Diagonal Sequence Research Articles

A commutator approach to absolute continuity for unbounded Jacobi operators

This paper uses commutator equations to study the absolute continuity of spectral measures associated with certain subclasses of unbounded self-adjoint Jacobi matrix operators determined by properties of the diagonal and subdiagonal sequences. If the diagonal sequence is the zero sequence, properties of the difference sequence of the subdiagonal determine the choice of a bounded operator for the commutator equation. The structure of the resulting commutator leads to results on absolute continuity.

On signed diagonal flip sequences

Eliahou (1999) [1] and Kryuchkov (1992) [3] conjectured a proposition that Gravier and Payan (2002) [2] proved to be equivalent to the Four Color Theorem. It states that any triangulation of a polygon can be transformed into another triangulation of the same polygon by a sequence of signed diagonal flips. It is well known that any pair of polygonal triangulations are connected by a sequence of (non-signed) diagonal flips. In this paper we give a sufficient and necessary condition for a diagonal flip sequence to be a signed diagonal flip sequence.

How well does the Hermite–Padé approximation smooth the Gibbs phenomenon?

In order to reduce the Gibbs phenomenon exhibited by the partial Fourier sums of a periodic function $f$, defined on $[-\pi ,\pi ]$, discontinuous at 0, Driscoll and Fornberg considered so-called singular Fourier-Padé approximants constructed from the Hermite-Padé approximants of the system of functions $(1,g_{1} (z),g_{2} (z))$, where $g_{1} (z)=\log (1-z)$ and $g_{2} (z)$ is analytic, such that $\operatorname {Re}(g_{2} (e^{it}))=f (t)$. Convincing numerical experiments have been obtained by these authors, but no error estimates have been proven so far. In the present paper we study the special case of Nikishin systems and their Hermite-Padé approximants, both theoretically and numerically. We obtain rates of convergence by using orthogonality properties of the functions involved along with results from logarithmic potential theory. In particular, we address the question of how to choose the degrees of the approximants, by considering diagonal and row sequences, as well as linear Hermite-Padé approximants. Our theoretical findings and numerical experiments confirm that these Hermite-Padé approximants are more efficient than the more elementary Padé approximants, particularly around the discontinuity of the goal function $f$.

An infinite dimensional Schur–Horn Theorem and majorization theory

The main result of this paper is the extension of the Schur–Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if ∑ j = 1 n ξ j ⩽ ∑ j = 1 n η j for every n if and only if ξ = Q η for some orthostochastic matrix Q. When ξ and η are summable, requiring additionally equality of their infinite series obtains the same conclusion, extending a theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices.

Limb kinematics during locomotion in the two-toed sloth ( Choloepus didactylus, Xenarthra) and its implications for the evolution of the sloth locomotor apparatus

In order to gain insight into the function of the extant sloth locomotion and its evolution, we conducted a detailed videoradiographic analysis of two-toed sloth locomotion (Xenarthra: Choloepus didactylus). Both unrestrained as well as steady-state locomotion was analyzed. Spatio-temporal gait parameters, data on interlimb coordination, and limb kinematics are reported. Two-toed sloths displayed great variability in spatio-temporal gait parameters over the observed range of speeds. They increase speed by decreasing the durations of contact and swing phases, as well as by increasing step length. Gait utilization also varies with no strict gait sequence or interlimb timing evident in slow movements, but a tendency to employ diagonal sequence, diagonal couplet gaits in fast movements. In contrast, limb kinematics were highly conserved with respect to ‘normal’ pronograde locomotion. Limb element and joint angles at touch down and lift off, element and joint excursions, and contribution to body progression of individual elements are similar to those reported for non-cursorial mammals of small to medium size. Hands and feet are specialized to maintain firm connection to supports, and do not contribute to step length or progression. In so doing, the tarsometatarsus lost its role as an individual propulsive element during the evolution of suspensory locomotion. Conservative kinematic behavior of the remaining limb elements does not preclude that muscle recruitment and neuromuscular control for limb pro- and retraction are also conserved. The observed kinematic patterns of two-toed sloths improve our understanding of the convergent evolution of quadrupedal suspensory posture and locomotion in the two extant sloth lineages.

CBIR Feature Vector Dimension Reduction with Eigenvectors of Covariance Matrix using Row, Column and Diagonal Mean Sequences

Because of the rising demand from wide range of applications the need of faster and better image retrieval techniques is growing day by day. Dimension reduction of CBIR feature vectors has gained momentum for swift image retrieval. The paper presents few novel techniques for image retrieval based on principal component analysis (PCA). Here feature vectors are eigenvectors of covariance matrix obtained using the row mean, column mean, forward diagonal mean, backward diagonal mean and mean combinations of database images. Instead of taking all pixels of database images for PCA, proposed CBIR methods use mean vectors, thus dimension of feature vectors used for image retrieval is reduced resulting in faster retrieval. The proposed CBIR techniques are tested on two different image databases, general image database (1000 images spread across 11 categories) and COIL image database (1080 images spread across 15 object categories). For each proposed CBIR technique 55 queries are fired on general image database, 75 queries are fired on COIL image database and net average precision and recall are computed. The experimental results show that proposed CBIR techniques gives the better performance in terms of higher precision and recall values with lesser computational complexity than the conventional PCA based CBIR using complete image data.

