some projection methods with the bb step sizes for variational inequalities since the appearance of the barzilai borwein bb step sizes strategy for unconstrained optimization problems it received more and more attention of the researchers it was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints in this paper we further extend the bb step sizes to more general variational inequality vi problems i e we adopt them in projection methods under the condition that the underlying mapping of the vi problem is strongly monotone and lipschitz continuous and the modulus of strong monotonicity and the lipschitz constant satisfy some further conditions we establish the global convergence of the projection methods with bb step sizes a series of numerical examples are presented which demonstrate that the proposed methods are convergent under mild conditions and are more efficient than some classical projection like methods

supplements to known inequalities for some special functions abstract we establish some inequalities for the bessel functions of the first kind j x and for the modified bessel functions i x and k x some of these results are obtained using the method of giordano and laforgia which is based essentially on the arithmetic geometric mean inequality other results are established as a consequence of the classical holder and gram inequalities

inequalities involving generalized interpolation polynomials the generalized vandermonde determinant and the generalized interpolation polynomial are defined we establish inequalities involving generalized interpolation polynomials under the proper hypotheses our main result is j 1nv n x j u a v n x a y j 2n n 1 i 0 n 12 j 1 n 12 1i j n n 1 2 v n x n q x 1 q qx n q a j 1n y j v n xx n a the approach used is able to combine algebraic and analytical theories

amos type bounds for modified bessel function ratios we systematically investigate lower and upper bounds for the modified bessel function ratio formula see text by functions of the form formula see text in case formula see text is positive for all formula see text or equivalently where formula see text or formula see text is a negative integer for formula see text we give an explicit description of the set of lower bounds and show that it has a greatest element we also characterize the set of upper bounds and its minimal elements if formula see text the minimal elements are tangent to formula see text in exactly one point formula see text and have formula see text as their lower envelope we also provide a new family of explicitly computable upper bounds finally if formula see text is a negative integer we explicitly describe the sets of lower and upper bounds and give their greatest and least elements respectively

differentiability and dimension of some fractal fourier series taking into account that the frequencies of fourier expansions are integers it is not surprising that some arithmetical results play an important role in many theorems and examples in harmonic analysis this relation is even clearer from the historical point of view and hardy and littlewood can be considered at the same time as founders of a substantial part of harmonic analysis and analytic number theory for instance in ha li they considered

fractal wavelet dimensions and localization in this paper we want to give a new definition of fractal dimensions as small scale behavior of theq energy of wavelet transforms this is a generalization of previous multi fractal approaches with this particular definition we will show that the 2 dimension correlation dimension of the spectral measure determines the long time behavior of the time evolution generated by a bounded self adjoint operator acting in some hilbert space it will be proved that for we have mathop lim inf limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 and that mathop lim sup limits t to infty frac log int 0 t d omega left left langle psi left e ia omega right phi right rangle right 2 log t kappa 2 wherek 2 are the upper and lower correlation dimensions of the spectral measure associated with and a quantitative version of the rage theorem shall also be given

continuous cascades in the wavelet space as models for synthetic turbulence we introduce a wide family of stochastic processes that are obtained as sums of self similar localized wave forms with multiplicative intensity in the spirit of the richardson cascade picture of turbulence we establish the convergence and the minimum regularity of our construction we show that its continuous wavelet transform is characterized by stochastic self similarity and multifractal scaling properties this model constitutes a stationary grid free extension of w cascades introduced in the past by arneodo bacry and muzy using a wavelet orthogonal basis moreover our approach generically provides multifractal random functions that are not invariant by time reversal and therefore is able to account for skewed multifractal models and for the so called leverage effect in that respect it can be well suited to providing synthetic turbulence models or to reproducing the main observed features of asset price fluctuations in financial markets

