fractals in physics squig clusters diffusions fractal measures and the unicity of fractal dimensionality the three topics discussed in this paper are largely independent part 1 fractal squig clusters are introduced and it is shown that their properties can match to a remarkable extent those of percolation clusters at criticality physics on these new geometric shapes should prove tractable as background the author s theories of squig intervals and squig trees are reviewed and restated in more versatile form part 2 the notion of latent fractal dimensionality is introduced and motivated by the desire to simplify the algebra of dimensionality scaling noises are touched upon a common formalism is presented for three forms of anomalous diffusion the ant in the fractal labyrinth fractional brownian motion and levy stable motion the fractal dimensionalities common to diverse shapes generated by diffusion are given in table i as functions of the latent dimensionalities of the support of the motion and of the diffusion itself part 3 it is argued that every fractal point set has a unique fractal dimensionality but it is pointed out that many fractals involve diverse combinations of many fractal point sets such is in particular the case for fractal measures and for fractal graphs often called hierarchical lattices the fractal measures that the author had introduced in the early 1970s are described including new developments

a regular model for cluster numbers and structure just below percolation threshold a two dimensional deterministic fractal model is proposed to imitate the geometric texture just below percolation threshold the model is constructed for bond percolation via a rule of bond occupation on the square lattice it show the typical percolation behaviour as a function of a parameter p p is the bond concentration the model describes the approach towards the threshold below criticality the critical bond concentration pc and the correlation length exponent nu are found the scaling property of the cluster size distribution is studied below the threshold the critical concentration and critical exponents are shown to agree with those derived from the regular model proposed for the geometric texture above the threshold the scaling relations for the critical exponents are shown to be satisfied

a regular model for bond percolation the approach towards the threshold a two dimensional regular model is constructed for bond percolation via a rule of bond deletions on the square lattice the model describes the approach towards the threshold the critical bond concentration pc and the correlation length exponent nu are found the fractal geometry of the infinite percolating network and the scaling property of the cluster size distribution are studied above the threshold the infinite percolating network is shown to be reconstructed by the mandelbrot koch curve proposed as the non random model for the infinite cluster at criticality

minimax mixed integer symmetric duality for multiobjective variational problems abstract a mond weir type multiobjective variational mixed integer symmetric dual program over arbitrary cones is formulated applying the separability and generalized f convexity on the functions involved weak strong and converse duality theorems are established self duality theorem is proved a close relationship between these variational problems and static symmetric dual minimax mixed integer multiobjective programming problems is also presented

a note on pseudo invexity and symmetric duality a pair of symmetric dual nonlinear programming problems is presented and duality theorems are established under pseudoinvexity type assumptions on the kernel function this formulation removes certain inconsistencies in a recently introduced primal dual pair and gives the correct proof of various duality theorems

minimax and symmetric duality for nonlinear multiobjective mixed integer programming abstract we formulate two pairs of symmetric duality for nonlinear multiobjective mixed integer programs for arbitrary cones by using the concept of efficiency we establish the weak strong converse and self duality theorems for our symmetric models several known results are obtained as special cases

second order symmetric duality for nonlinear minimax mixed integer programs abstract wolfe type second order minimax mixed integer dual programs are formulated and a symmetric duality theorem is e established under separability and bonvexity boncavity of the kernel function k x y mond weir type symmetric duality is also discussed under weaker bonvexity assumptions moreover self duality theorems for these pairs are obtained assuming k x y to be skew symmetric

on the free boundary problem for stationary compressible navier stokes equations we consider the equations which describe a stationary motion of a viscous compressible barotropic fluid in a bounded domain in 3 with a free boundary determined by the surface tension by means of some a priori estimates we prove the existence of rotationally symmetric solutions in reality with some additional symmetry for a sufficiently small external force and in the case of rotationally symmetric force and domain where also we need more symmetry respectively

second order symmetric duality in mathematical programming with f convexity abstract under second order f convexity f concavity and second order f pseudoconvexity f pseudoconcavity appropriate second order duality results for pair of wolfe and mond weir type second order symmetric dual nonlinear programming problems are established these second order duality results are then used to formulate wolfe type and mond weir type second order minimax mixed integer dual programs and symmetric duality theorem is established under separability and second order f convexity f concavity of the kernel function f x y second order symmetric dual fractional mixed integer programs are studied using the above programs moreover second order self duality theorems for the above pairs are obtained assuming f x y to be skew symmetric

nonlinear resistor fractal networks topological distances singly connected bonds and fluctuations the authors consider a fractal network of nonlinear resistors with the voltage v behaving as a power of the current i mod v mod r mod i mod alpha the resistance between two points at a distance l is r l varies as lzeta alpha they prove that zeta 0 describes the scaling of the topological chemical distance while zeta infinity describes that of the number of singly connected red bonds for random resistors they also consider the width of the resistance distribution delta r varies as l zeta 2 alpha values for zeta and zeta 2 are explicitly derived for two model fractals and delta r r is found to grow with l for the sierpinski gasket and alpha 1 612 the relevance of the results to percolation clusters is discussed

