The value of foresight

Suppose you have one unit of stock, currently worth 1, which you must sell before time $T$. The Optional Sampling Theorem tells us that whatever stopping time we choose to sell, the expected discounted value we get when we sell will be 1. Suppose however that we are able to see $a$ units of time into the future, and base our stopping rule on that; we should be able to do better than expected value 1. But how much better can we do? And how would we exploit the additional information? The optimal solution to this problem will never be found, but in this paper we establish remarkably close bounds on the value of the problem, and we derive a fairly simple exercise rule that manages to extract most of the value of foresight.     

Properties of the probability density function of the non-central chi-squared distribution

In this paper we consider the probability density function (pdf) of a non-central χ² distribution with arbitrary number of degrees of freedom. For this function we prove that can be represented as a finite sum and we deduce a partial derivative formula. Moreover, we show that the pdf is log-concave when the degrees of freedom is greater or equal than 2. At the end of this paper we present some Turán-type inequalities for this function and an elegant application of the monotone form of l'Hospital's rule in probability theory is given.     

Complexity of approximation of functions of few variables in high dimensions

In DeVore et al. (2011) [7] we considered smooth functions on [0,1]^N which depend on a much smaller number of variables ? or continuous functions which can be approximated by such functions. We were interested in approximating those functions when we can calculate point values at points of our choice. The number of points we needed for non-adaptive algorithms was higher than that in the adaptive case. In this paper we improve on DeVore et al. (2011) [7] and show that in the non-adaptive case one can use the same number of points (up to a multiplicative constant depending on ?) that we need in the adaptive case.     

On the first eigenvalue of a fourth order Steklov problem

We prove some results about the first Steklov eigenvalue d ₁ of the biharmonic operator in bounded domains. Firstly, we show that Fichera’s principle of duality (Fichera in Atti Accad Naz Lincei 19:411–418, 1955) may be extended to a wide class of nonsmooth domains. Next, we study the optimization of d ₁ for varying domains: we disprove a long-standing conjecture, we show some new and unexpected features and we suggest some challenging problems. Finally, we prove several properties of the ball.     

On Nonstructure of Elementary Submodels of an Unsuperstable Homogeneous Structure

In the first part of this paper we let M be a stable homogeneous model and we prove a nonstructure theorem for the class of all elementary submodels of M, assuming that M is ‘unsuperstable’ and has Skolem functions. In the second part we assume that M is an unstable homogeneous model of large cardinality and we prove a nonstructure theorem for the class of all elementary submodels of M.     

Construction of systems of sets related to the plurality functions

In the present paper we provide a way of construction of all m-elements consistent systems. These kinds of families of sets are the parameters determining the solutions of some functional equation, which express the consistency condition appearing in characterizing the plurality functions. First, we formulate the idea of extending p-elements family to such m-tuples families which are m-elements consistent systems. Then we study some of their properties and we use them in the constructing of extending p-elements families to m-elements consistent systems and all such systems. Finally, we deal with m-elements consistent systems which satisfy an additional condition.     

On special types of semiholonomic 3-jets

First we summarize some properties of the nonholonomic r-jets from the functorial point of view. In particular, we describe the basic properties of our original concept of nonholonomic r-jet category. Then we deduce certain properties of the Weil algebras associated with nonholonomic r-jets. Next we describe an algorithm for finding the nonholonomic r-jet categories. Finally we classify all special types of semiholonomic 3-jets.     

Regularities of multifractal measures

First, we prove the decomposition theorem for the regularities of multifractal Hausdorff measure and packing measure in ” ^d . This decomposition theorem enables us to split a set into regular and irregular parts, so that we can analyze each separately, and recombine them without affecting density properties. Next, we give some properties related to multifractal Hausdorff and packing densities. Finally, we extend the density theorem in [6] to any measurable set.     

Classification of quintic eutactic forms

From the classical Voronoi algorithm, we derive an algorithm to classify quadratic positive definite forms by their minimal vectors; we define some new invariants for a class, for which several conjectures are proposed. Applying the algorithm to dimension 5 we obtain the table of the 136 classes in this dimension, we enumerate the 118 eutactic quintic forms, and we verify the Ash formula.

     

Galois G-covering of quotients of linear categories

In this paper, we introduce the notion of G-liftable ideals, which extends the liftable ideas defined by Assem and Le Meur. We characterize the G-liftable ideals and construct the Galois G-coverings of quotients of categories associated to the G-liftable ideals. In particular, we study the behavior of G-liftable admissible ideals under Galois G-coverings. Furthermore, we show that the ideals generated by finite dimensional projective modules over a locally bounded linear categories are admissible G-liftable ideals. As an application, we provide a reduction technique for dealing with the existence of Serre functors in the stable categories of Gorenstein projective objects.     

