operators preserving orthogonality are isometries we prove that every operator preserving orthogonality in a real banach space is an isometry multiplied by a constant

dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

mappings approximately preserving orthogonality in normed spaces abstract we answer many open questions regarding approximately orthogonality preserving mappings in birkhoff james sense in normed spaces in particular we show that every approximately orthogonality preserving linear mapping in chmielinski sense is necessarily a scalar multiple of an e isometry thus whenever e isometries are close to isometries we obtain stability an example is given showing that approximately orthogonality preserving mappings are in general far from scalar multiples of isometries that is stability does not hold

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

the three dimensional dilute ising magnet the dilute ising model with a p 0 8 fraction of the sites occupied by spins is simulated on l3 systems for l up to 300 using a single cluster algorithm of wolff in the range of reduced temperature 0 002 t tc tc 0 03 the susceptibility appears to fit a simple power with an effective exponent eff 1 36

summability of the perturbative expansion for a zero dimensional disordered spin model we show analytically that the perturbative expansion for the free energy of the zero dimensional quenched disordered ising model is borel summable in a certain range of parameters provided that the summation is carried out in two steps first in the strength of the original coupling of the ising model and subsequently in the variance of the quenched disorder this result is illustrated by some high precision calculations of the free energy obtained by a straightforward numerical implementation of our sequential summation method

linear mappings approximately preserving orthogonality abstract we consider the class of linear mappings between real or complex inner product spaces such that for each two orthogonal vectors in the domain their values are orthogonal in the target space by orthogonality 0 1 we mean the relation u v u v u v

critical behavior of the specific heat for pure and site diluted simple cubic ising systems the exponent of the specific heatc is determined for the pure and the site diluted simple cubic ising model concentrationx 0 0 2 0 4 of nonmagnetic sites by a finite size scaling analysis of the peak value cmax l for systems of linear dimensionsl 8 16 32 and 64 the cmax values are obtained by the ferrenberg swendsen algorithm using monte carlo data from a fully vectorized multi spin coding program we obtain 0 11 for x 0 and a crossover to a negative value upon dilution with 0 029 4 both forx 0 2 andx 0 4

an identity for δ e approximately orthogonality preserving mappings abstract suppose t is a hilbert space operator given 0 1 we define e t to be the smallest e for which t is e approximately orthogonality preserving and then obtain an exact formula for e t in terms of t and the minimum modulus m t of t for two nonzero operators t s it follows from the formula that t is e s e aop if and only if s is e t e aop where e t e 0 t finally we show that an operator t is e aop if and only if there exists a special aop operator s such that ts is e aop theorem 3 8

random walks in cones we study the asymptotic behavior of a multidimensional random walk in a general cone we find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone the main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments for the proof of tail asymptotics and integral limit theorems we use a strong approximation of random walks by brownian motion for the proof of local limit theorems we suggest a rather simple approach which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks we also discuss some possible applications of our results to ordered random walks and lattice path enumeration

critical crossover phenomena in disordered ising systems the author has investigated the critical behaviour of site disordered ising systems over a large range of dilution 0 6 or p or 1 0 investing about 1015 single spin flips with conventional local dynamics critical temperatures have been determined with high precision using the cumulant method the finite size simulations lead to concentration dependent exponents which seem to violate universality the author shows that the concentration dependence is induced by a crossover phenomenon which governs a large interval of the critical region the author finds that the scaling functions of systems with different disorder converge asymptotically to universal behaviour consistent with the fixed point of weak disorder

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

approximately orthogonality preserving mappings on c modules abstract we study orthogonality preserving and approximately orthogonality preserving mappings in the setting of inner product c modules in particular if v and w are inner product c modules over the c algebra a any scalar multiple of an a linear isometry is an a linear orthogonality preserving mapping the converse does not hold in general but it holds if a contains k h the c algebra of all compact operators on a hilbert space h furthermore we give the estimate of t x t y t 2 x y for an a linear approximately orthogonality preserving mapping t v w when v and w are inner product c modules over a c algebra containing k h in the case a k h and v w are hilbert we also prove that an a linear approximately orthogonality preserving mapping can be approximated by an a linear orthogonality preserving mapping

