Behaviour of Exponential Splines as Tensions Increase without Bound

Schweikert (J. Math. Phys.45(1966), 312–317) showed that for sufficiently high tensions an exponential spline would have no more changes in sign of its second derivative than there were changes in the sign of successive second differences of its knot sequence. Späth (Computing4(1969), 225–233) proved the analogous result for first derivatives, assuming uniform tension throughout the spline. Later, Pruess (J. Approx. Theory17(1976), 86–96) extended Späth's result to the case where the inter-knot tensionsp_imay not all be the same but tend to infinity at the same asymptotic growth rate, in the sense thatp_i∈Θ(p₁) for alli. This paper extends Pruess's result by showing his hypothesis of uniform boundedness of the tensions to be unnecessary. A corollary is the fact that for high enough minimum interknot tension, the exponential spline through monotone knots will be a ²monotone curve. In addition, qualitative bounds on the difference in slopes between the interpolating polygon and the exponential spline are developed, which show that Gibbs-like behaviour of the spline's derivative cannot occur in the neighbourhood of the knots.     

Exact Enumeration of 1342-Avoiding Permutations: A Close Link with Labeled Trees and Planar Maps

Solving the first nonmonotonic, longer-than-three instance of a classic enumeration problem, we obtain the generating functionH(x) of all 1342-avoiding permutations of lengthnas well as anexactformula for their numberS_n(1342). While achieving this, we bijectively prove that the number of indecomposable 1342-avoiding permutations of lengthnequals that of labeled plane trees of a certain type onnvertices recently enumerated by Cori, Jacquard, and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte (Can. J. Math.33(1963), 249–271). Moreover,H(x) turns out to be algebraic, proving the first nonmonotonic, longer-than-three instance of a conjecture of Noonan and Zeilberger (Adv. Appl. Math.17(1996), 381–407). We also prove thatconverges to 8, so in particular, lim_n→∞(S_n(1342)/S_n(1234))=0.     

Summability of Lagrange type interpolation series

We consider certain aspects of the theory of interpolation via entire functions of exponential type, which include sampling node sequences χ which can be highly irregular and data sequences {y(_x)}_xn∈χ which are not necessarily bounded. Under appropriate conditions, we show that there is an entire function of a suitable exponential type which uniquely interpolates the data and indicate the validity of certain summabilty methods for the corresponding Lagrange type interpolation series. Some of our results significantly extend the work of Schoenberg,Cardinal interpolation and spline functions VII: The behavior of cardinal spline interpolation as their degree tends to infinity, J. Analyse Math.27 (1974), 205–229.     

Efficient Preprocessing of Simple Binary Pattern Forests

Multipattern matching in trees is fundamental to a variety of programming language systems. A bottleneck in this connection has been the combinatorial problem of constructing the immediate subsumption tree for a simple binary pattern forest. We reduce the complexity of this problem fromO(n²) time andO(n²) space toO(n log n) time andO(n) space. Such a result was conjectured possible in 1982 by C. Hoffmann and J. O'Donnell (J. Assoc. Comput. Mach.29(1) (1982), 68–95) and in 1992 finding it was called a main open problem by J. Cai, R. Paige, and R. Tarjan (J. Theoret. Comput. Sci.106(1992), 21–60).     

On the Number of Blocks in a Generalized Steiner System

We considert-designs withλ=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. Fort=2, this inequality is the well known De Bruijn–Erd?s inequality. Fort>2 it has the same order of magnitude as the Wilson–Petrenjuk inequality for Steiner systems with constant block size. The point of this note is that the inequality is very easy to derive and does not seem to be known. A stronger inequality was derived in 1969 by Woodall (J. London Math. Soc.(2)1, 509–519), but it requires Lagrange multipliers in the proof.     

