A simple yet powerful procedure for an echo attenuation in signals is introduced. The presented method involves no external reference signal. It is based on comb FIR filtering. To the advantages of the described method belong the simplicity and performance which are beneficial in real time implementations. For illustration, a simulation of the procedure is included. The efficiency of the presented method is demonstrated by a real time implementation on a digital signal processor.
Let $C[0,T]$ denote the space of real-valued continuous functions on the interval $[0,T]$ with an analogue $w_\varphi $ of Wiener measure and for a partition $ 0=t_0< t_1< \cdots < t_n <t_{n+1}= T$ of $[0, T]$, let $X_n\: C[0,T]\to \mathbb R^{n+1}$ and $X_{n+1} \: C [0, T]\to \mathbb R^{n+2}$ be given by $X_n(x) = ( x(t_0), x(t_1), \cdots , x(t_n))$ and $X_{n+1} (x) = ( x(t_0), x(t_1), \cdots , x(t_{n+1}))$, respectively. \endgraf In this paper, using a simple formula for the conditional $w_\varphi $-integral of functions on $C[0, T]$ with the conditioning function $X_{n+1}$, we derive a simple formula for the conditional $w_\varphi $-integral of the functions with the conditioning function $X_n$. As applications of the formula with the function $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions of the form $F_m(x) = \int _0^T (x(t))^m d t$ for $x\in C[0, T]$ and for any positive integer $m$. Moreover, with the conditioning $X_n$, we evaluate the conditional $w_\varphi $-integral of the functions in a Banach algebra $\mathcal S_{w_\varphi }$ which is an analogue of the Cameron and Storvick's Banach algebra $\mathcal S$. Finally, we derive the conditional analytic Feynman $w_\varphi $-integrals of the functions in $\mathcal S_{w_\varphi }$.
We show that a transformation method relating planar first-order differential systems to second order equations is an effective tool for finding non-liouvillian first integrals. We obtain explicit first integrals for a subclass of Kukles systems, including fourth and fifth order systems, and for generalized Liénard-type systems.
Measurement of leaf area is commonly used in many horticultural research experiments, but it is generally destructive, requiring leaves to be removed for measurement. Determining the individual leaf area (LA) of bedding plants like pot marigold (Calendula officinalis L.), dahlia (Dahlia pinnata), sweet William (Dianthus barbatus L.), geranium (Pelargonium × hortorum), petunia (Petunia × hybrida), and pansy (Viola wittrockiana) involves measurements of leaf parameters such as length (L) and width (W) or some combinations of these parameters. Two experiments were carried out during spring 2010 (on two pot marigold, four dahlia, three sweet William, four geranium, three petunia, and three pansy cultivars) and summer 2010 (on one cultivar per species) under greenhouse conditions to test whether a model could be developed to estimate LA of bedding plants across cultivars. Regression analysis of LA versus L and W revealed several models that could be used for estimating the area of individual bedding plants leaves. A linear model having LW as the independent variable provided the most accurate estimate (highest R2, smallest mean square error, and the smallest predicted residual error sum of squares) of LA in all bedding plants. Validation of the model having LW of leaves measured in the summer 2010 experiment coming from other cultivars of bedding plants showed that the correlation between calculated and measured bedding plants leaf areas was very high. Therefore, these allometric models could be considered simple and useful tools in many experimental comparisons without the use of any expensive instruments. and F. Giuffrida ... [et al.].
Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of $X$-valued $\sigma $-additive Baire measures on $T$ is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map $u\: C_0(T) \rightarrow X$ when $c_0 \lnot \subset X$ are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].
In 1932 Whitney showed that a graph G with order n ≥ 3 is 2-connected if and only if any two vertices of G are connected by at least two internally-disjoint paths. The above result and its proof have been used in some Graph Theory books, such as in Bondy and Murty’s well-known Graph Theory with Applications. In this note we give a much simple proof of Whitney’s Theorem.
The nonhomogeneous backward Cauchy problem $$u_t +Au(t) = f(t),\quad u(T) = \varphi$$, where $A$ is a positive self-adjoint unbounded operator which has continuous spectrum and $f$ is a given function being given is regularized by the well-posed problem. New error estimates of the regularized solution are obtained. This work extends earlier results by N. Boussetila and by M. Denche and S. Djezzar.
We present a simplified integral of functions of several variables. Although less general than the Riemann integral, most functions of practical interest are still integrable. On the other hand, the basic integral theorems can be obtained more quickly. We also give a characterization of the integrable functions and their primitives.