coding theorems for shannon s cipher system with correlated source outputs and common information source coding problems are treated for shannon s 1949 cipher system with correlated source outputs x y several cases are considered based on whether both x and y only x or only y must be transmitted to the receiver whether both x and y only x or only y must be kept secret or whether the security level is measured by sup 1 spl sol sub k h x sup k spl verbar w sup 1 spl sol sub k h y sup k spl verbar w or sup 1 spl sol sub k h x sup k y sup k spl verbar w where w is a cryptogram the admissible region of cryptogram rate and key rate for a given security level is derived for each case furthermore two new kinds of common information of x and y say c sub 1 x y and c sub 2 x y are considered c sub 1 x y is defined as the rate of the attainable minimum core of x sup k y sup k by removing each private information from x sup k y sup k as much as possible while c sub 2 x y is defined as the rate of the attainable maximum core v sub c such that if one loses v sub c then each uncertainty of x sup k and y sup k becomes h v sub c it is proved that c sub 1 x y i x y and c sub 2 x y min spl lcub h x h y spl rcub c sub 1 x y justifies the author s intuitive feeling that the mutual information represents a common information of x and y

on secret sharing systems a secret sharing system permits a secret to be shared among n trustees in such a way that any k of them can recover the secret but any k 1 have complete uncertainty about it a linear coding scheme for secret sharing is exhibited which subsumes the polynomial interpolation method proposed by shamir and can also be viewed as a deterministic version of blakley s probabilistic method bounds on the maximum value of n for a given k and secret size are derived for any system linear or nonlinear the proposed scheme achieves the lower bound which for practical purposes differs insignificantly from the upper bound the scheme may be extended to protect several secrets methods to protect against deliberate tampering by any of the trustees are also presented

randomness in distribution protocols randomness is a useful computation resource due to its ability to enhance the capabilities of other resources its interaction with resources such as time space interaction with provers and its role in several areas of computer science has been extensively studied in this paper we give a systematic analysis of the amount of randomness needed by secret sharing schemes and secure key distribution schemes we give both upper and lower bounds on the number of random bits needed by secret sharing schemes the bounds are tight for several classes of secret sharing schemes for secure key distribution schemes we provide a lower bound on the amount of randomness needed thus showing the optimality of a recently proposed key distribution protocol

two phase entropy solutions of a forward backward parabolic equation this article deals with the cauchy problem for a forward backward parabolic equation which is of interest in physical and biological models considering such an equation as the singular limit of an appropriate pseudoparabolic third order regularization we consider the framework of entropy solutions namely weak solutions satisfying an additional entropy inequality inherited by the higher order equation moreover we restrict the attention to two phase solutions that is solutions taking values in the intervals where the parabolic equation iswell posed proving existence and uniqueness of such solutions

generating random bits from an arbitrary source fundamental limits suppose we are given a random source and want to use it as a random number generator at what rate can we generate fair bits from it we address this question in an information theoretic setting by allowing for some arbitrarily small but nonzero deviation from ideal random bits we prove our results with three different measures of approximation between the ideal and the obtained probability distributions the variational distance the d bar distance and the normalized divergence two different contexts are studied fixed length and variable length random number generation the fixed length results of this paper provide an operational characterization of the inf entropy rate of a source defined in han and verdu see ibid vol 39 no 3 p 752 772 1993 and the variable length results characterize the liminf of the entropy rate thereby establishing a pleasing duality with the fundamental limits of source coding a feature of our results is that we do not restrict ourselves to ergodic or to stationary sources

the generalized cattaneo partial slip plane contact problem i theory the cattaneo problem is considered for a general plane contact between elastically similar materials i e a monotonically increasing tangential load starting from zero with normal loading held fixed instead of the classical argument on the displacement field in the stick zone of cattaneo solution we attack the problem implicitly from the governing integral equations in the stick zones after discussing and solving the full stick case we move to the more realistic for finite friction case of partial slip we show that upon isolating the effect of full sliding the equalities and inequalities governing the corrective solution for the corrective shearing tractions in the stick zone are exactly the same as those governing the solution of the normal contact problem with a lower load but the same rotation as the actual one this analogy permits us to deduce several general properties and gives a general procedures for solving partial slip cattaneo problems as frictionless normal indentation ones therefore the general solutions for single multiple and periodic contacts is given a comprehensive set of explicit results is given in the part ii of the paper

