Tableau complexes

Let X, Y be finite sets and T a set of functions X → Y which we will call “ tableaux”. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch’s set-valued semistandard tableaux. One vertex decomposition of this “Young tableau complex” parallels Lascoux’s transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials.     

A loosely Bernoulli counterexample machine

In Rudolph’s paper on minimal self joinings [7] he proves that a rank one mixing transformation constructed by Ornstein [5] can be used as the building block for many ergodic theoretical counterexamples. In this paper we show that Ornstein’s transformation can be altered to create a general method for producing zero entropy, loosely Bernoulli counter-examples. This paper answers a question posed by Ornstein, Rudolph, and Weiss [6].     

Radicals in semigroups

The general theories of radicals in rings and groups initiated by Amitsur [1] and Kurosh [6, 7] and developed by a number of authors in the past decade, were brought under a common roof by Hoehnke [3] in a theory of radicals in general algebras. In a concurrent paper [4], Hoehnke deals more specifically with radicals in semigroups. This lecture is an account of the simpler aspects of Hoehnke’s theory as it applies to semigroups, but based on Tully’s theory of radicals in semigroups, as set forth in §11.6 of [2]. An address delivered at the Symposium on Topological Semigroups, University of Florida, Gainesville, Florida, April 7–11, 1969.     

A proof of Ewell’s octuple product identity

We give a new proof of Ewell’s octuple product identity in [7] by using a general theorem developed by the first author in [3].     

Favard’s interpolation problem in one or more variables

Given scattered data on the real line, Favard [4] constructed an interpolant which depends linearly and locally on the data and whose nth derivative is locally bounded by the nth divided differences of the data times a constant depending only on n. It is shown that the (n —1)th derivative of Favard’s interpolant can be likewise bounded by divided differences, and that one can bound at best two consecutive derivatives of any interpolant by the corresponding divided differences. In this sense, Favard’s univariate interpolant is the best possible. Favard’s result has been extended [8] to a special case in several variables, and here the extent to which this can be repeated in a more general setting is proven exactly.     

Old and new moving-knife schemes

In general, moving-knife schemes seem to be easier to come by than pure existence results (like Neyman’s [N] theorem) but harder to come by than discrete algorithms (like the Dubins-Spanier [DS] last-diminisher method). For envy-free allocations for four or more people, however, the order of difficulty might actually be reversed. Neyman’s existence proof (for anyn) goes back to 1946, the discovery of a discrete algorithm for alln ≥ 4 is quite recent [BT1, BT2, BT3], and a moving-knife solution forn = 4 was found only as this article was being prepared (see [BTZ]). We are left with this unanswered question: Is there a moving-knife scheme that yields an envyfree division for five (or more) players?     

Some characterization theorems in rotatory magneto thermohaline convection

The present paper extends the results of Banerjeeet al [2] for the hydromagnetic thermohaline convection problems of Veronis’ [9] and Stern’s [8] types to include the effect of a uniform vertical rotation.     

Application of functional analysis to solve certain class of constrained time optimal control problems (III)

In this paper the authors have used certain fundamental concept of functional Analysis to tackle a class of constrained time optimal control problems. The authors claim that the technique used is much more general than that of Grimmell [6] so that the system need not be finite dimensional. Moreover Butkovskiy’s [1] moment problem technique can not be applied to solve the problems involving the types of constraints as considered in this paper. It is extremely difficult to solve such problems by the application of Pontryagin’s maximum principle. Presented at 5th International Conference onMathematical Modelling held at University of California, Barkley, 29–31 July, 1985.     

Intrinsic ergodicity of smooth interval maps

We generalize the technique of Markov Extension, introduced by F. Hofbauer [10] for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s [1] Spectral Decomposition, and a strengthened version of Y. Yomdin’s [23] and S. E. Newhouse’s [14] results on differentiable mappings and local entropy. In this way, we reduce the study ofC ^r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC ^∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.     

