Laura Pezza

On the Galerkin Method Based on a Particular Class of Scaling Functions

The aim of this paper is to provide a large class of scaling functions for which the convergence analysis for the Galerkin method developed in [9] is applicable, whereas in that paper the only scaling functions considered for practical applications are B-splines and a few of the orthonormal Daubechies scaling functions. The functions considered here, were recently introduced in [12] where it was proved that they satisfy many properties making them interesting for the applications. In particular, here we show that the use of these functions has some advantages with respect to other basis functions.

Variable Length Unordered Codes

In an unordered code, no code word is contained in any other code word. Unordered codes are all unidirectional error detecting (AUED) codes. In the binary case, it is well known that among all systematic codes with <i>k</i> information bits, Berger codes are optimal unordered codes with <i>r</i>=[log₂(<i>k</i>+1)] ≅ log₂<i>k</i> check bits. This paper gives some new theory on variable length unordered codes and introduces a new class of systematic (instantaneous) unordered codes with variable length check symbols. The average redundancy of the new codes presented here is <i>r</i> ≅ (1/2)log₂<i>k</i>+<i>c</i>, where <i>c</i> ∈ (1.0470,1.1332) ⊆ <b>IR</b> and <i>k</i> ∈ <b>IN</b> is the number of information bits. When <i>k</i> is large, it is shown that such redundancy is at most 0.6069 bits off the redundancy of an optimal systematic unordered code design with fixed length information symbols and variable length check symbols; and, at most 2.8075 bits off the redundancy of an optimal variable length unordered code design. The generalization is also given for the nonbinary case and it is shown that similar results hold true.

A multiscale collocation method for fractional differential problems

We introduce a multiscale collocation method to numerically solve differential problems involving both ordinary and fractional derivatives of high order. The proposed method uses multiresolution analyses (MRA) as approximating spaces and takes advantage of a finite difference formula that allows us to express both ordinary and fractional derivatives of the approximating function in a closed form. Thus, the method is easy to implement, accurate and efficient. The convergence and the stability of the multiscale collocation method are proved and some numerical results are shown.

On efficient second-order spectral-null codes using sets of m 1 -balancing functions

A new efficient coding scheme is given for second-order spectral-null (2-OSN) codes. The new method applies the Knuths optimal parallel decoding scheme for balanced (i.e., 1-OSN) codes to the random walk method introduced by Tallini and Bose to design 2-OSN codes. If k ∈ IN is the length of a 1-OSN code then the new 2-OSN coding scheme has length n = k+r ∈ IN with an extra redundancy of r ≳ 2 log2 k + (1/2) log2 log2 k - 0.674 check bits. The whole coding process requires O(n log n) bit operations and 0(n) bit memory elements.

Efficient Non-Recursive Design of Second-Order Spectral-Null Codes

A new efficient design of second-order spectralnull (2-OSN) codes is presented. The new codes are obtained by applying the technique used to design parallel decoding balanced (i.e., 1-OSN) codes to the random walk method introduced by some of the authors for designing 2-OSN codes. This gives new non-recursive efficient code designs, which are less redundant than the code designs found in the literature. In particular, if k ∈ IIN is the length of a 1-OSN code then the new 2-OSN coding scheme has length n = k + r ∈ IIN with an extra redundancy of r ≃ 2 log2 k + (1/2) log2 log2 k - 0.174 check bits, with k and r even and n multiple of 4. The whole coding process requires O(k log k) bit operations and O(k) bit memory elements.

A fractional spline collocation-Galerkin method for the time-fractional diffusion equation

Abstract The aim of this paper is to numerically solve a diffusion differential problem having time derivative of fractional order. To this end we propose a collocation-Galerkin method that uses the fractional splines as approximating functions. The main advantage is in that the derivatives of integer and fractional order of the fractional splines can be expressed in a closed form that involves just the generalized finite difference operator. This allows us to construct an accurate and efficient numerical method. Several numerical tests showing the effectiveness of the proposed method are presented.

A Fractional Spline Collocation Method for the Fractional-order Logistic Equation

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivatives, i.e., the fractional-order logistic equation. The use of the fractional B-splines allows us to express the fractional derivatives of the approximating function in an analytical form. Thus, the fractional collocation method is easy to implement, accurate, and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.

Analysis the Throughput of Type-I Hybrid ARQ Protocol Using t-AEC/AAED over the m (=2)-ary Varshamov Channel

In many communication media, optical systems, and some VLSI systems, only one errors type, either 0 → 1 or 1 → 0, can occur in any data word and the decoder knows a priori the error type. These types of errors are called asymmetric errors. Asymmetric Varshamov channels could be used to model some physical systems such as multilevel flash memories where there is an exponential behaviour in the real distance between the sent and received symbols and so, the number of errors between these symbols should be measured according to the L1 distance. In this paper, we introduce a Varshamov error model for the general m(≥2)-ary Z-Channel with taking into account the error magnitude, but with concerning to the asymmetric case. Also, we analyze the throughput performance for Type-I selective-repeat hybrid ARQ protocols using t-Error-Correcting/All Asymmetric Error Detecting (t-AEC/AAED) codes over a noticeable Varshamov error model for the general Z-Channel.

On systematic variable length unordered codes

In an unordered code no codeword is contained in any other codeword. Unordered codes are All Unidirectional Error Detecting (AUED) codes. In the binary case, it is well known that among all systematic codes with k information bits, Berger codes are optimal unordered codes with r = ⌈log2(k+1)⌉ check bits. This paper gives some new theory on variable length unordered codes and introduces a new class of systematic unordered codes with variable length check symbols. The average redundancy of these new codes is r ≈ (1/2) log2(πek/2) = (1/2) log2 k + 1.047, where k∈IN is the number of information bits. It is also shown that such codes are optimal in the class of systematic unordered codes with fixed length information symbols and variable length check symbols. The generalization to the non-binary case is also given.