Trivial Exceptions Research Articles

Growth of Local Height Functions Along Orbits of Self-Morphisms on Projective Varieties

Abstract We consider the limit $$ \begin{align*} & \lim_{n\to \infty} \sum_{v\in S} \lambda_{Y,v}(f^{n}(x))/h_{H}(f^{n}(x)) \end{align*}$$where $f \colon X \longrightarrow X$ is a surjective self-morphism on a smooth projective variety $X$ over a number field, $S$ is a finite set of places, $ \lambda _{Y,v}$ is a local height function associated with a proper closed subscheme $Y \subset X$, and $h_{H}$ is an ample height function on $X$. We give a geometric condition that ensures that the limit is zero, unconditionally when $\dim Y=0$ and assuming Vojta’s conjecture when $\dim Y\geq 1$. In particular, we prove (one is unconditional, one is assuming Vojta’s conjecture) dynamical Lang–Siegel type theorems, that is, the relative sizes of coordinates of orbits on ${{\mathbb {P}}}^{N}$ are asymptotically the same with trivial exceptions. These results are higher dimensional generalization of Silverman’s classical result.

The Nowicki conjecture for free metabelian Lie algebras

Let [Formula: see text] be the polynomial algebra in [Formula: see text] variables over a field [Formula: see text] of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations [Formula: see text] (known as Weitzenböck derivations), the algebra of constants [Formula: see text] is finitely generated. When the Weitzenböck derivation [Formula: see text] acts on the polynomial algebra [Formula: see text] in [Formula: see text] variables by [Formula: see text], [Formula: see text], [Formula: see text], Nowicki conjectured that [Formula: see text] is generated by [Formula: see text] and [Formula: see text] for all [Formula: see text]. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free [Formula: see text]-generated metabelian Lie algebra [Formula: see text], with few trivial exceptions, the algebra [Formula: see text] is not finitely generated. However, the vector subspace [Formula: see text] of the commutator ideal [Formula: see text] of [Formula: see text] is finitely generated as a [Formula: see text]-module. In this paper, we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the [Formula: see text]-module [Formula: see text].

Bipartite Kneser graphs are Hamiltonian

The Kneser graph K(n,k) has as vertices all k-element subsets of [n]:={1,2,…,n} and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph H(n,k) has as vertices all k-element and (n−k)-element subsets of [n] and an edge between any two vertices where one is a subset of the other. It has long been conjectured that all connected Kneser graphs and bipartite Kneser graphs (apart from few trivial exceptions) have a Hamilton cycle. The main contribution of this work is proving this conjecture for bipartite Kneser graphs. We also establish the existence of long cycles in Kneser graphs (visiting almost all vertices), generalizing and improving upon previous results on this problem.

Operations between sets in geometry

An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in n -dimensional Euclidean space \mathbb R^n . It is proved that if n\ge 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and associative if and only if it is L_p addition for some 1\le p\le\infty . It is also demonstrated that if n\ge 2 , an operation * between compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and has the identity property (i.e., K*\{o\}=K=\{o\}*K for all compact convex sets K , where o denotes the origin) if and only if it is Minkowski addition. Some analogous results for operations between star sets are obtained. Various characterizations are given of operations that are projection covariant, meaning that the operation can take place before or after projection onto subspaces, with the same effect. Several other new lines of investigation are followed. A relatively little-known but seminal operation called M -addition is generalized and systematically studied for the first time. Geometric-analytic formulas that characterize continuous and \operatorname{GL}(n) -covariant operations between compact convex sets in terms of M -addition are established. It is shown that if n\ge 2 , an o -symmetrization of compact convex sets (i.e., a map from the compact convex sets to the origin-symmetric compact convex sets) is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, and translation invariant if and only if it is of the form \lambda DK for some \lambda\ge 0 , where DK=K+(-K) is the difference body of K . The term “polynomial volume” is introduced for the property of operations * between compact convex or star sets that the volume of rK*sL , r,s\ge 0 , is a polynomial in the variables r and s . It is proved that if n\ge 2 , with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, \operatorname{GL}(n) covariant, associative, and has polynomial volume if and only if it is Minkowski addition.

Kodaira dimension and symplectic sums

Modulo trivial exceptions, we show that symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T2-bundles over T2 each having b1 = 2.

