Solution of a problem of Ulam

In this paper we solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist” and establish results involving a product of powers of norms [[5.]; [5.]; [5.]]. There has been much activity on a similar “ -isometry” problem of Ulam [ [1.], 633–636; [2.], 263–277; [4.]]. This work represents an improvement and generalization of the work of [3.], 222–224].     

Corrigendum to “A note on the Loewenstein–Prelec theory of intertemporal choice” [Math. Social Sci. 52 (1) (2006) 99–108]

In a critique of the Loewenstein and Prelec [Loewenstein G., Prelec D., 1992. Anomalies in intertemporal choice: Evidence and an interpretation. The Quarterly Journal of Economics 107, 573–597] theory of intertemporal choice, [al-Nowaihi, A., Dhami, S., 2006. A note on the Loewenstein–Prelec theory of intertemporal choice. Mathematical Social Sciences 52, 99–108] point out four errors. One of the alleged errors was that the elasticity of the value function in prospect theory is decreasing. But it is in fact increasing. We provide a correction and a formal proof. As a corollary, we show that the elasticity of the value function is bounded between zero and one. Nevertheless, all the remaining points in [al-Nowaihi, A., Dhami, S., 2006. A note on the Loewenstein–Prelec theory of intertemporal choice. Mathematical Social Sciences 52, 99–108] remain valid     

Cross-validation and the smoothing of orthogonal series density estimators

We describe a class of smoothed orthogonal series density estimates, including the classical sequential-series introduced by [6], Soviet Math. Dokl. 3 1559–1562) and [16], Ann. Math. Statist. 38 1261–1265), and [23], Ann. Statist 9 146–156) two-parameter smoothing. The Bowman-Rudemo method of least-squares cross-validation (1982, Manchester-Sheffield School of Probability and Statistics Research Report 84/AWB/1; 1984, Biometrika 71 353–360; [14], Scand. J. Statist. 9 65–78), is suggested as a practical way of choosing smoothing parameters automatically. Using techniques of [18], Ann. Statist. 12 1285–1297), that method is shown to perform asymptotically optimally in the case of cosine and Hermite series estimators. The same argument may be used for other types of series.     

A new proof of some polynomial inequalities related to pseudo-splines

Pseudo-splines of type I were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1–46] and [Selenick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163–181] and type II were introduced in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104]. Both types of pseudo-splines provide a rich family of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. In [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], Dong and Shen gave a regularity analysis of pseudo-splines of both types. The key to regularity analysis is Proposition 3.2 in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], which also appeared in [A. Cohen, J.P. Conze, Régularité des bases d'ondelettes et mesures ergodiques, Rev. Mat. Iberoamericana 8 (1992) 351–365] and [I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992] for the case l=N−1. In this note, we will give a new insight into this proposition.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

Stability of transversally isotropic shells coupled with an elastic foundation

The stability of shells coupled with an elastic Winkler foundation is investigated. It is assumed that the shell is made of a material (glass-reinforced plastic) with low resistance to shear, as a result of which generalized theories that take transverse shear strains into account [1–4] must be used in the stability calculations. The solution obtained is compared with the corresponding solution obtained on the basis of the classical Kirchhoff-Love theory [8].Lvov Polytechnic Institute. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 669–673, July–August, 1969.     

On the applicability of two-dimensional applied theories to problems of the stability of axially loaded cylindrical shells made of materials with low shear stiffness

The applicability of two-dimensional applied theories of the Kirchhoff-Love, Timoshenko, and Ambartsumyan types to problems of the stability of shells with low shear stiffness [1] is discussed. The critical loads given by these theories are compared with those recently obtained [6–8] by solving the problem of the stability of a cylindrical shell on the basis of general solutions [3, 4] of the three-dimensional linearized equations of the theory of elasticity [5].Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. L'vov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 1, pp. 141–143, January–February, 1970.     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

State of stress and strain of plates with two equal openings in the presence of high-elastic deformation

The apparatus of the nonlinear theory of elasticity [1–3] is used to investigate the high-elastic deformation of a plate with two equal circular openings and a massive block with two cylindrical channels. Computer-calculated stress concentration factors are given for compressible and incompressible materials.Translated from Mekhanika Polimerov, No. 4, pp. 687–692, July–August, 1969.     

Selections and countably-approachable points

The present paper improves a result of Gutev [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by characterizing the countably-approachable points in sense of [V. Gutev, Approaching points by continuous selections, J. Math. Soc. Japan (4) (2006) 1203–1210] by a natural extreme-like condition in the spirit of [V. Gutev, T. Nogura, Vietoris continuous selections and disconnectedness-like properties, Proc. Amer. Math. Soc. 129 (2001) 2809–2815; V. Gutev, T. Nogura, Selection pointwise-maximal spaces, Topology Appl. 146–147 (2005) 397–408]. This demonstrates the natural relationship between different extreme-like points with respect to continuous selections for the Vietoris hyperspace of nonempty closed subsets.     

