Points Of Indeterminacy Research Articles

Double Dirichlet series associated with arithmetic functions II

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the Möbius function, and so on. We consider the analytic behaviour around the non-positive integer points on singularity sets which are points of indeterminacy. In particular, we show a certain reciprocity law of their residues. Also on this occasion we correct some inaccuracies in our previous paper.

A short note on the Painlevé XXV–Ermakov equation

Solutions of the second member of the Riccati chain and of the corresponding third order linear differential equation are related to solutions of the so-called Painlevé XXV–Ermakov equation via the Schwarzian derivative. The reduction to the generalised Ermakov equation is shown to arise naturally from the Painlevé XXV–Ermakov equation. Specifically, the first order system of ordinary differential equations, equivalent to the Painlevé XXV–Ermakov equation, is analysed by resolving points of indeterminacy of the vector field over P1×P1.

On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

Bratteli diagrams via the De Concini–Procesi theorem

An AF algebra [Formula: see text] is said to be an AF[Formula: see text] algebra if the Murray–von Neumann order of its projections is a lattice. Many, if not most, of the interesting classes of AF algebras existing in the literature are AF[Formula: see text] algebras. We construct an algorithm which, on input a finite presentation (by generators and relations) of the Elliott semigroup of an AF[Formula: see text] algebra [Formula: see text], generates a Bratteli diagram of [Formula: see text] We generalize this result to the case when [Formula: see text] has an infinite presentation with a decidable word problem, in the sense of the classical theory of recursive group presentations. Applications are given to a large class of AF algebras, including almost all AF algebras whose Bratteli diagram is explicitly described in the literature. The core of our main algorithms is a combinatorial-polyhedral version of the De Concini–Procesi theorem on the elimination of points of indeterminacy in toric varieties.

SPACE OF INITIAL VALUES OF A MAP WITH A QUARTIC INVARIANT

AbstractWe compactify and regularise the space of initial values of a planar map with a quartic invariant and use this construction to prove its integrability in the sense of algebraic entropy. The system has certain unusual properties, including a sequence of points of indeterminacy in $\mathbb {P}^{1}\!\times \mathbb {P}^{1}$ . These indeterminacy points lie on a singular fibre of the mapping to a corresponding QRT system and provide the existence of a one-parameter family of special solutions.

Un'ontologia dell'opera musicale, ripartendo da Ingarden

The recent philosophical debate about the musical ontology, between Platonist and Nominalist positions, has to face several unsurmountable objections. The aesthetics of Roman Ingarden, in spite of intrinsic limitations, can be considered in order to relaunch the research about the ontological status of musical works. Three Ingarden's ideas seem particularly useful: the work of music as pure intentional object, as scheme with many points of indeterminacy and as object founded on score, performance or recording. A comparison arises with the theory of documentality proposed by Maurizio Ferraris.

Darboux points and integrability analysis of Hamiltonian systems with homogeneous rational potentials

We study integrability in the Liouville sense for natural Hamiltonian systems with a homogeneous rational potential V(q). Strong necessary conditions for the integrability of such systems were obtained by analysis of differential Galois groups of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions on eigenvalues of the Hessians V″(d) calculated at non-zero solutions d of the equation gradV(d)=d. Such solutions are called proper Darboux points.It was recently proved that for generic polynomial homogeneous potentials, there exist universal relations between eigenvalues of Hessians of the potential taken at all proper Darboux points. The existence of such relations for rational potentials seems to be a difficult question. One of the reasons is the presence of points of indeterminacy of the potential and its gradient. Nevertheless, for two degrees of freedom we prove that such a relation exists. This result is important because it allows us to show that the set of admissible values for Hessian eigenvalues at a proper Darboux point for potentials satisfying the necessary conditions for integrability is finite. In turn, this provides a tool for classification of integrable rational potentials.It was also recently shown that for polynomial homogeneous potentials, additional necessary conditions for integrability can be deduced from the existence of improper Darboux points, that is, points d that are non-zero solutions of gradV(d)=0. These new conditions also take the form of arithmetic restrictions imposed on eigenvalues of V″(d). In this paper we prove that for rational potentials, improper Darboux points give the same necessary conditions for integrability.