a self adaptive projection method with improved step size for solving variational inequalities in this paper we propose a new projection method for solving variational inequality problems which can be viewed as an improvement of the method of han and lo d r han hong k lo two new self adaptive projection methods for variational inequality problems computers mathematics with applications 43 2002 1529 1537 by adopting a new step size rule the method is as simple as han and lo s methods d r han hong k lo two new self adaptive projection methods for variational inequality problems computers mathematics with applications 43 2002 1529 1537 and other extra gradient type methods which uses only function evolutions and projections onto the feasible set we prove that under the condition that the underlying function is co coercive the sequence generated by the method converges to a solution of the variational inequality problem globally some preliminary computational results are reported which illustrated that the new method is more efficient than han and lo s method d r han hong k lo two new self adaptive projection methods for variational inequality problems computers mathematics with applications 43 2002 1529 1537

more on the analysis of local regularity through wavelets in this paper we want to extend the results of pointwise analysis through wavelet transforms to the class of functions where the local fluctuation is bounded by any submultiplicative function this generalizez the results obtained before in the well known case of holder regularity

functional calculus using wavelet transforms it is shown how the wavelet transform may be used to compute for a function s the symbol s a for any not necessarily self adjoint operator a whose spectrum is contained in the upper half plane for self adjoint operators it is shown that this functional calculus coincides with the usual one in particular it is shown how the exponential eita can be written in terms of the resolvent rz a z 1 of a as follows eita 1 c 0 da an 2 dbg at eitbrb ian a with c 2i 0 d i n 1g e and n n and the integral is understood as the cesaro limit this shows explicitly how the behavior for large t is determined by the behavior of rz at iz 1 t

inequalities involving modified bessel functions of the first kind ii the intrinsic properties including logarithmic convexity concavity of the modified bessel functions of the first kind and some other related functions are obtained several inequalities involving functions under discussion are established

a partial parallel splitting augmented lagrangian method for solving constrained matrix optimization problems alternating directions methods adms are very effective for solving convex optimization problems with separable structure however when these methods are applied to solve convex optimization problems with three separable operators their convergence results have not been established as yet in this paper we consider a class of constrained matrix optimization problems the problem is first reformulated into a convex optimization problem with three separable operators then it is solved by a proposed partial parallel splitting method the proposed method combines the parallel splitting augmented lagrangian method psalm and the alternating directions method adm and it is referred to as padalm in short the main difference between padalm and psalm is that in padalm two operators are handled first by a parallel method then the third operator and the former two are dealt with by an alternating method finally the convergence result for padalm is established and numerical results are provided to show the efficacy of padalm and its superiority over psalm

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

a globally convergent newton method for solving strongly monotone variational inequalities variational inequality problems have been used to formulate and study equilibrium problems which arise in many fields including economics operations research and regional sciences for solving variational inequality problems various iterative methods such as projection methods and the nonlinear jacobi method have been developed these methods are convergent to a solution under certain conditions but their rates of convergence are typically linear in this paper we propose to modify the newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length the purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate it is then shown that under the strong monotonicity assumption the method is globally convergent and under some additional assumptions the rate of convergence is quadratic limited computational experience indicates the high efficiency of the proposed method

convergence tests on constant dirichlet series by means of the theory of mathematical analysis this paper derives the convergence tests for the following two kinds of constant dirichlet series and an improper integral n 1 p n f j n d n n 1 1 n 1 f j n d n and 1 p t f j t d t dt as applications we give two assertions to demonstrate the effectiveness of these convergence tests

continuous wavelet transforms on the sphere in this very short paper we shall construct a continuous wavelet analysis based on dilations translations and rotations on the sphere it is the analog of the construction proposed by murenzi in his thesis 1990 on r2 at small scale we shall recover the euclidian structure of the sphere at large scale we obtain that the wavelet transform decays rapidly because the sphere is compact

on the wavelet transformation of fractal objects the wavelet transformation is briefly presented it is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half plane near the boundary of its domain of analyticity as an example a weierstrass like fractal function is considered for which the wavelet transform is related to a jacobi theta function some of the scalings of this theta function are analyzed and give some information about the scaling behavior of this fractal

