Abstract. In this paper, some monotoneity and concavity properties of the gamma, beta and psi functions are obtained, from which several asymptotically sharp inequalities follow. Applying these properties, the authors improve some well-known results for the volume Ωn of the unit ball B n ⊂ R n,thesurface area ωn−1 of the unit sphere S n−1, and some related constants. 1.

We further the study of order-preserving symmetric encryption (OPE), a primitive for allowing efficient range queries on encrypted data, recently initiated (from a cryptographic perspective) by Boldyreva et al. (Eurocrypt ’09). First, we address the open problem of characterizing what encryption via a random order-preserving function (ROPF) leaks about underlying data (ROPF being the “ideal object ” in the security definition, POPF, satisfied by their scheme.) In particular, we show that, for a database of randomly distributed plaintexts and appropriate choice of parameters, ROPF encryption leaks neither the precise value of any plaintext nor the precise distance between any two of them. The analysis here introduces useful new techniques. On the other hand, we show that ROPF encryption leaks approximate value of any plaintext as well as approximate distance between any two plaintexts, each to an accuracy of about square root of the domain size. We then study schemes that are not order-preserving, but which nevertheless allow efficient range queries and achieve security notions stronger than POPF. In a setting where the entire database is known in advance of key-generation (considered in several prior works), we show that recent constructions of “monotone minimal perfect hash functions ” allow to efficiently achieve (an adaptation of) the notion

We examine the capacity of beamforming over a single-user, multi-antenna link taking into account the overhead due to channel estimation and limited feedback of channel state information. Multi-input single-output (MISO) and multi-input multi-output (MIMO) channels are considered subject to block Rayleigh fading. Each coherence block contains L symbols, and is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a Minimum Mean Squared Error estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2B i.i.d. random vectors, and sends the corresponding B bits back to the transmitter. We derive bounds on the beamforming capacity for MISO and MIMO channels and characterize the optimal (rate-maximizing) training and feedback overhead (T and B) as L and the number of transmit antennas Nt both become large. The optimal Nt is limited by the coherence time, and increases as L / logL. For the MISO channel the optimal T/L and B/L (fractional overhead due to training and feedback) are asymptotically the same, and tend to zero at the rate 1 / logNt. For the MIMO channel the optimal feedback overhead B/L tends to zero faster (as 1 / log 2 Nt).