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.     

On the convergence of the secant method under the gamma condition

We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.      

On Shannon’s entropy power inequality

Summary We prove that the entropy power inequality follows from Blachman’s argument [1] if the densities have finite moments of order α, for some α>0, whenever Shannon’s variational approach can be applied if α>=2.

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Sommario Si dimostra che la disuguaglianza esponenziale dell’entropia può essere dedotta da alcuni precedenti risultati di Blachman [1] se le densità di probabilità hanno momenti finiti di ordine α, per qualche α>0. Si dimostra inoltre che l’argomento variazionale di Shannon può essere applicato se α≥2.

     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

The Blok–Ferreirim theorem for normal GBL-algebras and its application

Generalized basic logic algebras (GBL-algebras for short) have been introduced in [JT02] as a generalization of Hájek’s BL-algebras, and constitute a bridge between algebraic logic and ℓ-groups. In this paper we investigate normal GBL-algebras, that is, integral GBL-algebras in which every filter is normal. For these structures we prove an analogue of Blok and Ferreirim’s [BF00] ordinal sum decomposition theorem. This result allows us to derive many interesting consequences, such as the decidability of the universal theory of commutative GBL-algebras, the fact that n-potent GBL-algebras are commutative, and a representation theorem for finite GBL-algebras as poset sums of GMV-algebras, a result which generalizes Di Nola and Lettieri’s [DL03] representation of finite BL-algebras. Presented by J. G. Raftery. Received May 23, 2007; accepted in final form February 20, 2008.     

Retract rational fields and cyclic Galois extensions

In [23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point was the connection between these problems and Noether’s problem. In [24], a similar sort of study was begun for central simple algebras, with a connection to the center of generic matrices. In [25], the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field. The author is grateful for NSF support under grant #MCS79-04473.     

On the Converse of Aliprantis and Burkinshaw's Theorem

We prove a complete converse of Aliprantis and Burkinshaw’s Theorem [2]. Also we obtain a generalization of Wickstead’s Theorem [9] about this converse, and we give some interesting consequences. revised 4 April, 18 October, and 26 December 2005     

Acceleration property for the E-algorithm and an application to the summation of series

The E-algorithm is the most general sequence transformation actually known, since it contains as particular cases almost all the sequence transformations discovered so far: Richardson polynomial extrapolation, Shanks’ transformation, summation processes, Germain-Bonne transformation, Levin’s generalized transformations, the processp and rational extrapolation. In [10] some results concerning the columns of the E-algorithm were proved. In this paper, by adding conditions about determinants, we prove that the diagonal of this algorithm also accelerates the convergence of the initial sequence.     

A generalization of Quillen’s lemma and its application to the Weyl algebras

Quillen’s lemma [17] is generalized to modules of arbitrary Krull dimension. This leads to some generalizations of the results of [5] and [12] for the Weyl algebras of index > 1. Work supported by the CNRS.     

Spectral synthesis on multivariate polynomial hypergroups

In a recent paper [16] we presented some results concerning spectral analysis and spectral synthesis on polynomial hypergroups in a single variable. Now we show that using Hilbert’s Nullstellensatz, the Noether-Lasker-theorem and the Ehrenpreis-Palamodov-theorem the ideas of [16] can be extended to prove spectral analysis and spectral synthesis on any multivariate polynomial hypergroup. Author’s address: Institute of Mathematics, University of Debrecen, Egyetem tér 1, 4032 Debrecen, Hungary     

Corrected Euler-Maclaurin’s formulae

The aim of this paper is to derive corrected Euler-Maclaurin’s formulae, i.e. open type quadrature formulae where the integral is approximated not only with the values of the function in points (5a+b)/6, (a+b)/2 and (a+5b)/6, but also with values of the first derivative in end points of the interval. These formulae will have a higher degree of exactness than the ones obtained in [2]. Using the derived formulae, a number of inequalities for various classes of functions are obtained.