critical behaviour of 3d systems with long range correlated quenched defects a field theoretic description of the critical behaviour of systems with quenched defects obeying power law correlations x a for large separations x is given directly for three dimensional systems and for different values of the correlation parameter 2 a3 a renormalization analysis of the scaling function in the two loop approximation is carried out and the fixed points corresponding to the stability of various types of critical behaviour are identified the obtained results essentially differ from results evaluated by a double expansion the critical exponents in the two loop approximation are calculated with the use of the pade borel summation technique

on mappings approximately preserving orthogonality abstract we answer a question posed by chmielinski whether a linear map which approximately preserves orthogonality must be close to an orthogonality preserving one furthermore we give a short proof of the stability of the orthogonality equation on finite dimensional hilbert spaces

conditioned limit theorems for products of random matrices consider the product g n g n dots g 1 of the random matrices g 1 dots g n in gl left d mathbb r right and the random process g n v g n dots g 1 v in mathbb r d starting at point v in mathbb r d smallsetminus left 0 right it is well known that under appropriate assumptions the sequence left log left vert g n v right vert right n geq 1 behaves like a sum of i i d r v s and satisfies standard classical properties such as the law of large numbers law of iterated logarithm and the central limit theorem denote by mathbb b the closed unit ball in mathbb r d and by mathbb b c its complement for any v in mathbb b c define the exit time of the random process g n v from mathbb b c by tau v min left n geq 1 g n v in mathbb b right we establish the asymptotic as n rightarrow infty of the probability of the event left tau v n right and find the limit law for the quantity frac 1 sqrt n log left vert g n v right vert conditioned that tau v n

critical behavior of the pure and random bond two dimensional triangular ising ferromagnet we investigate the effects of quenched bond randomness on the critical properties of the two dimensional ferromagnetic ising model embedded in a triangular lattice the system is studied in both the pure and disordered versions by the same efficient two stage wang landau method in the first part of our study we present the finite size scaling behavior of the pure model for which we calculate the critical amplitude of the specific heat s logarithmic expansion for the disordered system the numerical data and the relevant detailed finite size scaling analysis along the lines of the two well known scenarios logarithmic corrections versus weak universality strongly support the field theoretically predicted scenario of logarithmic corrections a particular interest is paid to the sample to sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality

isosceles orthogonality preserving property and its stability for real normed spaces we consider the class of linear operators preserving approximately the relation of isosceles orthogonality we show some general properties of such mappings next we examine whether an approximately orthogonality preserving mapping admits an approximation by an orthogonality preserving one in regard to this we generalize some results obtained earlier for inner product spaces with standard orthogonality relation

critical exponents of the dilute ising model from four loop expansions four loop expansions of the coefficients of the callan symanzik equation for 3d and 2d dilute ising models are calculated fixed point coordinates and critical exponents are estimated by two methods of summation of double series a generalisation of the pade borel approximation and the first confluent form of the e algorithm of wynn summation of the double series for the 2d dilute ising model gives exponents very close to exponents of the pure ising model in accordance with the exact solution

spitzer s condition and ladder variables in random walks spitzer s condition holds for a random walk if the probabilities n p n 0 converge in cesaro mean to where 0 1 we answer a question which was posed both by spitzer 12 and by emery 5 by showing that whenever this happens it is actually true that n converges to this also enables us to give an improved version of a result in doney and greenwood 4 and show that the random walk is in a domain of attraction without centering if and only if the first ladder epoch and height are in a bivariate domain of attraction

aop mappings and the distance to the scalar multiples of isometries in this paper linear e orthogonality preserving mappings are studied we define e t as the smallest e for which t is e orthogonality preserving and then derive an exact formula for e t in terms of t and the minimum modulus m t of t we see that e orthogonality preserving mappings for some e 1 are exactly the operators that are bounded from below we improve an upper bound in the stability equation given in 7 theorem 2 3 which was thought to be sharp

stability of the orthogonality preserving property in finite dimensional inner product spaces we establish a stability property for inner product preserving not necessarily linear mappings then as a consequence we show that a linear mapping defined on a finite dimensional inner product space which approximately preserves orthogonality can be approximated by a linear orthogonality preserving one