on one dimensional forward backward diffusion equations with linear convection and reaction abstract we study the initial neumann boundary value problem for a class of one dimensional forward backward diffusion equations with linear convection and reaction the diffusion flux function is assumed to contain two forward diffusion phases we prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases furthermore we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant the results are new even for the pure forward backward diffusion problem without convection and reaction our primary approach relies on a combination of the convex integration and baire s category methods for a related nonlocal differential inclusion

coding theorems on the threshold scheme for a general source in this paper coding theorems on the t m threshold scheme for a general source are discussed where m means the number of the shares and t means a threshold the t m threshold scheme treated in this paper encrypts n source outputs xn to m shares at once and is required to satisfy the two conditions that 1 xn is reproduced from arbitrary t shares and 2 almost no information of xn is revealed from any t 1 shares it is shown that the t m threshold scheme must satisfy certain inequalities including the limit inferiors in probability one of the inequalities is closely related to the minimum length of the fair random bits needed to a dealer for realizing the t m threshold scheme in addition it is shown that a certain variation of shamir s threshold scheme meets the two conditions the same approach can be taken to the problems of shannon s cipher system with the perfect secrecy and fixed length source coding with vanishing decoding error probability it is shown that the same kind of inequalities which indicate the converse coding theorems hold in both two cases

new analytical and numerical results for two dimensional contact profiles recently a generalized coulomb law for elastic bodies in contact has been developed by the author which assumes that the tangential traction is the difference of the slip stress of the contact and the stick area whereby each stick area corresponds to a smaller contact area it holds for multiple contact regions also several applications for elastic half planes half spaces thin and thick layers and impact problems have been published for plane contact of equal bodies with friction it provides exact solutions and the interior stress field can be expressed with analytical results in closed form in this article a singular superposition of flat punch solutions is outlined in which the punches are aligned with an edge of the contact area it is shown that this superposition satisfies coulomb s inequalities directly and new results for the muskhelishvili potentials of several profiles are presented it is illustrated how problems of singularity and multi valuedness of complex functions can be solved in closed form and the chebyshev approximation used by earlier authors can be avoided for comparison some previous solutions for symmetric profiles are appended some results for the interior stress field the pressure the frictional traction and the surface displacements are compared with fem solutions of an equivalent problem the small differences between both methods show characteristic features of the fem model and the theoretical assumptions and are shortly explained further this example can be used as benchmark test for fem and bem programs

young measure solutions and instability of the one dimensional perona malik equation we apply a variational approach to the one dimensional version of the widely used perona malik equation in image processing we rephrase the problem into the one related to the quasiconvex hull of a graph in the space of 2 x 2 matrices m 2x2 we then use the solutions of some heat equations as the centre of the mass for the young measure valued solutions to construct the approximate solutions by using simple laminates the approximate solutions can be viewed as solutions of a perturbation problem by w 1 w p or w 1 w infinity functions the sequences of the approximate solutions generates young measure valued solutions our results also show that the solutions of the one dimensional perona malik equation are unstable under small w 1 w infinity perturbations

gradient estimates for the perona malik equation we consider the cauchy problem for the perona malik equation u t mathrm div left frac nabla u 1 nabla u 2 right in a bounded open set omega subseteq mathbb r n with neumann boundary conditions

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

existence of infinitely many solutions for a forward backward heat equation let 4 be a piecewise linear function which satisfies the condition s40 s 2 cs2 c 0 s e r and which is monotone decreasing on an interval a b c r it is shown that for f e c2 0 11 with max f a there exists a t 0 such that the initial boundary value problem ut 00 x x ux os t u ls t u 0 f has infinitely many solutions u satisfying 11ul u x u x u 1l 2 6 c f 4 on 0 1i x 0 t 0 introduction consider the initial boundary value problem ut u x x t e 0 1 x 0 t 1 u t uj l t 0 t e 0 t u x 0 f x x e 0 1 if 4 is strictly monotone increasing with 4 c 0 1 has a unique solution which is roughly speaking as smooth as the function o on the other hand if 4 c 0 this assumption allows 4 to have monotone decreasing parts e g the model cubic et s 1s3 s2 2s 4 s 3 42 2s a natural and interesting question is whether problem 1 has a solution if the usual assumption 4 0 is replaced by the weaker coercivity condition c under this hypothesis j bona j nohel and l wahlbin bnw hn obtained several a received by the editors april 2 1982 and in revised form may 14 1982 1980 mathematics subject classification primary 35k55 35k65