A Combinatorial Algorithm Related to the Geometry of the Moduli Space of Pointed Curves

As pointed out in Arbarello and Cornalba (J. Alg. Geom. 5 (1996), 705–749), a theorem due to Di Francesco, Itzykson, and Zuber (see Di Francesco, Itzykson, and Zuber, Commun. Math. Phys. 151 (1993), 193–219) should yield new relations among cohomology classes of the moduli space of pointed curves. The coefficients appearing in these new relations can be determined by the algorithm we introduce in this paper.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

Some argument inequalities for certain analytic functions

For analytic functions f(z) in the open unit disk U and convex functions g(z) in U, Nunokawa et al. [NUNOKAWA, M.—OWA, S.—NISHIWAKI, J.—KUROKI, K.—HAYAMI, T: Differential subordination and argumental property, Comput. Math. Appl. 56 (2008), 2733–2736] have proved one theorem which is a generalization of the result [POMMERENKE, CH.: On close-toconvex analytic functions, Trans. Amer. Math. Soc. 114 (1965), 176–186]. The object of the present paper is to generalize the theorem due to Nunokawa et al..     

On the cyclic homology of commutative algebras over arbitrary ground rings

Abstract

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).     

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations, by using a fixed point theorem in the mixed monotone operators （Ma, Y.： On a class of mixed monotone operators and a kind of two-point bounded value problem. Indian J. Math., 41（2）, 211-220 （1999]]. Some known results are operators.     

On the intersection of two subgeometries of PG(n, q)

The study of the intersection of two Baer subgeometries of PG(n, q), q a square, started in Bose et al. (Utilitas Math 17, 65–77, 1980); Bruen (Arch Math 39(3), 285–288, (1982). Later, in Svéd (Baer subspaces in the n-dimensional projective space. Springer-Verlag) and Jagos et al. (Acta Sci Math 69(1–2), 419–429, 2003), the structure of the intersection of two Baer subgeometries of PG(n, q) has been completely determined. In this paper, generalizing the previous results, we determine all possible intersection configurations of any two subgeometries of PG(n, q).      

Measures of Non-compactness of Operators on Banach Lattices

[Indag. Math.(N.S.) 2(2) (1991), 149–158; Uspehi Mat. Nauk 27(1(163)) (1972), 81–146] used representation spaces to study measures of non-compactness and spectral radii of operators on Banach lattices. In this paper, we develop representation spaces based on the nonstandard hull construction (which is equivalent to the ultrapower construction). As a particular application, we present a simple proof and some extensions of the main result of [J. Funct. Anal. 78(1) (1988), 31–55] on the monotonicity of the measure of non-compactness and the spectral radius of AM-compact operators. We also use the representation spaces to characterize d-convergence and discuss the relationship between d-convergence and the measures of non-compactness.     

Singularity and decay estimates for the conformal k-hessian equation via Liouville-type theorems

We study some connections between Liouville type theorems and local properties of nonnegative solutions to conformal k-hessian equations by making use of an elementary lemma for all positive functions in Li and Zhang (J. Anal. Math. 90 (2003), 27–87) and related Liouville type theorems in Li and Li (Acta. Math. 195 (2005), 117–154). Research of the second author is supported by Tianyuan Fund of Mathematics (10826061).     

Existence and uniqueness of an invariant measure for a chain of oscillators in contact with two heat baths