An extension of the Wang transform derived from Bühlmann’s economic premium principle for insurance risk

It is well known that the Wang transform [Wang, S.S., 2002. A universal framework for pricing financial and insurance risks. Astin Bull. 32, 213–234] for the pricing of financial and insurance risks is derived from Bühlmann’s economic premium principle [Bühlmann, H., 1980. An economic premium principle. Astin Bull. 11, 52–60]. The transform is extended to the multivariate setting by [Kijima M., 2006. A multivariate extension of equilibrium pricing transforms: The multivariate Esscher and Wang transforms for pricing financial and insurance risks, Astin Bull. 36, 269–283]. This paper further extends the results to derive a class of probability transforms that are consistent with Bühlmann’s pricing formula. The class of transforms is extended to the multivariate setting by using a Gaussian copula, while the multiperiod extension is also possible within the equilibrium pricing framework.     

On the thermodynamic limit for Hartree–Fock type models

We continue here our study [10–13] of the thermodynamic limit for various models of Quantum Chemistry, this time focusing on the Hartree–Fock type models. For the reduced Hartree–Fock models, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the Hartree–Fock model, and prove that it is well-posed.     

Standard Monomial Theory for Bott–Samelson Varieties

Bott–Samelson varieties are an important tool in geometric representation theory [1, 3, 10, 25]. They were originally defined as desingularizations of Schubert varieties and share many of the properties of Schubert varieties. They have an action of a Borel subgroup, and the projective coordinate ring of a Bott–Samelson variety splits into certain generalized Demazure modules (which also appear in other contexts [22, 23]). Standard Monomial Theory, developed by Seshadri and the first author [15, 16], and recently completed by the second author [20], gives explicit bases for the Demazure modules associated to Schubert varieties. In this paper, we extend the techniques of [20] to give explicit bases for the generalized Demazure modules associated to Bott–Samelson varieties, thus proving a strengthened form of the results announced by the first and third authors in [12] (see also [13]). We also obtain more elementary proofs of the cohomology vanishing theorems of Kumar [10] and Mathieu [25]; of the projective normality of Bott–Samelson varieties; and of the Demazure character formula.     

Creep of polymer materials under periodically varying loads

A method is proposed for constructing the creep curves of a material whose nonlinear memory properties are described by Rozovskii's nonlinear integral equation [2] (with allowance for the stress dependence of the relaxation time) under given periodic loading from known creep curves recorded at constant stress. In deriving the theoretical relation certain simplifying assumptions are made (the creep strain accumulated in 1–2 cycles is small, no vibration [4–6]). An experimental check shows that the proposed method can be used to predict the behavior of a material under periodic loading with an accuracy sufficient for practical purposes.Mekhanika Polimerov, Vol. 2, No. 3, pp. 330–336, 1966     

Even infinite-dimensional real Banach spaces

This article is a continuation of a paper of the first author [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] and C(K) examples due to Plebanek [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We extend results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis [S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1–32] provides examples of essentially incomparable complex structures which are not totally incomparable.     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. III

This paper is a continuation of [1]–[2].     

Contact problems for beams and plates with low shear stiffness

The low shear stiffness and strength of unidirectionally reinforced plates and beams predetermines the choice of calculation model in specific problems [4]. The contact problem is considered for a glass-reinforced plastic plate resting on a Winkler foundation; the deformation properties of the plate are described by the equations of an orthotropic material; the investigation is based on generalized applied theories of the Timoshenko and Ambartsumyan types [5–8], which permit the high shear compliance of thin-walled structures to be taken into account.L'vov Polytechnic Institute; Ivan Franko Institute of Petroleum and Gas. Translated from Mekhanika Polimerov, No. 4, pp. 715–720, July–August, 1970.     

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.     

On the poisedness of Bojanov–Xu interpolation

In this paper, we consider the bivariate Hermite interpolation introduced by Bojanov and Xu [SIAM J. Numer. Anal. 39(5) (2002) 1780–1793]. The nodes of the interpolation with Π_2k-δ, where δ=0 or 1, are the intersection points of 2k+1 distinct rays from the origin with a multiset of k+1-δ concentric circles. Parameters are the values and successive radial derivatives, whenever the corresponding circle is multiple. The poisedness of this interpolation was proved only for the set of equidistant rays [Bojanov and Xu, 2002] and its counterparts with other conic sections [Hakopian and Ismail, East J. Approx. 9 (2003) 251–267]. We show that the poisedness of this (k+1-δ)(2k+1) dimensional Hermite interpolation problem is equivalent to the poisedness of certain 2k+1 dimensional Lagrange interpolation problems. Then the poisedness of Bojanov–Xu interpolation for a wide family of sets of rays satisfying some simple conditions is established. Our results hold also with above circles replaced by ellipses, hyperbolas, and pairs of parallel lines.Next a conjecture [Hakopian and Ismail, J. Approx. Theory 116 (2002) 76–99] concerning a poisedness relation between the Bojanov–Xu interpolation, with set of rays symmetric about x-axis, and certain univariate lacunary interpolations is established. At the end the poisedness for a wide class of lacunary interpolations is obtained.