channel coding and lossy source coding using a generator of constrained random numbers stochastic encoders for channel coding and lossy source coding are introduced with a rate close to the fundamental limits where the only restriction is that the channel input alphabet and the reproduction alphabet of the lossy source code are finite random numbers which satisfy a condition specified by a function and its value are used to construct stochastic encoders the proof of the theorems is based on the hash property of an ensemble of functions where the results are extended to general channels sources and alternative formulas are introduced for channel capacity and the rate distortion region since an ensemble of sparse matrices has a hash property we can construct a code by using sparse matrices

coding theorems for a 2 2 threshold scheme with detectability of impersonation attacks coding theorems on a 2 2 threshold scheme with an opponent are discussed in an asymptotic setup where the opponent tries to impersonate one of the two participants a situation is considered where n secrets s n from a memoryless source is blockwisely encoded to two shares and the two shares are decoded to s n with permitting negligible decoding error we introduce correlation level of the two shares and characterize the minimum attainable rates of the shares and a uniform random number for realizing a 2 2 threshold scheme that is secure against the impersonation attack by the opponent it is shown that if the correlation level between the two shares equals to ell geq 0 the minimum attainable rates coincide with h s ell where h s denotes the entropy of the source and the maximum attainable exponent of the success probability of the impersonation attack equals to ell it is also shown that a simple scheme using an ordinary 2 2 threshold scheme attains all the bounds as well

the generalized cattaneo partial slip plane contact problem ii examples this second part of the paper uses the method devised in the part i to give explicit solution to several cases of cattaneo s plane contact problem where a monotonically increasing tangential load starting from zero is applied to the bodies in contact with normal loading held fixed the method consists in reducing the partial slip problem to a superposition of frictionless normal contact problems for which several results are available including some recent cases studied by the author therefore a comprehensive set of results is given for single multiple and periodical contacts

an accurate elasto plastic frictional tangential force displacement model for granular flow simulations displacement driven formulation we present in this paper the displacement driven version of a tangential force displacement tfd model that accounts for both elastic and plastic deformations together with interfacial friction occurring in collisions of spherical particles this elasto plastic frictional tfd model with its force driven version presented in l vu quoc l lesburg x zhang an accurate tangential force displacement model for granular flow simulations contacting spheres with plastic deformation force driven formulation journal of computational physics 196 1 2004 298 326 is consistent with the elasto plastic frictional normal force displacement nfd model presented in l vu quoc x zhang an elasto plastic contact force displacement model in the normal direction displacement driven version proceedings of the royal society of london series a 455 1991 4013 4044 both the nfd model and the present tfd model are based on the concept of additive decomposition of the radius of contact area into an elastic part and a plastic part the effect of permanent indentation after impact is represented by a correction to the radius of curvature the effect of material softening due to plastic flow is represented by a correction to the elastic moduli the proposed tfd model is accurate and is validated against nonlinear finite element analyses involving plastic flows in both the loading and unloading conditions the proposed consistent displacement driven elasto plastic nfd and tfd models are designed for implementation in computer codes using the discrete element method dem for granular flow simulations the model is shown to be accurate and is validated against nonlinear elasto plastic finite element analysis

on a class of forward backward parabolic equations formation of singularities abstract we study the formation of singularities for the problem u t u x x e u t x x in 0 t u e u t 0 in 0 t u u 0 0 in 0 where and t are positive constants a bounded interval u 0 a nonnegative radon measure on a nonmonotone and nonnegative function with 0 0 and an increasing bounded function we show that if u 0 is a bounded or continuous function singularities may appear spontaneously the class of singularities which can arise in finite time is remarkably large and includes infinitely many dirac masses and singular continuous measures

on structure of a set of solutions of dirichlet boundary value problem for stationary one dimensional forward backward parabolic equation on etudie la structure des ensembles de solutions pour des problemes de dirichlet pour l equation d du dx du dx dx udu dx on considere les proprietes de compacite des ensembles de solutions dans les espaces c c 0 x et w p 1 w p 1 0 x 1 p et on decrit les fermetures dans ces espaces