critical dynamics of cluster algorithms in the dilute ising model autocorrelation times for thermodynamic quantities att c are calculated from monte carlo simulations of the site diluted simple cubic ising model using the swendsen wang and wolff cluster algorithms our results show that for these algorithms the autocorrelation timesdecrease when reducing the concentration of magnetic sites from 100 down to 40 this is of crucial importance when estimating static properties of the model since the variances of these estimators increase with autocorrelation time the dynamical critical exponents are calculated for both algorithms observing pronounced finite size effects in the energy autocorrelation data for the algorithm of wolff we conclude that when applied to the dilute ising model cluster algorithms become even more effective than local algorithms for whichincreasing autocorrelation times are expected

static and dynamic critical properties of 3d systems with long range correlated quenched defects a field theoretic description of static and purely relaxational dynamic critical behaviour of systems with quenched defects obeying power law correlations x a for large separations x is given directly for three dimensional systems and for different values of the correlation parameter 2 a3 a renormalization analysis of the scaling functions in the two loop approximation is carried out and the fixed points corresponding to the stability of various types of critical behaviour are identified the obtained results essentially differ from results evaluated by a double expansion the static and dynamic critical exponents in the two loop approximation are calculated with the use of the pade borel summation technique

a conditioned limit theorem for random walk and brownian local time on square root boundaries we count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a local time whose inverse is a stable subordinator of a known index and a brownian motion the study of this limit process leads to some precise sample path properties of brownian motion these properties improve earlier results of dvoretsky and kahane on the existence of small oscillations in the brownian path

on the critical fluctuations in superconductors the field theoretical gauge model for a superconductor generalized to an n 2 component complex order parameter is considered the question of the order of the phase transition occurring in this model is discussed previous renormalization group calculations suggest that in such a model a fluctuation induced first order phase transition occurs we re examine previously obtained expressions for the renormalization group functions in the two loop approximation in three dimensions special attention is being paid to the fact that the corresponding series might be the asymptotic ones and therefore have zero radius of convergence we discuss the possible ways of analytical continuation of the series obtained comparing results obtained by direct calculations with those obtained by pade analysis and the pade borel resummation technique the conjecture is made that in the model under consideration there still exists a possibility for a second order phase transition

on mappings approximately transferring relations in finite dimensional normed spaces abstract we discuss the problem of stability of relations preserving property for finite dimensional normed spaces moreover we obtain similar results for other orthogonality relations next we show a counterexample for infinite dimensional normed spaces

nonequilibrium relaxation study on spin glass model ferromagnetic fm transition point and universality class of the j ising model on cubic lattice are studied by computer simulation using the nonequilibrium relaxation ner method at fm bond density p between multicritical one pmc and 1 static exponents take universal values which are different from those of p 1 or p pmc and they are consistent with the values of the random fixed point and of the simulations and experiments for diluted ising systems the dynamical exponent however seems to depend on p and not to be universal although it may be due to finite time correction the estimation of pmc on square lattice is also refined to be 0 8894 9 which is consistent with the nishimori and nemoto s conjecture 0 889972

generalized stability of isometries abstract let x and y be real banach spaces and let e p 0 a mapping f x y is called an e p isometry if f x f y x y e x y p holds for all x y x a pair x y is p stable with respect to isometries if there exists a function 0 0 with lime 0 e 0 such that for every surjective e p isometry f x y there is a surjective isometry u x y satisfying the estimate f x u x e x p x x we show that every pair of banach spaces x y is p stable for 0 p r 2 r 2 is not 1 stable when p 1 a superstability phenomenon occurs for finite dimensional banach spaces

linear mappings preserving ρ orthogonality abstract we define a orthogonality in a real normed space and we consider the class of linear mappings preserving this relation we show that a linear mapping preserving orthogonality has to be a similarity i e a scalar multiple of an isometry as a result we give a characterization of smooth spaces in terms of this orthogonality

on certain basis connected with operator and its applications abstract we assume that h k are the complex hilbert spaces and a linear operator t b h k is given the object of this paper is to define and consider the certain basis of h connected with the operator t b h k this basis will play an important role in describing the approximately orthogonality preserving operators next we will give some application of our results we give new results and more general ones namely we answer a question whether a linear operator which approximately preserves orthogonality must be close to an orthogonality preserving one