Thin Subset Research Articles

The Manin–Peyre conjecture for smooth spherical Fano threefolds

The Manin–Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin’s conjecture in its original form would turn out to be incorrect.

Density of rational points on some quadric bundle threefolds

We prove the Manin–Peyre conjecture for the number of rational points of bounded height outside of a thin subset on a family of Fano threefolds of bidegree (1, 2).

Geometric consistency of Manin's conjecture

We conjecture that the exceptional set in Manin's conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying that there is no counterexample to Manin's conjecture which arises from an incompatibility of geometric invariants.

On prime values of binary quadratic forms with a thin variable

In this paper, we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form x 2 + y 2 with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive positive definite binary quadratic form. In particular, for any positive definite binary quadratic form F and binary linear form G, there exist infinitely many ℓ , m ∈ Z such that both F ( ℓ , m ) and G ( ℓ , m ) are primes as long as there are no local obstructions.

Two remarks on sums of squares with rational coefficients

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields $K/\mathbb Q$ with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any $f$ as above necessarily has a (non-trivial) real zero. In the minimal open cases $(3,6)$ and $(4,4)$, we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.

BASIS THEOREMS FOR -SETS

AbstractWe prove the following two basis theorems for ${\rm{\Sigma }}_2^1$-sets of reals:(1)Every nonthin ${\rm{\Sigma }}_2^1$-set has a perfect ${\rm{\Delta }}_2^1$-subset if and only if it has a nonthin ${\rm{\Delta }}_2^1$-subset, and this is equivalent to the statement that there is a nonconstructible real.(2)Every uncountable ${\rm{\Sigma }}_2^1$-set has an uncountable ${\rm{\Delta }}_2^1$-subset if and only if either every real is constructible or $\omega _1^L$ is countable.We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect ${\rm{\Pi }}_2^1$-set with no nonempty ${\rm{\Pi }}_2^1$-thin subset, strengthening a result of Harrington [4].

Critical solutions of nonlinear equations: stability issues

It is known that when the set of Lagrange multipliers associated with a stationary point of a constrained optimization problem is not a singleton, this set may contain so-called critical multipliers. This special subset of Lagrange multipliers defines, to a great extent, stability pattern of the solution in question subject to parametric perturbations. Criticality of a Lagrange multiplier can be equivalently characterized by the absence of the local Lipschitzian error bound in terms of the natural residual of the optimality system. In this work, taking the view of criticality as that associated to the error bound, we extend the concept to general nonlinear equations (not necessarily with primal–dual optimality structure). Among other things, we show that while singular noncritical solutions of nonlinear equations can be expected to be stable only subject to some poor “asymptotically thin” classes of perturbations, critical solutions can be stable under rich classes of perturbations. This fact is quite remarkable, considering that in the case of nonisolated solutions, critical solutions usually form a thin subset within all the solutions. We also note that the results for general equations lead to some new insights into the properties of critical Lagrange multipliers (i.e., solutions of equations with primal–dual structure).

The dynamical look at the subsets of a group

<p>We consider the action of a group $G$ on the family $\mathcal{P}(G)$ of all subsets of $G$ by the right shifts $A\mapsto Ag$ and give the dynamical characterizations of thin, $n$-thin, sparse and scattered subsets.</p><p>For $n\in\mathbb{N}$, a subset $A$ of a group $G$ is called $n$-thin if $g_0A\cap\dots\cap g_nA$ is finite for all distinct $g_0,\dots,g_n\in G$.<br />Each $n$-thin subset of a group of cardinality $\aleph_0$ can be partitioned into $n$ $1$-thin subsets but there is a $2$-thin subset in some Abelian group of cardinality $\aleph_2$ which cannot be partitioned into two $1$-thin subsets. We eliminate the gap between $\aleph_0$ and $\aleph_2$ proving that each $n$-thin subset of an Abelian group of cardinality $\aleph_1$ can be partitioned into $n$ $1$-thin subsets.</p>

Thin subsets of topological groups

A subset A of a group G with the identity e is called thin if gA∩A and Ag∩A are finite for each g∈G∖{e}. We prove that each countable totally bounded topological group has a dense thin subset and a thin subset X such that e is the unique limit point of X. On the other hand, for each thin subset T of a countable Abelian group G, we construct a non-discrete group topology on G in which T is closed and discrete.

Extensions of convex and semiconvex functions and intervally thin sets

We call A ⊂ R N intervally thin if for all x , y ∈ R N and ε > 0 there exist x ′ ∈ B ( x , ε ) , y ′ ∈ B ( y , ε ) such that [ x ′ , y ′ ] ∩ A = ∅ . Closed intervally thin sets behave like sets with measure zero (for example such a set cannot “disconnect” an open connected set). Let us also mention that if the ( N − 1 ) -dimensional Hausdorff measure of A is zero, then A is intervally thin. A function f is preconvex if it is convex on every convex subset of its domain. The consequence of our main theorem is the following: Let U be an open subset of R N and let A be a closed intervally thin subset of U. Then every preconvex function f : U ∖ A → R can be uniquely extended ( with preservation of preconvexity) onto U. In fact we show that a more general version of this result holds for semiconvex functions.

An absorption theorem for minimal AF equivalence relations on Cantor sets

We prove that a ‘small’ extension of a minimal AF equivalence relation on a Cantor set is orbit equivalent to the AF relation. By a ‘small’ extension we mean an equivalence relation generated by the minimal AF equivalence relation and another AF equivalence relation which is defined on a closed thin subset. The result we obtain is a generalization of the main theorem in [GMPS2]. It is needed for the study of orbit equivalence of minimal Z d -systems for d>2 [GMPS3], in a similar way as the result in [GMPS2] was needed (and sufficient) for the study of minimal Z 2 -systems [GMPS1].

