We consider the following problem. At each point of discrete time the learner must make a prediction; he is given the predictions made by a pool of experts. Each prediction and the outcome, which is disclosed after the learner has made his prediction, determine the incurred loss. It is known that, under weak regularity, the learner can ensure that his cumulative loss never exceeds cL+ a ln n, where c and a are some constants, n is the size of the pool, and L is the cumulative loss incurred by the best expert in the pool. We find the set of those pairs (c; a) for which this is true.

We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that nontrivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata. 1

In this work, we consider a distributed source coding problem with a joint distortion criterion depending on the sources and the reconstruction. This includes as a special case the problem of computing a function of the sources to within some distortion and also the classic Slepian-Wolf problem [12], Berger-Tung problem [5], Wyner-Ziv problem [4], Yeung-Berger problem [6] and the Ahlswede-Korner-Wyner problem [3], [13]. While the prevalent trend in information theory has been to prove achievability results using Shannon’s random coding arguments, using structured random codes offer rate gains over unstructured random codes for many problems. Motivated by this, we present a new achievable ratedistortion region (an inner bound to the performance limit) for this problem for discrete memoryless sources based on “good” structured random nested codes built over abelian groups. We demonstrate rate gains for this problem over traditional coding schemes using random unstructured codes. For certain sources and distortion functions, the new rate region is strictly bigger than the Berger-Tung rate region, which has been the best known achievable rate region for this problem till now. Further, there is no known unstructured random coding scheme that achieves these rate gains. Achievable performance limits for single-user source coding using abelian group codes are also obtained as parts of the proof of the main coding theorem. As a corollary, we also prove that nested linear codes achieve the Shannon rate-distortion bound in the single-user setting. Note that while group codes retain some structure, they are more general than linear codes which can only be built over finite fields which are known to exist only for certain sizes.

Recent results on the information theory of steganography suggest, and under some conditions prove, that the detectability of payload is proportional to the square of the number of changes caused by the embedding. Assuming that result in general, this paper examines the implications for an embedder when a payload is to be spread amongst multiple cover objects. A number of variants are considered: embedding with and without adaptive source coding, in uniform and nonuniform covers, and embedding in both a fixed number of covers (so-called batch steganography) as well as establishing a covert channel in an infinite stream (sequential steganography, studied here for the first time). The results show that steganographic capacity is sublinear, and strictly asymptotically greater in the case of a fixed batch than an infinite stream. In the former it is possible to describe optimal embedding strategies; in the latter the situation is much more complex, with a continuum of strategies which approach the unachievable asymptotic optimum.

We point out that the optimal pebbling number of the n-cube is ( 4 3 ) n+O(log n) , and explain how to approximate the optimal pebbling number of the nth cartesian power of any graph in a similar way.

Let ^ be a compact body of positive volume in W, star-shaped with respect to an interior point, taken to be the origin. For subsets Q of R n, the functional sup lattices A represents the minimum density with which Q. can be covered by a lattice A of translates of < S. We obtain an upper bound on «9L(#, Z n). If the attributes of # are supplemented with convexity, write 3fC instead. We also bound above the classical minimum la nice-covering density of Jtf. No symmetry conditions are imposed on # and Jf. 1.