Predicting the future of scholarly publishing

Conclusions  What should we conclude? Should we steadfastly maintain the status quo? Do we avoid technology altogether? Of course not.We should experiment; we should try out new things; we should tinker with technology and find better ways to communicate. This article is based on a talk given at the Conference on Electronic Information and Communication, Tsinghua University, China, August 29-31, 2002. A version of this paper will appear in the proceedings of that conference.     

Investigation of properties of the solutions of hyperbolic stochastic PDEs

In this paper we investigate the compact support property of the solutions of hyperbolic Stochastic PDE (SPDE) providing that initial condition function is deterministic and has compact support property. First, to approach this problem, we consider semi-SPDE. It turns out that in the semi-SPDE case solution u (t, x) preserve compact support property. When we consider SPDE, we use the stochastic differential-difference equations (SDDE) approach. It turns out that in SPDE case solution u (t, x) does not preserve compact support property. So, if we compare the semi-SPDE and SPDE then it becomes obvious that differentiation in space in SPDE plays crucial role and influence the behavior of the solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Measurability of the ellipses’ family and of the hyperbolae’s family in the affine spaceA 3

In this work we prove the measurability of the ellipses’ family in the affine spaceA ₃ and consequently we give its elementary measure; then we prove that also the hyperbolae’s family inA ₃ is measurable and we give its elementary measure, too.     

On the product of twists of rank two and a conjecture of Larsen

In this note we present examples of elliptic curves and infinite parametric families of pairs of integers (d,d′) such that, if we assume the parity conjecture, we can show that E ^d ,E ^d′ and E ^dd′ are all of positive even rank over ℚ. As an application, we show examples where a conjecture of M. Larsen holds.      

Average densities of frequency bands of wavelets

A function is called a wavelet if its integral translations and dyadic dilations form an orthonormal basis for L ²(?). The support of the Fourier transform of a wavelet is called its frequency band. In this paper, we study the relation between diameters and measures of frequency bands of wavelets, precisely say, we study the ratio of the measure to the diameter. This reflects the average density of the frequency band of a wavelet. In particular, for multiresolution analysis (MRA) wavelets, we do further research. First, we discuss the relation between diameters and measures of frequency bands of scaling functions. Next, we discuss the relation between frequency bands of wavelets and the corresponding scaling functions. Finally, we give the precise estimate of the measure of frequency bands of wavelets. At the same time, we find that when the diameters of frequency bands tend to infinity, the average densities tend to zero.     

Invariants of modular groups

For a faithful linear representation of a finite group G over a field of characteristic p, we study the ring of invariants. We especially study the polynomial and Cohen–Macaulay properties of the invariant ring. We first show that certain quotient rings of the invariant ring are polynomial rings by which we prove that the Hilbert ideal conjecture is true for a class of groups. In particular, we prove that the conjecture is true for vector invariant rings of Abelian reflection p-groups. Then we study the relationships between the invariant ring of G and that of a subgroup of G. Finally, we study the invariant rings of affine groups and show that, over a finite field, if an affine group contains all translations then the invariant ring is isomorphic to the invariant ring of a linear group.     

Constancy of p-harmonic maps of finite q-energy into non-positively curved manifolds

We investigate p-harmonic maps, p ≥ 2, from a complete non-compact manifold into a non-positively curved target. First, we establish a uniqueness result for the p-harmonic representative in the homotopy class of a constant map. Next, we derive a Caccioppoli inequality for the energy density of a p-harmonic map and we prove a companion Liouville type theorem, provided the domain manifold supports a Sobolev–Poincaré inequality. Finally, we obtain energy estimates for a p-harmonic map converging, with a certain speed, to a given point.      

-extension of Lie triple systems

In this paper, we introduce the notion of T^*-extension of a Lie triple system. Then we show that T^*-extension is compatible with nilpotency, solvability, and it preserves in certain sense the decomposition properties. In addition, we investigate the equivalence of T^*-extensions using cohomology. Finally, we show that every finite-dimensional nilpotent metrised Lie triple system over an algebraically closed field is the T^*-extension of an appropriate quotient system.     

Completions of local morphisms and valuations

In this note we study injective local morphisms of local excellent domains. In particular we are interested in the problem when the –adic topology onS restricts to a topology on R that is linearly equivalent to the –adic topology. Using a valuative criterion, we prove this in case R is analytically irreducible and is essentially of finite type, and we recover and extend a weak version of Gabrielov's rank condition. Received July 30, 1999; in final form November 16, 1999 / Published online December 8, 2000     

The semilocal convergence of a generalization of Brent's and Brown's methods

In this paper, we discuss the semilocal convergence of Martínez's generalization of Brent's and Brown's methods. Through a careful investigation of the algorithm structure, we convert Martínez's generalized method into an approximate Newton method with a special error term. Based on such equivalent variation, we prove the semilocal convergence theorem of Martínez's generalized method. This is a complementary result to the convergence theory of Martínez's generalized method.