Small exotic 4–manifolds

Modulo trivial exceptions, we show that smoothly nontrivial symplectic sums of symplectic 4-manifolds along surfaces of positive genus are never rational or ruled, and we enumerate each case in which they have Kodaira dimension zero (i.e., are blowups of symplectic 4-manifolds with torsion canonical class). In particular, a symplectic four-manifold of Kodaira dimension zero arises by such a surgery only if it is diffeomorphic to a blowup either of the K3 surface, the Enriques surface, or a member of a particular family of T^2-bundles over T^2 each having b_1=2.

Trivial and Real Exceptions in Generics

Trivial and Real Exceptions in Generics

The structure and existence of 2-factors in iterated line graphs

We prove several results about the structure of 2-factors in it- erated line graphs. Specifically, we give degree conditions on G that ensure L 2 (G) contains a 2-factor with every possible number of cycles, and we give a sufficient condition for the existence of a 2-factor inL 2 (G) with all cycle lengths specified. We also give a characterization of the graphs G where L k (G) A spanning cycle in a graph G is called a hamiltonian cycle and if such a cycle exists, we say that G is hamiltonian. Similarly, a 2-factor in a graph G is a 2-regular spanning subgraph, or equivalently a partition of V (G) into cycles. Hamiltonicity and the existence of 2-factors in graphs have been widely studied. A good reference for the current state of such problems is (4). We say that two edges in a graph G are adjacent if they share an end-vertex. The line graph of G, denoted L(G) is the graph with V (L(G)) = E(G) and E(L(G)) = {eiej | ei and ej are adjacent in G}. We define the i th iterated line graph of G recursively with L 1 (G) = L(G) and L i+1 (G) = L(L i (G)). Chartrand (2) was one of the first to study properties of iterated line graphs, proving that for every graph G (with a few trivial exceptions), L k (G) is hamiltonian for k sufficiently large. Since this first paper, many cycle-structural properties of iterated line graphs have been studied, including when L k (G) is k-ordered (10), pancyclic (12)), k-ordered hamiltonian ((8)), and characterizations of G when L k (G) is hamiltonian ((11)). In this paper, we extend Chartrand's result by giving degree conditions on G to ensure that L 2 (G) contains a 2-factor with every possible number of cycles. We also give a sufficient condition for the existence of a 2-factor in L 2 (G) with all cycle lengths specified. Finally in section 5, we give a characterization of the graphs G where L k (G) contains a 2-factor. 2. Preliminaries In the majority of this paper, we consider only undirected, loopless graphs with- out multiple edges. The main results in the last section will consider multigraphs as well. We will use nk to denote the number of vertices in the k th iteration of the line graph, andk to denote �(L k (G)).

Prolegomena to Realistic Monetary Macroeconomics: A Theory of Intelligible Sequences

This paper sets out a rigorous basis for the integration of Keynes-Kaleckian macroeconomics (with constant or increasing returns to labor, multipliers, mark-up pricing, etc.) with a model of the financial system (comprising banks, loans, credit money, equities, etc.), together with a model of inflation. Central contentions of the paper are that, with trivial exceptions, there are no equilibria outside financial markets, and the role of prices is to distribute the national income, with inflation sometimes playing a key role in determining the outcome. The model deployed here describes a growing economy that does not spontaneously find a steady state even in the long run, but which requires active management of fiscal and monetary policy if full employment without inflation is to be achieved. The paper outlines a radical alternative to the standard narrative method used by post-Keynesians as well as by Keynes himself.

Some Realism About Parochialism: The Economic Analysis of Legal Ethics

This essay reviews Professor Randal Graham's Legal Ethics, the first teaching text to present legal ethics from an explicitly economic point of view. The essay argues that Graham is right to focus on an economic approach, which has not been used as much to analyze legal ethics as it has been to analyze other subjects, such as antitrust, corporations, securities, intellectual property, and tort law. The essay argues that Graham is too soft on professional ethics. He defends, implicitly or explicitly, professional licensing, administrative discipline, and the attorney-client privilege, which have only weak (if any) support in economic theory. For example, Professor Graham employs the lemons model defense of licensing, which is a poor fit for a profession in which reputation plays such an important role. The essay concludes by acknowledging that economic analysis, and utilitarianism in general, does not tell us where preferences come from. Nor, with some trivial exceptions that need no special ethical rules (don't steal, etc.), does it say whether professors should try to alter students' preferences. There is an irreducible normative element to legal ethics, though not a single irreducible approach that could actually be taught to students. The essay concludes that economic analysis should be the dominant approach to legal ethics, though not the only one, and that it should be applied rigorously to the assumptions and pretensions of the profession.