wavelet spectra compared to fourier spectra the relation between fourier spectra and spectra obtained from wavelet analysis is established small scale asymptotic analysis shows that the wavelet spectrum is meaningful only when the analyzing wavelet has enough vanishing moments these results are related to regularity theorems in besov spaces for the analysis of infinitely regular signals a new wavelet with an infinite number of cancellations is proposed

a modified projection method for solving co coercive variational inequalities abstract this paper presents a modified projection method for solving variational inequalities which can be viewed as an improvement of the method of yan han and sun x h yan d r han w y sun a modified projection method with a new direction for solving variational inequalities applied mathematics and computation 211 2009 118 129 by adopting a new prediction step under the same assumptions we establish the global convergence of the proposed algorithm some preliminary computational results are reported

on the determination of box dimensions by means of wavelet transforms sufficient conditions are established for the determination of box dimensions of graphs from the decrease of the corresponding wavelet transforms as an application wavelet weierstrass functions are constructed

general inversion formulas for wavelet transforms this article is the continuation of a series of articles about group theory and wavelet analysis a grossmann j morlet and t paul j math phys 26 2473 1985 as is well known in the case of the afine group the reconstruction wavelet and the analyzing wavelet need not be identic in this article it is shown that this holds for arbitrary groups in addition it is shown that even for nonadmissible analyzing wavelets the wavelet transform may be inverted accordingly the image of the wavelet transform can be characterized by many different reproducing kernels

self affine fractal functions and wavelet series abstract we consider functions represented by series g g c g g 1 x of wavelet type where g is a group generated by affine functions l 1 l n and is piecewise affine by means of those functions we characterize the class of self affine fractal functions previously studied by barnsley et al we compute their global and local holder exponents and investigate points of non differentiability wavelet representations for various continuous nowhere differentiable and singular functions are presented another application is the construction of functions with prescribed local holder exponents at each point

multifractal behavior of polynomial fourier series abstract we study the spectrum of singularities of a family of fourier series with polynomial frequencies in particular we prove that they are multifractal functions the case of degree two was treated by s jaffard in 1996 higher degrees require completely different ideas essentially because harmonic analysis techniques poisson summation are useless to study the oscillation at most of the points we introduce a new approach involving special diophantine approximations with prime power denominators and fine analytic and arithmetic aspects of the estimation of exponential sums to control the holder exponent in thin cantor like sets

pointwise analysis of riemann s nondifferentiable function we will show how to analyse the local regularity of functions with the help of the wavelet transform these results will be applied to the function of riemann where we show the existence of a dense set of points where this function is differentiable on another dense set we show the existence of local singularities of cusp type on a third set we show differentiability to the right left on the remaining set the functions will be shown to be not differentiable

a new modified goldstein levitin polyakprojection method for variational inequality problems in this paper we first show that the adjustment parameter in the step size choice strategy of the modified goldstein levitin polyak projection method proposed by he et al for asymmetric strongly monotone variational inequality problems can be bounded away from zero by a positive constant under this observation we propose a new step size rule which seems to be more practical and robust than the original one we show that the new modified method is globally convergent under the same conditions and report some computational results to illustrate the method

two new self adaptive projection methods for variational inequality problems abstract in this paper we propose two new projection methods for solving variational inequality problems vi the method is simple it uses only function evaluation and projection onto the feasible set under the conditions that the underlying function is continuous and satisfies some generalized monotonicity assumption the methods are proven to converge to a solution of variational inequality globally some preliminary computational results are reported to illustrate the efficiency of the methods