In this note we consider a chain of N$N$ oscillators, whose ends are in contact with two heat baths at different temperatures. Our main result is the exponential convergence to the unique invariant probability measure (the stationary state). We use the Lyapunov’s function technique of Rey-Bellet and coauthors [Luc Rey-Bellet, Statistical mechanics of anharmonic lattices, in: Advances in Differential Equations and Mathematical Physics (Birmingham, AL, 2002), in: Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 283–298. MR MR1991548 (2005a:82068) [11]; Luc Rey-Bellet, Lawrence E. Thomas, Fluctuations of the entropy production in anharmonic chains, Ann. Henri Poincaré 3 (3) (2002) 483–502. MR MR1915300 (2003g:82060); Luc Rey-Bellet, Lawrence E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Comm. Math. Phys. 225 (2) (2002) 305–329. MR MR1889227 (2003f:82052); Luc Rey-Bellet, Lawrence E. Thomas, Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators, Comm. Math. Phys. 215 (1) (2000) 1–24. MR MR1799873 (2001k:82061) [12]; Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Comm. Math. Phys. 201 (3) (1999) 657–697. MR MR1685893 (2000d:82025); Jean-Pierre Eckmann, Claude-Alain Pillet, Luc Rey-Bellet, Entropy production in nonlinear, thermally driven Hamiltonian systems, J. Statist. Phys. 95 (1–2) (1999) 305–331. MR MR1705589 (2000h:82075)], with different model of heat baths, and adapt these techniques to two new case recently considered in the literature by Bernardin and Olla [Cédric Bernardin, Stefano Olla, Fourier’s law for a microscopic model of heat conduction, J. Statist. Phys. 121 (3–4) (2005) 271–289. MR MR2185330] and Lefevere and Schenkel [R. Lefevere, A. Schenkel, Normal heat conductivity in a strongly pinned chain of anharmonic oscillators, J. Stat. Mech. Theory Exp. 2006 (02) (2006) L02001].     

Some New Inverse-type Hilbert-Pachpatte Integral Inequalities

In this paper,some new generalizations of inverse type Hilbert-Pachpatte integral inequalities are proved.The results of this paper reduce to those of Pachpatte(1998,J.Math.Anal.Appl.226,166-179)and Zhan and Debnath(2001,J.Math.Anal.Appl.262,411-418).     

Common extension of bilateral series for Andrews’ q-Bailey and q-Gauss sums

By means of the modified Abel lemma on summation by parts, we establish a common bilateral series extension of the q-Bailey and q-Gauss sums discovered by Andrews (Duke Math. J. 40 (1973), 525–528).     

Little q-Legendre Polynomials and Irrationality of Certain Lambert Series

Certain q-analogs h _p(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erds (J. Indiana Math. Soc. 12, 1948, 63–66). In 1991–1992 Peter Borwein (J. Number Theory 37, 1991, 253–259; Proc. Cambridge Philos. Soc. 112, 1992, 141–146) used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs ln _p (2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger (Adv. Appl. Math. 20, 1998, 275–283) used the qEKHAD symbolic package to find q-WZ pairs that provide a proof of irrationality similar to Apéry's proof of irrationality of (2) and (3). They also obtain an upper bound for the measure of irrationality, but better upper bounds were earlier given by Bundschuh and Väänänen (Compositio Math. 91, 1994, 175–199) and recently also by Matala-aho and Väänänen (Bull. Australian Math. Soc. 58, 1998, 15–31) (for ln _p (2)). In this paper we show how one can obtain rational approximants for h _p(1) and ln _p (2) (and many other similar quantities) by Padé approximation using little q-Legendre polynomials and we show that properties of these orthogonal polynomials indeed prove the irrationality, with an upper bound of the measure of irrationality which is as sharp as the upper bound given by Bundschuh and Väänänen for h _p(1) and a better upper bound as the one given by Matala-aho and Väänänen for ln _p (2).     

Inverse Backscattering Problem for the Acoustic Equation in Even Dimensions

We show that the sound speedc(x) of the acoustic wave equation in any even dimension can be uniquely determined by the backscattering data provided that it is close to a constant. In the three-dimensional case, P. Stefanov and G. Uhlmann (SIAM J. Math. Anal.28,1997, 1191–1204) have proved a similar result. Their method takes advantage of the inversion formula for the Radon transform in odd dimensions being a local operator. This is not true in even dimensions. Moreover, the odd-dimensional Lax and Phillips modified Radon transform fails to work in even dimensions. In this paper, we overcome these difficulties and prove an even-dimensional version of Stefanov and Uhlmann's result.     

A shorter proof of Kanter’s Bessel function concentration bound

We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I₀(x) + I₁(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Čekanavičius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).