Some Realism about Parochialism: The Economic Analysis of Legal Ethics

This essay reviews Professor Randal Graham's Legal Ethics, the first teaching text to present legal ethics from an explicitly economic point of view. The essay argues that Graham is right to focus on an economic approach, which has not been used as much to analyze legal ethics as it has been to analyze other subjects, such as antitrust, corporations, securities, intellectual property, and tort law. The essay argues that Graham is too soft on professional ethics. He defends, implicitly or explicitly, professional licensing, administrative discipline, and the attorney-client privilege, which have only weak (if any) support in economic theory. For example, Professor Graham employs the lemons model defense of licensing, which is a poor fit for a profession in which reputation plays such an important role. The essay concludes by acknowledging that economic analysis, and utilitarianism in general, does not tell us where preferences come from. Nor, with some trivial exceptions that need no special ethical rules (don't steal, etc.), does it say whether professors should try to alter students' preferences. There is an irreducible normative element to legal ethics, though not a single irreducible approach that could actually be taught to students. The essay concludes that economic analysis should be the dominant approach to legal ethics, though not the only one, and that it should be applied rigorously to the assumptions and pretensions of the profession.

Lower powers of elliptic units

In the previous paper [Sch2] it has been shown that ray class fields over quadratic imaginary number fields can be generated by simple products of singular values of the Klein form defined below. In the present article the second named author has constructed more general products that are contained in ray class fields thereby correcting Theorem 2 of [Sch2]. An algorithm for the computation of the algebraic equations of the numbers in Theorem 1 of this paper has been implemented in a KASH program by the first named author, who also calculated the list of examples at the end of this article. As in the special cases treated in [Sch2] these examples exhibit again that the coefficients of the algebraic equations are rather small. Moreover, apart from trivial exceptions, all numbers computed so far turn out to be generators of the corresponding ray class field, thereby suggesting the conjecture formulated more precisely after Theorem 1.

A Characterization of Tutte Invariants of 2-Polymatroids

This paper develops a theory of Tutte invariants for 2-polymatroids that parallels the corresponding theory for matroids. It is shown that such 2-polymatroid Invariants arise in the enumeration of a wide variety of combinatorial structures including matchings and perfect matchings in graphs, weak colourings in hypergraphs, and common bases in pairs of matroids. The main result characterizes all such invariants proving that, with some trivial exceptions, every 2-polymatroid Tutte invariant can be easily expressed in terms of a certain two-variable polynomial that is closely related to the Tutte polynomial of a matroid.

Existentially closed torsion-free nilpotent groups of class three

We denote by and the classes of torsion-free nilpotent groups of nilpotency class at most two and three, respectively. In this paper we show that most of the known results about existentially closed (e.c.) groups in remain true in : Up to isomorphism, there exist only countably many countable e.c. groups and they are distinguished by the ranks of their centers. An e.c. group is finitely (infinitely) generic if and only if the center has dimension one (≥ω). Apart from trivial exceptions, e.c, algebraically closed, and “closed with respect to systems of equations in one unknown” are equivalent.Let K be a class of groups. A group G ϵ K is called existentially closed in K if G contains a solution of any finite system Σ of equations and inequations with constants in G and one or more unknowns, provided that Σ has a solution in some in K. If Σ may contain equations (in at most n unknowns) only, G is called algebraically closed (n-unknown closed [1]). We use the abbreviations e.c, a.c and n-u.c. throughout. For more background on these terms the reader is referred to [3] and [4]. We also assume a basic knowledge of generic structures. The main group theoretic notions needed here are explained in §2; [2] and [11] are the general references for this field.

