has been devoted to the algorithmic aspects of the frequent itemset (FI) mining problem, leading to a phenomenal number of algorithms and associated implementations, each of which claims supremacy. Meanwhile, it is generally well agreed that developing a unifying theory is one of the most important issues in data mining research. Hence, our primary motivation for this work is to introduce a high-level formalism for this basic problem, which induces a unified vision of the algorithmic approaches presented so far. The key distinctive feature of the introduced model is that it combines, in one fashion, both the qualitative and the quantitative aspects of this basic problem. In this paper, we propose a new model for the FI-mining task based on formal series. In fact, we encode the itemsets as words over a sorted alphabet and express this problem by a formal series over the counting semiring $$(\mathbb N,+,\times ,0,1)$$ , whose range represents the itemsets, and the coefficients are their supports. The aim is threefold: First, to define a clear, unified and extensible theoretical framework through which we can state the main FI-approaches. Second, to prove a convenient connection between the determinization of the acyclic weighted automaton that represents a transaction dataset and the computation of the associated collection of FI. Finally, to devise a first algorithmic transcription, baptized Wafi, of our model by means of weighted automata, which we evaluate against representative leading algorithms. The obtained results show the suitability of our formalism. [ABSTRACT FROM AUTHOR]

It is easy to show that, in the Tractatus, color ascriptions cannot be elementary propositions. But it is not so easy to determine which kind of analysis could be made of judgments of perception like "a is red". Wittgenstein gives us a kind of hint on aphorism 6.3751. He asks the reader to keep in mind the way we deal with the mutual exclusion of colors within the realm of physics. Even so, it is difficult to determine what exactly he is trying to say. I offer an interpretation that links the color exclusion problem with the treatment given to numbers and the notion of "series of forms" (Formenreihe). Some passages of the Philosophische Bemerkungen are used to bring some support to this line of thought.