The Long Empty Canyon: A Study of the Old/New Legal Problems of the Nile Basin

The Nile River Basin witnesses a long history of tension and negotiation among riparian states. There are two legal frameworks govern the Nile Basin. Firstly, the private legal framework reflected in legal history on the Nile. The most legal active period among Nile Basin states was the period between 1890th and 1930th. The legal solutions to the Nile Basin problems came to an end with the end of the colonization in Africa, especially the Nile riparian states. During this period, the tension among liberal states took a different shape. Harmon and Nyerere doctrine were introduced among the riparian states. This led to the refutation of most of the private legal framework from most of the independent states. Thus, riparian states started to explore new legal ground to regulate their relationship. On the other hand, the public legal framework represented in the work of the International Law Association, which started with Helsinki rules in 1966, and the 1997 UN Convention. Many scholars argue that the legal solution is the best one for the Nile question, based on the previous frameworks. However, this note argued that the international legal framework governing the international rivers generally and the Nile specifically cannot offer a solution to the disputes over the water of the Nile. This note discusses both the legal frameworks of the Nile on one hand. On the other hand, it highlights the points of indeterminacy of both frameworks to solve the Nile dilemma. It argues that the solutions of the present and future disputes through legal tools are not enough. This note goes beyond the most proposed recommendation to form a comprehensive treaty as the solution to the riparian problems. It asserts that the law is not a tool to end the states tension, rather than it is a tool to persevere good faith and prevent future dispute. A main role of the extra legal solutions must be played. It based its argument on substantive and formulate dilemma in the previous frameworks.

Questions inspired by Mikael Passare’s mathematics

Mikael Passare (1959–2011) was a brilliant mathematician. His PhD thesis from 1984 was a breakthrough in the theory of residues in several complex variables. Some time before 1998 he started to work on amoebas and coamoebas. In discussions with him during the last 30 years many questions have emerged—not all of them were resolved at the time of his premature death. The purpose of the paper is to save from oblivion some of the mathematical ideas of Mikael Passare. In the article some of these unanswered questions are presented, always preceded by a discussion leading up to the question. Some of the questions might present challenges to his nine former PhD students, to his many collaborators around the globe—and to anybody interested. Is there an associative algebra of residue currents and principal-value distributions? Is there an interesting non-associative algebra of such currents? Meromorphic extension using two parameters often leads to points of indeterminacy—what is the natural choice at such points? Several questions have bearing on tropical mathematics. Is it possible to build an axiomatic theory for tropical geometry? There are also questions on tropical polynomials as limits of classical polynomials. Can the absolute values of the coefficients of a polynomial be retrieved from its growth function? Some questions are concerned with digital convexity. Finally, there is a question on the constant term in powers of a Laurent polynomial.

The Haar theorem for lattice-ordered abelian groups with order-unit

Given a finite union $P$ of rational simplexes, we assign to $P$ numerical invariants $\lambda_{0}, \lambda_{1},\ldots,\lambda_{\dim P};$ each $\lambda_{i}$ is the suitably normalized volume of the $i$-dimensional part of $P$. We then prove that every finitely generated projective lattice-ordered abelian group $G$ with order-unit $u$ has a faithful invariant positive linear functional $s: G\to \mathbb R$. For each $g\in G$, $s(g)$ is the integral of $g$ over the maximal spectrum of $G$, the latter being canonically identified with a rational polyhedron $P$. Volume elements are measured by the $\lambda_{i}$'s. The proof uses the polyhedral versions of the Wlodarczyk-Morelli theorem on decompositions of birational toric maps in blow-ups and blow-downs, and of the De Concini-Procesi theorem on elimination of points of indeterminacy.