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

a generalization of fourier trigonometric series in this paper by using the extended sturm liouville theorem for symmetric functions we introduce the differential equation f n t n a 1 1 n 2 2 a a 1 cos 2t 1 1 n 2 f n t 0 as a generalization of the differential equation of trigonometric sequences sinnt n 1 and cosnt n 0 for a 0 and obtain its explicit solution in a simple trigonometric form we then prove that the obtained sequence of solutions is orthogonal with respect to the constant weight function on 0 p and compute its norm square value explicitly one of the important advantages of this generalization is to find some new infinite series a practical example is given in this sense

new bounds for the generalized marcum q function in this paper we study the generalized marcum q function q nu a b where a nu 0 and b ges 0 our aim is to extend the results of corazza and ferrari ieee trans inf theory vol 48 pp 3003 3008 2002 to the generalized marcum q function in order to deduce some new tight lower and upper bounds the key tools in our proofs are some monotonicity properties of certain functions involving the modified bessel function of the first kind and some classical inequalities i e the cauchy buniakowski schwarz and chebyshev integral inequalities these bounds are shown to be very tight for large b i e the relative errors of our bounds converge to zero as b increases both theoretical analysis and numerical results are provided to show the tightness of our bounds

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 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 turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

wavelet expansions and asymptotic behavior of distributions abstract we develop a distribution wavelet expansion theory for the space of highly time frequency localized test functions over the real line s 0 r s r and its dual space s 0 r namely the quotient of the space of tempered distributions modulo polynomials we prove that the wavelet expansions of tempered distributions converge in s 0 r a characterization of boundedness and convergence in s 0 r is obtained in terms of wavelet coefficients our results are then applied to study local and non local asymptotic properties of schwartz distributions via wavelet expansions we provide abelian and tauberian type results relating the asymptotic behavior of tempered distributions with the asymptotics of wavelet coefficients

modified self adaptive projection method for solving pseudomonotone variational inequalities abstract in this paper a self adaptive projection method with a new search direction for solving pseudomonotone variational inequality vi problems is proposed which can be viewed as an extension of the methods in b s he x m yuan j z zhang comparison of two kinds of prediction correction methods for monotone variational inequalities computational optimization and applications 27 2004 247 267 and x h yan d r han w y sun a self adaptive projection method with improved step size for solving variational inequalities computers mathematics with applications 55 2008 819 832 the descent property of the new search direction is proved which is useful to guarantee the convergence under the relatively relaxed condition that f is continuous and pseudomonotone the global convergence of the proposed method is proved numerical experiments are provided to illustrate the efficiency of the proposed method

on the norm of an integral operator and applications abstract in this paper the norm of an integral operator t l r 0 l r 0 r 1 is obtained as applications a new bilinear integral operator inequality with the norm and the equivalent forms are given and some new hilbert s type inequalities with the best constant factors as the particular cases are established

tight bounds for the generalized marcum q function abstract in this paper we study the generalized marcum q function of order 0 real defined by q a b 1 a 1 b t e t 2 a 2 2 i 1 a t dt where a 0 b 0 and i stands for the modified bessel function of the first kind our aim is to improve and extend some recent results of wang to the generalized marcum q function in order to deduce some sharp lower and upper bounds in both cases b a and b a we give the best possible upper bound for q a b the key tools in our proofs are some monotonicity properties of certain functions involving the modified bessel function of the first kind these monotonicity properties are deduced from some results on modified bessel functions which have been used in wave mechanics and finite elasticity

on cubic lacunary fourier series for 2 4 we analyze the behavior near the rational points x p q of n 1 n exp ixn 3 considered as a function of x we expand this series into a constant term a term on the order of x p q 1 3 a term linear in x p q a chirp term on the order of x p q 2 1 4 and an error term on the order of x p q 2 at every such rational point the left and right derivatives are either both finite and equal or both infinite in contrast with the quadratic series where the derivative is often finite on one side and infinite on the other however in the cubic series again in contrast with the quadratic case the chirp term generally has a different set of frequencies and amplitudes on the right and left sides finally we show that almost every irrational point can be closely approximated in a suitable diophantine sense by rational points where the cubic series has an infinite derivative this implies that when 97 1 4 2 212 both the real and imaginary parts of the cubic series are differentiable almost nowhere

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac