The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation

The definition and properties of an abstract and very general nonparametric integral of the Calculus of Variations is presented. In harmony with the Lewy-McShane approach, the nonparametric integral ∝ f, for set functions ? taking their values in a Banach space E, is defined in terms of its associated parametric integral. For the latter use is made of the abstract parametric integral proposed by Cesari in Rⁿ and then extended to Banach spaces by Breckenridge, Warner, and the authors. A condition (c) is shown to be relevant for the existence of the integral, and is preserved by the nonlinear operation f. Also, for f nonnegative, a Tonelli-type theorem is proved in the sense that the so defined Weierstrass integral ∝ f is always larger than or equal to the corresponding Lebesgue integral, and equality holds if and only if absolute continuity conditions hold. In the proof a suitable martingale is associated and a convergence theorem for martingales is applied. Applications to the calculus of variations will follow.     

Positivity of Riesz functionals and solutions of quadratic and quartic moment problems

We employ positivity of Riesz functionals to establish representing measures (or approximate representing measures) for truncated multivariate moment sequences. For a truncated moment sequence y, we show that y lies in the closure of truncated moment sequences admitting representing measures supported in a prescribed closed set K⊆Rn if and only if the associated Riesz functional Ly is K-positive. For a determining set K, we prove that if Ly is strictly K-positive, then y admits a representing measure supported in K. As a consequence, we are able to solve the truncated K-moment problem of degree k in the cases: (i) (n,k)=(2,4) and K=R²; (ii) n?1, k=2, and K is defined by one quadratic equality or inequality. In particular, these results solve the truncated moment problem in the remaining open cases of Hilbert's theorem on sums of squares.     

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation property is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the weak separation property is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a weak tangent that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the ‘equality/maximal’ dichotomy does not extend to this setting.     

Recent Developments In Harmonic Approximation,With Applications

A theorem of J.L. Walsh (1929) says that if E is a compact subset of Rⁿ with connected complement and if u is harmonic on a neighbourhood of E, then u can be uniformly approximated on E by functions harmonic on the whole of Rⁿ. In Part I of this article we survey some generalizations of Walsh’s theorem from the period 1980–94. In Part II we discuss applications of Walsh’s theorem and its generalizations to four diverse topics: universal harmonic functions, the Radon transform, the maximum principle, and the Dirichlet problem.     

Quasi-injective dimension

Following our previous work about quasi-projective dimension [11], in this paper, we introduce quasi-injective dimension as a generalization of injective dimension. We recover several well-known results about injective and Gorenstein-injective dimensions in the context of quasi-injective dimension such as the following. (a) If the quasi-injective dimension of a finitely generated module M over a local ring R is finite, then it is equal to the depth of R. (b) If there exists a finitely generated module of finite quasi-injective dimension and maximal Krull dimension, then R is Cohen-Macaulay. (c) If there exists a nonzero finitely generated module with finite projective dimension and finite quasi-injective dimension, then R is Gorenstein. (d) Over a Gorenstein local ring, the quasi-injective dimension of a finitely generated module is finite if and only if its quasi-projective dimension is finite.     

Finite Gorenstein representation type implies simple singularity

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.     

Pairwise compatibility for 2-simple minded collections

In τ-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a τ-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the τ-cluster morphism category of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.     

Some characterizations of Krull domains

In this paper, we define the v-finiteness for a length function L_v on the set of all v-ideals of an integral domain R and show that R is a Krull domain if and only if every proper integral v-ideal of R has v-finite length and L_v((AB)_v)=L_v(A)+L_v(B) for every pair of proper integral v-ideals A and B in R. We also give Euclidean-like characterizations of factorial, Krull, and π-domains. Finally we define the notion of quasi-∗-invertibility and show that if every proper prime t-ideal of an integral domain R is quasi-t-invertible, then R is a Krull domain.     

Squares and their centers

We study the relationship between the size of two sets B, S ? R², when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, Hausdorff dimension, packing dimension, or upper or lower box dimension. Perhaps surprisingly, the results vary depending on the notion of size under consideration. For example, we construct a compact set B of Hausdorff dimension 1 which contains the boundary of an axes-parallel square with center in every point in [0, 1]², prove that such a B must have packing and lower box dimension at least 7/4, and show by example that this is sharp. For more general sets of centers, the answers for packing and box counting dimensions also differ. These problems are inspired by the analogous problems for circles that were investigated by Bourgain, Marstrand and Wolff, among others.     