A degenerate Newton’s map in two complex variables: Linking with currents

Little is known about the global structure of the basins of attraction of Newton’s method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of P(x,y) = x(1 x) and Q(x,y) = y 2 + Bxy y. There are invariant circles S0 and S1 within the lines x = 0 and x = 1 which are superattracting in the x-direction and hyperbolically repelling within the vertical line. We show that S0 and S1 have local super-stable manifolds, which when pulled back under iterates of N form global super-stable spaces W0 and W1. By blowing-up the points of indeterminacy p and q of N and all of their inverse images under N we prove that W0 and W1 are real-analytic varieties. We define linking between closed 1-cycles in Wi (i = 0,1) and an appropriate closed 2 current providing a homomorphism lk : H1(Wi, Z) ! Q. If Wi intersects the critical value locus of N, this homomorphism has dense image, proving that H1(Wi, Z) is infinitely generated. Using the Mayer-Vietoris exact sequence and an algebraic trick, we show that the same is true for the closures of the basins of the roots W(ri).

Response: Theses on Literary Theory

(1) The mannerism ›I’m doing theory‹, popular especially in fashionable circles of American and globalized post-structuralism, has little to do with scientifically sound literary theory: the latter assumes a commitment to clarifying basic terms, employing logically coherent argumentation, and critically testing any hypothetically formulated generalizations. (2) There can be no turning back for literary theory from the achievement of analytic philosophy and its linguistic turn, the effect of which was to make those seeking to construct theories on the basis of rational argument engage in critical analysis of the language they use. (3) There have been several attempts to create a rule-based model of literature (with terms such as the ›grammar of poetry‹, the ›sign system of literature‹, ›poetic conventions‹, and ›poetic competence‹). Various such efforts have been dominant at one time or another; they have also been optimistically put forward in the context of linguistic poetics with an analytic/critical orientation, semiotics, and literary theory aligned with structuralism or systems theory. All such efforts, however, have failed. And they are condemned to lasting failure for compelling reasons: Any literary rule can always be individually lifted and is to this extent merely a quasi-norm with text-internal status or limited scope in the context of literary history. (4) Only with a deviation-based model of poetics is it possible to generalize successfully about literature as an art form. In such a model, literary texts, text events, and textual strategies are described as violations of otherwise binding rules of language, communication, semiotics, linguistic structure, and systems of social action (as cases of exallaxai, priem ostranenija as alienation, desautomatization, actualisace, foregrounding, ecart/Abweichung as functional deviation). Central importance then lies with the resultant poetic blanks (Frege’s Leerstellen), points of indeterminacy (Ingarden’s Unbestimmtheitsstellen), or openness (Eco’s Opera aperta), and their fundamental appellative function (Buhler) or Appellstruktur (Iser) with respect to the ›creative reader‹ participating individually in the making of the work. (5) It will certainly be possible to say this in an infinite variety of different and perhaps better ways in future. But as far as the heart of the matter is concerned, we will not be able to do substantively better than the two central te-

Ergodic properties of rational mappings with large topological degree

Let X be a projective manifold and f : X ?? X a rational mapping with large topological degree, dt > ?Ek.1(f) := the (k . 1)th dynamical degree of f. We give an elementary construction of a probability measure ?Ef such that d.n t (fn).?? ?? ?Ef for every smooth probability measure ?? on X. We show that every quasiplurisubharmonic function is ?Ef -integrable. In particular ?Ef does not charge either points of indeterminacy or pluripolar sets, hence ?Ef is f-invariant with constant jacobian f.?Ef = dt?Ef . We then establish the main ergodic properties of ?Ef : it is mixing with positive Lyapunov exponents, preimages of ?hmost?h points as well as repelling periodic points are equidistributed with respect to ?Ef . Moreover, when dimC X . 3 or when X is complex homogeneous, ?Ef is the unique measure of maximal entropy.

Equivariant resolution of points of indeterminacy

We prove an equivariant form of Hironaka's theorem on elimination of points of indeterminacy. Our argument uses canonical resolution of singularities and an extended version of Sumihiro's equivariant Chow lemma.

RESOLUTION THEOREMS FOR COMPACT COMPLEX SPACES WITH A SUFFICIENTLY LARGE FIELD OF MEROMORPHIC FUNCTIONS

The Chow lemma and theorems on the resolution of singularities and of the points of indeterminacy of meromorphic mappings are proved for n-dimensional compact complex spaces with n algebraically independent meromorphic functions. It is established that any such space may be made into a projective algebraic variety by a finite number of monoidal transformations with nonsingular centers.