Graded Betti numbers of powers of ideals

Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive \ZZ-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. More precisely, in the case of \ZZ-grading, \ZZ^2 can be splitted into a finite number of regions such that each region corresponds to a polynomial that depending to the degree $(\mu, t)$, \dim_k \left(\tor_i^S(I^t, k)_{\mu} \right) is equal to one of these polynomials in $(\mu, t)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a \ZZ^d-graded algebra, for a positive grading.Comment: 20 page

Dielectric Metasurfaces for Complete Control of Phase and Polarization with Subwavelength Spatial Resolution and High Transmission

Metasurfaces are planar structures that locally modify the polarization, phase, and amplitude of light in reflection or transmission, thus enabling lithographically patterned flat optical components with functionalities controlled by design. Transmissive metasurfaces are especially important, as most optical systems used in practice operate in transmission. Several types of transmissive metasurfaces have been realized, but with either low transmission efficiencies or limited control over polarization and phase. Here we show a metasurface platform based on high-contrast dielectric elliptical nano-posts which provides complete control of polarization and phase with sub-wavelength spatial resolution and experimentally measured efficiency ranging from 72% to 97%, depending on the exact design. Such complete control enables the realization of most free-space transmissive optical elements such as lenses, phase-plates, wave-plates, polarizers, beam-splitters, as well as polarization switchable phase holograms and arbitrary vector beam generators using the same metamaterial platform.Comment: Nature Nanotechnology (2015

The Approximate Capacity Region of the Gaussian Z-Interference Channel with Conferencing Encoders

A two-user Gaussian Z-Interference Channel (GZIC) is considered, in which encoders are connected through noiseless links with finite capacities. In this setting, prior to each transmission block the encoders communicate with each other over the cooperative links. The capacity region and the sum-capacity of the channel are characterized within 1.71 bits per user and 2 bits in total, respectively. It is also established that properly sharing the total limited cooperation capacity between the cooperative links may enhance the achievable region, even when compared to the case of unidirectional transmitter cooperation with infinite cooperation capacity. To obtain the results, genie-aided upper bounds on the sum-capacity and cut-set bounds on the individual rates are compared with the achievable rate region. In the interference-limited regime, the achievable scheme enjoys a simple type of Han-Kobayashi signaling, together with the zero-forcing, and basic relaying techniques. In the noise-limited regime, it is shown that treating interference as noise achieves the capacity region up to a single bit per user.Comment: 25 pages, 6 figures, submitted to IEEE Transactions on Information Theor

Improving Energy Efficiency in MANETs by Multi-Path Routing

Some multi-path routing algorithm in MANET, simultaneously send information to the destination through several directions to reduce end-to-end delay. In all these algorithms, the sent traffic through a path affects the adjacent path and unintentionally increases the delay due to the use of adjacent paths. Because, there are repetitive competitions among neighboring nodes, in order to obtain the joint channel in adjacent paths. The represented algorithm in this study tries to discover the distinct paths between source and destination nodes with using Omni directional antennas, to send information through these simultaneously. For this purpose, the number of active neighbors is counted in each direction with using a strategy. These criterions are effectively used to select routes. Proposed algorithm is based on AODV routing algorithm, and in the end it is compared with AOMDV, AODVM, and IZM-DSR algorithms which are multi-path routing algorithms based on AODV and DSR. Simulation results show that using the proposed algorithm creates a significant improvement in energy efficiency and reducing end-to-end delay