Mathematical theory of signal analysis vs. complex analysis method of harmonic analysis

We present recent work of harmonic and signal analysis based on the complex Hardy space approach.

Recent advances in the syntheses of pyrroles

The pyrrole moiety is an important structural motif in functional materials,natural products,and pharmaceuticals.More and more synthetic strategies toward pyrroles have emerged,where various efficient building blocks are developed and these synthons enable the syntheses of pyrroles with different numbers of components.However,no review specifically summarizes the syntheses of pyrroles according to the type and number of employed building blocks.To aid researchers to design appropriate substrates for pyrrole synthesis,herein we summarized the advances in pyrrole syntheses and classified these reactions into four categories according to the number of employed components,which may shed light on developing more efficient synthetic methods toward substituted pyrroles.

Protease supplementation reduced the heat increment of feed and improved energy and nitrogen partitioning in broilers fed maize-based diets with supplemental phytase and xylanase

An experiment was conducted to explore the effects of digestible amino acid(dAA)concentrations and supplemental protease on live performance and energy partitioning in broilers.Ross 308 male broilers(n?288)were distributed into 24 floor pens and offered 1 of 4 dietary treatments with 6 replicates from 1 to 35 d of age.Dietary treatments consisted of a 2×2 factorial arrangement with dAA concentrations(standard and reduced[34 g/kg below standard])and supplemental protease(without or with)as the main factors.At 1,15,28,and 35 d of age,feed and broilerswereweighed to determine live performance.From20 to 23 d of age,a total of 32 birds(2 birds/chamber,4 replicates)were placed in closed-calorimeter chambers to determine respiratory exchange(heat production,HP),apparent metabolisable energy(AME),retained energy(RE),and net energy(NE).From 29 to 35 d of age,supplemental protease in the reduced-dAA diet decreased broiler feed conversion ratio(FCR)by 5.6 points,whereas protease supplementation in the standard-dAA diet increased FCR by 5.8 points.The indirect calorimetry assay revealed that supplemental protease decreased(P<0.05)the heat increment of feed(HIF)by 0.22 MJ/kg.Also,from 20 to 23 d of age,broilers offered the reduced-dAA diet with supplemental protease had a higher daily body weight gain(BWG)(t10.4%),N intake(t7.1%),and N retention(t8.2%)than those offered the standard-dAA with supplemental protease.Broilers offered the reduced-dAA without supplemental protease exhibited a 3.6%higher AME-to-crude protein(CP)ratio than those offered other treatments.Protease supplementation in the standard-and reduced-dAA diets resulted in 2.7%and 5.6%lower AME intake-to-N retention ratios,respectively,compared with the unsupplemented controls.Reduced-dAA increased(P<0.05)AME intake(t4.8%),RE(t9.8%),NE intake(t5.8%),NE intake-to-CP ratio(t3.0%),and RE fat-to-RE ratio(t8.6%).Protease supplementation increased(P<0.05)respiratory quotient(t1.2%)and N retention-to-N intake ratio(t2.2%),NE-to-AME ratio(t1.9%),and reduced HP(3.6%),heat increment(7.4%),and NE intake-to-N retention(2.5%).In conclusion,protease positively affected FCR and energy partitioning in broilers;responses were most apparent in diets with reduced-dAA concentrations.

Positive-instantaneous frequency and approximation

Positive-instantaneous frequency representation for transient signals has always been a great concern due to its theoretical and practical importance,although the involved concept itself is paradoxical.The desire and practice of uniqueness of such frequency representation(decomposition)raise the related topics in approximation.During approximately the last two decades there has formulated a signal decomposition and reconstruction method rooted in harmonic and complex analysis giving rise to the desired signal representations.The method decomposes any signal into a few basic signals that possess positive instantaneous frequencies.The theory has profound relations to classical mathematics and can be generalized to signals defined in higher dimensional manifolds with vector and matrix values,and in particular,promotes kernel approximation for multi-variate functions.This article mainly serves as a survey.It also gives two important technical proofs of which one for a general convergence result(Theorem 3.4),and the other for necessity of multiple kernel(Lemma 3.7).Expositorily,for a given real-valued signal f one can associate it with a Hardy space function F whose real part coincides with f.Such function F has the form F=f+iHf,where H stands for the Hilbert transformation of the context.We develop fast converging expansions of F in orthogonal terms of the form F=∑k=1^(∞)c_(k)B_(k),where B_(k)'s are also Hardy space functions but with the additional properties B_(k)(t)=ρ_(k)(t)e^(iθ_(k)(t)),ρk≥0,θ′_(k)(t)≥0,a.e.The original real-valued function f is accordingly expanded f=∑k=1^(∞)ρ_(k)(t)cosθ_(k)(t)which,besides the properties ofρ_(k)andθ_(k)given above,also satisfies H(ρ_(k)cosθ_(k))(t)ρ_(k)(t)sinρ_(k)(t).Real-valued functions f(t)=ρ(t)cosθ(t)that satisfy the conditionρ≥0,θ′(t)≥0,H(ρcosθ)(t)=ρ(t)sinθ(t)are called mono-components.If f is a mono-component,then the phase derivativeθ′(t)is defined to be instantaneous frequency of f.The above described positive-instantaneous frequency expansion is a generalization of the Fourier series expansion.Mono-components are crucial to understand the concept instantaneous frequency.We will present several most important mono-component function classes.Decompositions of signals into mono-components are called adaptive Fourier decompositions(AFDs).Wc note that some scopes of the studies on the ID mono-components and AFDs can be extended to vector-valued or even matrix-valued signals defined on higher dimensional manifolds.We finally provide an account of related studies in pure and applied mathematics.

A class of iterative greedy algorithms related to Blaschke product

Mobius transforms,Blaschke products and starlike functions as typical conformal mappings of one complex variable give rise to nonlinear phases with non-negative phase derivatives with the latter being de ned by instantaneous frequencies of signals they represent.The positive analytic phase derivative has been a widely interested subject among signal analysts(see Gabor(1946)).Research results of the positive analytic frequency and applications appears in the literature since the middle of the 20th century.Of the positive frequency study a directly related topic is positive frequency decomposition of signals.The mainly focused methods of such decompositions include the maximal selection method and the Blaschke product unwinding method,and joint use of the mentioned methods.In this paper,we propose a class of iterative greedy algorithms based on the Blaschke product and adaptive Fourier decomposition.It generalizes the Blaschke product unwinding method by subtracting constants other than the averages of the remaining functions,aiming at larger winding numbers,and subtracting n-Blaschke forms of the remaining functions,aiming at generating larger numbers of zero-crossings,to fast reduce energy of the remaining terms.Furthermore,we give a comprehensive and rigorous proof of the converging rate in terms of the zeros of the remainders.Finite Blaschke product methods are proposed to avoid the in nite phase derivative dilemma,and to avoid the computational diculties.