The exterior algebra ΛkRn and area minimization

The structure of the exterior algebra Λ^kRⁿ is studied in low dimensions, and consequences are drawn for k-dimensional area-minimizing surfaces in Rⁿ. For a general form ø∈Λ²R⁴, Section 3 gives explicit formulas for the comass 6ø6 and the face of the Grassmannian exposed by ø. Section 4 classifies the faces of the Grassmannian of the 3-planes in R⁶ and hence the associated geometries of area-minimizing surfaces (there are four types). Section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the Cartesian product of area-minimizing surfaces is area-minimizing. New examples of area-minimizing integral currents with singularities follow.     

A new approach to the stability theory of functional differential systems

In this article we study the behaviour of dominant Fredholm eigenvalues for the Helmholtz operator in a regular bounded open set Ω in R^m relative to some larger set Ω′ if the latter is altered. It is shown that if the frequency is suitably chosen, then the dominant Fredholm eigenvalues decrease when Ω′ is decreased. This property was so far merely established for the Fredholm eigenvalues for the Laplacian (Kress and Roach, J. Math. Anal. Appl.55 (1976), 102–111). The results obtained will be applied to improve the convergence of a Neumann-Liouville bounded integral operator series, which serves as a tool in determining the solution of the Dirichlet problem.     

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

A centro-projective inequality

We give a new integral formula for the centro-projective area of a convex body, which was first defined by Berck–Bernig–Vernicos. We then use the formula to prove that it is bounded from above by the centro-projective area of an ellipsoid and that equality occurs if and only if the convex set is an ellipsoid.     

Score vectors of Kotzig tournaments

An n-tournament T_n is said to be a Kotzig tournament if the n subtournaments of T_n of order n − 1 are isomorphic. And a nonnegative integral vector R is said to be potentially Kotzig if there exists some Kotzig tournament T_n such that its score vector is R. In this paper, a criterion is found for determining whether a non-negative integral vector R is potentially Kotzig.     

Nonnoetherian coordinate rings with unique maximal depictions

A depiction of a nonnoetherian integral domain R is a special coordinate ring that provides a framework for describing the geometry of R. We show that if R is noetherian in codimension 1, then R has a unique maximal depiction T. In this case, the geometric dimensions of the points of Spec R may be computed directly from T. If in addition R has a normal depiction S, then S is the unique maximal depiction of R.     

Supervised box clustering

In this work we address a technique for effectively clustering points in specific convex sets, called homogeneous boxes, having sides aligned with the coordinate axes (isothetic condition). The proposed clustering approach is based on homogeneity conditions, not according to some distance measure, and, even if it was originally developed in the context of the logical analysis of data, it is now placed inside the framework of Supervised clustering. First, we introduce the basic concepts in box geometry; then, we consider a generalized clustering algorithm based on a class of graphs, called incompatibility graphs. For supervised classification problems, we consider classifiers based on box sets, and compare the overall performances to the accuracy levels of competing methods for a wide range of real data sets. The results show that the proposed method performs comparably with other supervised learning methods in terms of accuracy.     

A periodic review inventory system with two supply modes

We describe a periodic review inventory system in which there are two modes of resupply, namely a regular mode and an emergency mode. Orders placed through the emergency channel have a shorter supply lead time but are subject to higher ordering costs compared to orders placed through the regular channel. We analyze this problem within the framework of an order-up-to-R inventory control policy. At each epoch, the inventory manager must decide which of the two supply modes to use and then order enough units to raise the inventory position to a level R. We show that given any non-negative order-up-to level, either only the regular supply mode is used, or there exists an indifference inventory level such that if the inventory position at the review epoch is below the indifference inventory level, the emergency supply mode is used. We also develop procedures for solving for the two policy parameters, i.e., the order-up-to level and the indifference inventory level.     

A bound for the number of tournaments with specified scores

Let $\text{T}$(R) denote the set of all tournaments with score vector R = (r₁, r₂,…, r_n). R. A. Brualdi and Li Qiao (“Proceedings of the Silver Jubilee Conference in Combinatorics at Waterloo,” in press) conjectured that if R is strong with r₁ ≤ r₂ ≤ … ≤ r_n, then |$\text{T}$(R)| ≥ 2^n?2 with equality if and only if R = (1, 1, 2,…, n ? 3, n ? 2, n ? 2). In this paper their conjecture is proved, and this result is used to establish a lower bound on the cardinality of $\text{T}$(R) for every R.     

Lazy bases: a minimalist constructive theory of Noetherian rings

We give a constructive treatment of the theory of Noetherian rings. We avoid the usual restriction to coherent rings; we can even deal with non‐discrete rings. We introduce the concept of rings with certifiable equality which covers discrete rings and much more. A ring R with certifiable equality can be fitted with a partial ideal membership test for ideals of R. Lazy bases of ideals of R [X ] are introduced in order to derive a partial ideal membership test for ideals of R [X ]. It is then proved that if R is Noetherian, then so is R [X ]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)