fractal models for diffusion controlled aggregation a two dimensional fractal model is constructed for diffusion controlled deposition on a surface the fractal geometry of the deposit and the power law behaviour of the quantities characterising the non equilibrium cluster size distribution are shown to be consequences of the competition generally present in a nonlinear growth process a qualitative agreement with previous numerical results is found and the scaling laws for the critical exponents of the problem are shown to be satisfied exactly

on a pair of nonlinear mixed integer programming problems abstract we consider maximin and minimax nonlinear mixed integer programming problems which are nonsymmetric in duality sense under weaker pseudo convex pseudo concave assumptions we show that the supremum infimum of the maximin problem is greater than or equal to the infimum supremum of the minimax problem as a particular case this result reduces to the weak duality theorem for minimax and symmetric dual nonlinear mixed integer programming problems further this is used to generalize available results on minimax and symmetric duality in nonlinear mixed integer programming

regular random fractals and the n parameter model two models of fractal objects are presented the first model represents a new class of fractals which is intermediate to statistical fractals and exact fractals in that each configuration of the random ensemble has an exact and adjustable fractal dimension the second model has a continuously adjustable backbone dimensions and can be used to directly generate conducting backbones

symmetric duality for minimax nonlinear mixed integer programming abstract a pair of symmetric dual minimax integer programming problems is formulated and an appropriate duality theorem is established under the pseudoconvexity pseudoconcavity assumption on the kernel function k x y this model removes certain inconsistencies in an earlier model and helps to study self duality as well

technical note on symmetric duality in multiobjective programming some remarks on recent results abstract certain shortcomings are pointed out in some recent works of das and nanda european journal of operational research 97 1997 167 171 and kim et al european journal of operational research 107 1998 686 691 which show that these studies are highly restricted

a hierarchical model for scaling structure in generalised diffusion limited aggregations deterministic fractal models are presented to be exactly solvable for the growth probability distribution on the surface of the cluster in a hierarchical lattice the recursion relations of the electric field on the growth bond are obtained for diffusion limited aggregation and the dielectric breakdown models a hierarchy of generalised dimensions d q is calculated to describe the growth probability by using the recursion relations the partition of q 1 d q into a density of singularities f q with singularity strength alpha q is made and the alpha f spectra are studied for different dielectric breakdown models the scaling of the highest growth probability pmax on the growth bond is analytically derived and the dependence of the fractal dimensions is found on the parameter eta describing the different dielectric breakdown models

exact fractals with adjustable fractal and fracton dimensionalities a family of exact fractals is presented by adjusting two external parameters a wide range of fractal and fracton dimensionalities can be achieved this includes the case of fracton dimensionality of 2 which is critical for diffusion

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

squig sheets and some other squig fractal constructions squig intervals are a class of hierarchically constructed fractals introduced by the author they can be visualized as the final outcome upon a straight interval of a suitable cascade of local perturbative eddies ruled by two processes called decimation and separation their theory is summarized and their scope is extended in several new directions especially by introducing new forms of separation squig intervals are generalized in two dimensions with fractal dimensions ranging from 1 2886 to 1 589 squig sheets are constructed in three dimensional space with fractal dimensions ranging from 8 3 up they should prove useful in modeling the fractal surfaces associated with turbulence and related phenomena squig intervals are constructed in three dimensions nonsymmetric eddies and the resulting squigs are tackled squig trees and intervals are drawn on unconventional lattices either in the plane or in a prescribed fractal surface peyriere sm systems are mentioned their study includes the proof that the informal renormalization argument involving a transfer matrix is exact for squigs

symmetric duality for minimax multiobjective variational mixed integer programming problems with partial invexity a pair of symmetric dual multiobjective variational mixed integer programs for the polars of arbitrary cones are formulated which some primal and dual variables are constrained to belong to the set of integers under the separability with respect to integer variables and partial invexity assumptions on the functions involved we prove the weak strong converse and self duality theorems to related minimax efficient solution these results include some of available results

renormalisation group approach to multifractal structure in random resistor networks just above the percolation threshold a renormalisation group method is presented to analyse the multifractal structure of the current distribution in random linear resistor networks just above the percolation threshold the recursion relation for the current distribution is given under the renormalisation transformation the distribution function of the current is derived with the use of the recursion relation the fragmentation of the current is described as a random multiplicative process an infinite set of exponents is calculated to scale each of the moments of the current distribution the alpha f spectrum is derived from the legendre transform of these exponents the authors exponents obtained by the relatively small cell renormalisation are very close to the previous simulation data

diffusion on fractal lattices and the fractal einstein relation finitely ramified fractal lattices show anomalous diffusion with r2 varies as t2h there is a hierarchy of transit times which are shown by monte carlo simulation to satisfy tau n varies as alpha n for large n where alpha b1h and b is the base of the lattice the lattice resistivity scales with size as rho varies as rn where r is characteristic of the lattice exact renormalisation group methods yield alpha and r analytically and verify the einstein relation for fractals alpha rn