Faithful consensus methods for n-trees

A class of new consensus methods for n-trees (hierarchical clusterings) is proposed. These methods apply systematically to an arbitrary collection of given classifications of a fixed set of taxa, and produce a single consensus classification. They are motivated by the desire that the consensus classification retain as much information as possible from the given classifications, even in the case of only approximate agreement among them. A focus of the paper is the concept of faithfulness of consensus methods; this concept explicates the informal notion of adequate retention of information referred to above, and is proposed as a desirable requirement for consensus methods in general. The new methods are all faithful; they have the additional property that they take hierarchical level into account. Other general properties of consensus methods are investigated, especially with reference to their relation with faithfulness. The most important of these properties is neutrality; loosely speaking a consensus method is neutral if all nontrivial clusters are treated equally in the conditions on the given classifications required to guarantee the appearance of a cluster in the consensus. A central result of the paper is an analogue of the classical impossibility theorem of K. Arrow: with trivial exceptions it is impossible to have a consensus method that is simultaneously faithful and neutral. Thus two intuitively very appealing general properties of consensus methods are seen to be incompatible.

Venn Diagrams as Cartographic Symbols

The subject of this article is the graphic representation of data which are shown, in formal logic, by a Venn diagram. The linguistic censuses of Alsace are used as an example. With trivial exceptions, circles cannot show three non-mutually-exclusive sets of attributes of a population if the areas are to be proportional to the size of the respective segments of the population. Regular polygons are also very limited in the data they can represent. Many-sided polygons have relatively stringent constraints, and even squares are logically and practically difficult. Equilateral triangles are more useful figures. They too are limited, and not all possible sets of values can be represented correctly by overlapping triangles. However, the limitations are much less severe and the calculation of the relative positions of the three triangles is much easier than for other polygons.

Decidability and density in two-symbol grammar forms

It is proved that form equivalence is decidable for context-free grammar forms with only one nonterminal and one terminal symbol. However it also proved that there are “many” grammatical families generated by such forms: with some trivial exceptions, the families are dense in the sense that between any two families one can “squeeze” in a third one. The results obtained are applied also to L forms.

On the complexity of the general coloring problem

A graph H is called an interpretation of a graph G if a morphic image of H is (isomorphic to) a subgraph of G . A graph H can be seen to be n -colorable iff H is an interpretation of K n (the complete graph on n vertices): hence colorability is a special case of the concept of interpretation. In this paper the complexity of the general coloring problem, i.e., of deciding for a given graph G whether some graph H is an interpretation of G , is investigated. It is shown that for many very simple undirected graphs G this question is NP -complete (this was previously known for the graphs K n only). In fact, it seems to be NP -complete for all but trivial exceptions. In the directed case, non-trivial graphs G for which the problem is in P , and rather simple graphs G for which the problem is NP -complete are presented. A characterization of all graphs G for which the problem at issue is in P remains an evasive open problem.

On the normal bundles of smooth rational space curves

in this note we consider smooth rational curves C of degree n in threedimensional projective space IP 3 (over a closed field of characteristic 0). To avoid trivial exceptions we shall always assume that n ~ 4 (this does not hold however for certain auxiliary curves we shall consider). Let N = N c be the normal bundle of C in IP 3. Since degel(IP3)=4, and d e g c l ( l P 0 = 2 , we have that d e g c l ( N ) = 4 n 2 . By a well-known theorem of Grothendieck the bundle N is a direct sum of two line bundles. Hence N ~ O c ( 2 n l a ) G O c ( 2 n 1 +a) for some non-negative a=a(C), which is uniquely determined by C. The question we would like to answer is an obvious one: which values of a occur? We shall show (Theorem 4 below) that a value a occurs if and only if 0_ =0, therefore Hi(C, N)=O. It follows [K, p. 150] that C represents a smooth point on the Chow variety Ch(3, 1, n) of effective cycles of dimension 1 and degree n in IP 3. Since the set of all smooth rational curves with a fixed degree is obviously connected, we see that the smooth C's represent a smooth, irreducible, 4n-dimensional (Zariski-)open subset S of Ch(3, 1, n). In a for thcoming paper [ E V ] we shall prove the following

Reconstruction of a Pair of Graphs from their Concatenations

A concatenation of two vertex disjoint graphs is defined to be the graph obtained by identifying a vertex of one graph with a vertex of the other graph. We show that an arbitrary pair of connected graphs can be uniquely reconstructed from the set of all their distinct concatenations, with only trivial exceptions.