Cloud Properties Simulated by a Single-Column Model. Part I: Comparison to Cloud Radar Observations of Cirrus Clouds

Abstract This study describes and demonstrates a new method for identifying deficiencies in how cloud processes are represented in large-scale models. Kilometer-scale-resolving cloud radar observations and cloud-resolving model (CRM) simulations were used to evaluate the representation of cirrus clouds in the single-column model (SCM) version of the National Centers for Environmental Prediction Global Forecast System model for a 29-day period during June and July 1997 at the Atmospheric Radiation Measurement Program site in Oklahoma. To produce kilometer-scale cirrus statistics from the SCM results, synthetic subgrid-scale (SGS) cloud fields were generated using the SCM’s cloud fraction and hydrometeor content profiles, and the SCM’s cloud overlap and horizontal inhomogeneity assumptions. Three sets of SCM synthetic SGS cloud fields were analyzed. Two NOSNOW sets were produced in which clouds did not include snow; one set used random overlap, the other, maximum/random. In the SNOW set, clouds included snow and random overlap was used. The three sets were sampled in the same way as the cloud-radar-detected cloud fields and the CRM-simulated cloud fields. The mean cirrus cloud occurrence frequency for the SCM NOSNOW cloud fields agrees with the observed value as well as the CRM’s does, while that for SCM SNOW cloud fields is only half that observed. In most aspects, the SCM’s cirrus properties differ significantly from the cloud radar’s and the CRM’s, which generally agree. In comparison, there are too many physically thin SCM NOSNOW cirrus layers (most occupy only a single model layer) and too many physically thick SCM SNOW cirrus layers (most are thicker than 4 km). For the optically thin subset of cirrus layers, 1) the mean, mode, and median ice water path, and layer-mean ice water content (IWC) values for the SCM are significantly larger than the observed and CRM values; 2) the SCM layer-mean IWCs decrease with cloud physical thickness, opposite to the observations and CRM results; and 3) the range of layer-mean effective radii in the SCM thin cirrus is too narrow.

On the solution sets of differential inclusions and the periodic problem in Banach spaces

In this paper, the topological structure of the solution set of a constrained semilinear differential inclusion in a Banach space E is studied. It is shown that the set of all mild solutions, with values in a closed and, in general, thin subset D⊂ E, is an R δ -set provided natural boundary conditions and appropriate geometrical assumptions on D (which hold, e.g. when D is convex) are satisfied. Applications to the periodic problem and to the existence of equilibria are given.

Fibré en droites et diviseurs de Cartier sur l'espace des cycles en géométrie analytique complexe

Let ( Z, S, X) be the triple defined by an analytic manifold Z of dimension n+ p, a reduced complex space S, and the graph X of an analytic family ( X s ) s∈ S of n-cycles on Z, parametrized by S. Denote by p 1 (resp. p 2) the canonical projection of S× Z onto S (resp. onto Z). Then, for every p−1-cycle Y in Z such that |X∩p 2 ∗(Y)| is S-finite and p 1(|X∩p 2 ∗(Y)|) is a thin subset of S, there corresponds a unique Cartier divisor such that the isomorphism class of the associated line bundle is given by integration on the analytic family ( X s ) s∈ S of the fundamental class of Y in Deligne cohomology. This cohomological correspondence satisfies nice functorial properties with respect to the arguments ( Z, S, X), admits a relative version and coincides with the Barlet–Magnusson geometric correspondence [2] in case Z is smooth.

Thin and bounded subsets of free topological groups

It is shown that if B is a thin subset of a free topological group F M ( X), then supp( B) is thin in F M ( X) where supp( B) is the minimal set A ⊂ X with the property B ⊂ gp( A), and gp( A) is the group envelope of A. If X is not a P-space, then B ⊂ F M ( X) is thin in F M ( X) iff supp( B) is bounded in X.

Special square sequences

If there is a special square sequence then there is a nonspecial one. A square sequence on a regular uncountable cardinal 0 is a sequence (Cc: a < 0) such that: (i) Cc is closed, unbounded in a and C+l = {a}; (ii) if a is a limit point of then Cc = Cfln a; (iii) if C is closed, unbounded in 0, then there is a limit point a of C such that C, Q C n a. By El(0) we denote the statement that a square sequence exists on 0 . Principles of this kind were first considered by Jensen [5]. They have been since then quite frequently used in recursive constructions of various set-theoretical or modeltheoretical structures [3]. Typically, the single linear construction of type 0 is achieved as a limit of 0 separate recursive constructions along the sets Ca 's. By (ii) those recursive steps cohere as if we were doing a single linear construction. The usefulness of this approach stems from the fact that Ca can be a very thin subset of a so the a th recursive step is thus simplified. In our case the thinness of the C 's is controlled by the condition (iii), but in many cases one a actually has that Ca has order type < a. The following is the most basic result concerning the existence of square sequences. Theorem 1 (Jensen). If 0 is not weakly compact in L, then there is a square sequence on 0 which is, moreover, constructible. One can actually show that the square sequence constructed in [5, ?6] satisfies Theorem 1 (see [10, ? 1]). Note that there is always a square sequence on cO . By the condition (ii) the square sequence (C(: a < 0) can essentially be identified with the following tree ordering <2 on 0: a <2 fl iff a is a limit point of Cf. Note that the condition (iii) becomes equivalent to the condition that the tree 0, <2 has no branches of size 0. We shall call (C(: a < 0) a special square Received by the editors February 8, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 03E05, 03E10, 03E55. (?) 1989 American Mathematical Society 0002-9939/89 $1.00+ $.25 per page