Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.

On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Identification of Mechanical Systems through Kautz Filter

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Tridiagonal Reproducing Kernels and Subnormality

We consider analytic reproducing kernel Hilbert spaces H with orthonormal bases of the form {(a(n) + b(n)z)z(n) : n >= 0}. If b(n) = 0 for all n, then H is a diagonal space and multiplication by z, M-z, is a weighted shift. Our focus is on providing extensive classes of examples for which M-z is a bounded subnormal operator on a tridiagonal space H where b(n) not equal 0. The Aronszajn sum of H and (1 - z)H where H is either the Hardy space or the Bergman space on the disk are two such examples.

Zero- one- and two-dimensional hydrogen-bonded structures in the 1:1 proton-transfer compounds of 4,5-dichlorophthalic acid with the monocyclic heteroaromatic Lewis bases 2-aminopyrimidine, nicotinamide and isonicotinamide

The structures of the anhydrous 1:1 proton-transfer compounds of 4,5-dichlorophthalic acid (DCPA) with the monocyclic heteroaromatic Lewis bases 2-aminopyrimidine, 3-(aminocarboxy) pyridine (nicotinamide) and 4-(aminocarbonyl) pyridine (isonicotinamide), namely 2-aminopyrimidinium 2-carboxy-4,5-dichlorobenzoate C4H6N3+ C8H3Cl2O4- (I), 3-(aminocarbonyl) pyridinium 2-carboxy-4,5-dichlorobenzoate C6H7N2O+ C8H3Cl2O4- (II) and the unusual salt adduct 4-(aminocarbonyl) pyridinium 2-carboxy-4,5-dichlorobenzoate 2-carboxymethyl-4,5-dichlorobenzoic acid (1/1/1) C6H7N2O+ C8H3Cl2O4-.C9H6Cl2O4 (III) have been determined at 130 K. Compound (I) forms discrete centrosymmetric hydrogen-bonded cyclic bis(cation--anion) units having both R2/2(8) and R2/1(4) N-H...O interactions. In compound (II) the primary N-H...O linked cation--anion units are extended into a two-dimensional sheet structure via amide-carboxyl and amide-carbonyl N-H...O interactions. The structure of (III) reveals the presence of an unusual and unexpected self-synthesized methyl monoester of the acid as an adduct molecule giving one-dimensional hydrogen-bonded chains. In all three structures the hydrogen phthalate anions are

One-dimensional hydrogen-bonded structures in the 1:1 proton-transfer salts of 4,5-dichlorophthalic acid with the aliphatic Lewis bases diisopropylamine and hexamethylenetetramine

The crystal structures of the 1:1 proton-transfer compounds of 4,5-dichlorophthalic acid with the aliphatic Lewis bases diisopropylamine and hexamethylenetetramine, viz. diisopropylaminium 2-carboxy-4,5-dichlorobenzoate (1) and hexamethylenetetraminium 2-carboxy-4,5-dichlorobenzoate hemihydrate (2), have been determined. Crystals of both 1 and 2 are triclinic, space group P-1, with Z = 2 in cells with a = 7.0299(5), b = 9.4712(7), c = 12.790(1)Å, α = 99.476(6), β = 100.843(6), γ = 97.578(6)o (1) and a = 7.5624(8), b = 9.8918(8), c = 11.5881(16)Å, α = 65.660(6), β = 86.583(4), γ = 86.987(8)o (2). In each, one-dimensional hydrogen-bonded chain structures are found: in 1 formed through aminium N+-H...Ocarboxyl cation-anion interactions. In 2, the chains are formed through anion carboxyl O...H-Obridging water interactions with the cations peripherally bound. In both structures, the hydrogen phthalate anions are essentially planar with short intra-species carboxylic acid O-H...Ocarboxyl hydrogen bonds [O…O, 2.381(3) Å (1) and 2.381(8) Å (2)].

Low-dimensional hydrogen-bonded structures in the 1:1 and 1:2 proton-transfer compounds of 4,5-dichlorophthalic acid with the aliphatic Lewis bases triethylamine, diethylamine, n-butylamine and piperidine

The structures of proton-transfer compounds of 4,5-dichlorophthalic acid (DCPA) with the aliphatic Lewis bases triethylamine, diethylamine, n-butylamine and piperidine, namely triethylaminium 2-carboxy-4,5-dichlorobenzoate C~6~H~16~N^+^ C~8~H~3~Cl~2~O~4~^-^ (I), diethylaminium 2-carboxy-4,5-dichlorobenzoate C~4~H~12~N^+^ C~8~H~3~Cl~2~O~4~^-^ (II), bis(n-butylaminium) 4,5-dichlorophthalate monohydrate 2(C~4~H~12~N^+^) C~8~H~2~Cl~2~O~4~^2-^ . H~2~O (III) and bis(piperidinium) 4,5-dichlorophthalate monohydrate 2(C~5~H~12~N^+^) C~8~H~2~Cl~2~O~4~^2-^ . H~2~O (IV)have been determined at 200 K. All compounds have hydrogen-bonding associations giving in (I) discrete cation-anion units, linear chains in (II) while (III) and (IV) both have two-dimensional structures. In (I) a discrete cation-anion unit is formed through an asymmetric R2/1(4) N+-H...O,O' hydrogen-bonding association whereas in (II), one-dimensional chains are formed through linear N-H...O associations by both aminium H donors. In compounds (III) and (IV) the primary N-H...O linked cation-anion units are extended into a two-dimensional sheet structure via amide N-H...O(carboxyl) and ...O(carbonyl) interactions. In the 1:1 salts [(I) and (II)], the hydrogen 4,5-dichlorophthalate anions are essentially planar with short intramolecular carboxylic acid O-H...O(carboxyl) hydrogen bonds [O...O, 2.4223(14) and 2.388(2)A respectively]. This work provides a further example of the uncommon zero-dimensional hydrogen-bonded DCPA-Lewis base salt and the one-dimensional chain structure type, while even with the hydrate structures of the 1:2 salts with the primary and secondary amines, the low dimensionality generally associated with 1:1 DCPA salts is also found.

Hydrogen bonding in proton-transfer compounds of 5-sulfosalicylic acid with a series of aliphatic nitrogen Lewis bases

The crystal structures of the proton-transfer compounds of 5-sulfosalicylic acid (3-carboxy-4-hydroxybenzenesulfonic acid) with the aliphatic nitrogen Lewis bases, hydroxylamine, triethylamine, pyrrolidine, morpholine, N-methylmorpholine and piperazine, viz. hydroxyammonium 3-carboxy-4-hydroxybenzenesulfonate (1), triethylaminium 3-carboxy-4-hydroxybenzenesulfonate (2), pyrrolidinium 3-carboxy-4-hydroxybenzenesulfonate monohydrate (3), morpholinium 3-carboxy-4-hydroxybenzenesulfonate monohydrate (4), N-methylmorpholinium 3-carboxy-4-hydroxybenzenesulfonate monohydrate (5) and piperazine-1,4-diium bis(3-carboxy-4-hydroxybenzenesulfonate) hexahydrate (6) have been determined and their comparative structural features and hydrogen-bonding patterns described. Crystals of 4 are triclinic, space group P-1 while the remainder are monoclinic with space group either P21/c (1 - 3) or P21/n (5, 6). Unit cell dimensions and contents are: for 1, a = 5.0156(3), b = 10.5738(6), c = 18.4785(9) Å, β = 96.412(5)o, Z = 4; for 2, a = 8.4998(4), b = 12.3832(6), c = 15.4875(9) Å, β = 102.411(5)o, Z = 4; for 3, a = 6.8755(2), b = 15.5217(4), c = 12.8335(3) Å, β = 92.074(2)o, Z = 4; for 4, a = 6.8397(2), b = 12.9756(5), c = 15.8216(6) Å, α = 90.833(3), β = 95.949(3), γ = 92.505(3)o, Z = 4; for 5, a = 7.0529(3), b = 13.8487(7), c = 15.6448(6) Å, β = 90.190(6)o, Z = 4; for 6, a = 7.0561(2), b = 15.9311(4), c = 12.2102(3) Å, β = 100.858(3)o, Z = 2. The hydrogen bonding generates structures which are either two-dimensional (2 and 5) or three-dimensional (1, 3, 4 and 6). Compound 6 represents the third reported structure of a salt of 5-sulfosalicylic acid having a dicationic piperazine species.

Mutually unbiased bases as submodules and subspaces

Mutually unbiased bases (MUBs) have been used in several cryptographic and communications applications. There has been much speculation regarding connections between MUBs and finite geometries. Most of which has focused on a connection with projective and affine planes. We propose a connection with higher dimensional projective geometries and projective Hjelmslev geometries. We show that this proposed geometric structure is present in several constructions of MUBs.

The molecular bases of training adaptation

Skeletal muscle is a malleable tissue capable of altering the type and amount of protein in response to disruptions to cellular homeostasis. The process of exercise-induced adaptation in skeletal muscle involves a multitude of signalling mechanisms initiating replication of specific DNA genetic sequences, enabling subsequent translation of the genetic message and ultimately generating a series of amino acids that form new proteins. The functional consequences of these adaptations are determined by training volume, intensity and frequency, and the half-life of the protein. Moreover, many features of the training adaptation are specific to the type of stimulus, such as the mode of exercise. Prolonged endurance training elicits a variety of metabolic and morphological changes, including mitochondrial biogenesis, fast-to-slow fibre-type transformation and substrate metabolism. In contrast, heavy resistance exercise stimulates synthesis of contractile proteins responsible for muscle hypertrophy and increases in maximal contractile force output. Concomitant with the vastly different functional outcomes induced by these diverse exercise modes, the genetic and molecular mechanisms of adaptation are distinct. With recent advances in technology, it is now possible to study the effects of various training interventions on a variety of signalling proteins and early-response genes in skeletal muscle. Although it cannot presently be claimed that such scientific endeavours have influenced the training practices of elite athletes, these new and exciting technologies have provided insight into how current training techniques result in specific muscular adaptations, and may ultimately provide clues for future and novel training methodologies. Greater knowledge of the mechanisms and interaction of exercise-induced adaptive pathways in skeletal muscle is important for our understanding of the aetiology of disease, maintenance of metabolic and functional capacity with aging, and training for athletic performance. This article highlights the effects of exercise on molecular and genetic mechanisms of training adaptation in skeletal muscle.