Classical multiple orthogonal polynomials of Angelesco system

In this paper, we analyze more carefully the Rodrigues formula of classical multiple orthogonal polynomials of Angelesco system and consider the sequence of them for the diagonal index case, and then find many properties of such a sequence of orthogonal polynomials. More precisely, we consider the diagonal index sequence { P n , n ( x ) } n = 0 ∞ of classical multiple orthogonal polynomials { P n 1 , n 2 ( x ) } n 1 , n 2 = 0 ∞ of Angelesco system. For such a sequence we give recurrence relations, differential-difference equations, and then we finally find a third order differential equation having { P n , n ( x ) } n = 0 ∞ as solutions.

Palmar and Plantar Pressure While Walking on a Horizontal Ladder and Single Pole in Macaca fuscata

To investigate biomechanical function in the hand and foot during quadrupedal locomotion in nonhuman primates, physical anthropologists and primatologists measure the pressure under them. We collected hand and foot pressure data while a Japanese macaque (Macaca fuscata), a semiterrestrial anthropoid, walked on 2 different simulated arboreal substrates, a horizontal ladder and a single pole, to explore differences in hand and foot use between the 2 substrates. The ladder rungs were perpendicular to the craniocaudal axis of the subject, and the pole was parallel to the subject’s craniocaudal axis. We tested the hypothesis that the pole was a more challenging substrate for the macaque than the ladder. Focusing on a diagonal sequence, diagonal couplets gait, we calculated gait characteristics and computed mean peak-pressure images of the hand and foot for each substrate from individual peak images via translation registration. We found several substrate differences that supported the hypothesis. The Japanese macaque walked at significantly slower speeds when traveling on the pole than on the ladder. Slower travel speed on the pole suggests that the Japanese macaque needed a wider support base to maintain balance on this substrate. Mean peak-pressure images suggest that the ladder invoked a more stepping-like behavior, but the pole invoked a more grasping-like behavior, especially of the foot. We show that the hand and foot use of the Japanese macaque would be adaptable to biomechanical challenges posed by different substrates.

The effect of substrate size on the locomotion and gait patterns of the kinkajou (Potos flavus)

Diagonal-sequence (DS) gaits, which are very rare among mammals, are common and well documented in primates and some arboreal marsupials. DS walking gaits have been reported in the kinkajou (Potos flavus), which shows ecological similarities with primates and arboreal opossums but lacks prehensile specializations of the hindfoot. Nevertheless, the actual frequency of DS gaits and the functional context in which these gaits occur in this highly arboreal mammal remain unknown. We examined the effect of substrate size on the locomotion and gait patterns of kinkajous by recording gaits in two individuals walking and running on poles of two different diameters and on a runway. Diagonality and limb duty factors were calculated for 534 gait cycles. Kinkajous relied mostly on DS gaits and trots during walking, and increased the diagonality of their gait patterns on thinner substrates. The proposed functional link between locomotion on thin branches and the presence of a grasping, primate-like hindfoot is not supported by these data. However, further analysis of kinkajou gait cycles showed that DS gaits may have advantages overlooked earlier. DS gaits, during walking, minimize the distance between two ipsilateral feet during short periods of unilateral bipedality, and per corollary maximize the distance between two contralateral feet during the much longer periods of diagonal bipedality. Such foot positioning during the gait cycle could be beneficial in walking on a relatively thin substrate and could explain why kinkajous adopt DS walking gaits, especially on thinner poles, despite lacking prehensile specializations of the hindfoot.

Gait dynamics of Cebus apella during quadrupedalism on different substrates

Primates are distinguished from many mammals by emphasizing arboreal lifestyles. Primate arboreal adaptations include specializations for enhancing balance and manipulative skills. Compliant gait and diagonal sequence (DS) footfalls are hypothesized mechanisms for improving balance during arboreal quadrupedalism (AQ), while simultaneously permitting vertical peak force reductions sustained by limbs, particularly forelimbs (FLs). Capuchin monkeys (Cebus apella) are arboreally-adapted quadrupeds that use both lateral sequence (LS) and DS footfalls. As tool-users, capuchins experience selective pressures for FL manipulative capabilities, which seemingly conflict with encountering substantial locomotor stresses. We evaluate kinetic and 3-D kinematic data from 172 limb contacts of two adult males on terrestrial and arboreal substrates to address questions about C. apella gait compliancy, kinematics of LS and DS footfalls during quadrupedalism on different substrates, and whether capuchins reduce FL vertical peak forces relative to hind limb (HL) forces more than other primates that use tools or those that do not. Lower vertical peak forces during AQ are consistent with compliant gait, but mixed kinematic results obscure how the reduction occurs. Forearm adduction angle is one consistent kinematic difference between terrestrial and arboreal quadrupedalism, which may implicate frontal plane movements in gait compliancy. Major differences between DS and LS gaits were not observed in kinetic or kinematic comparisons. Capuchins exhibit low FL/HL vertical peak force ratios like several anthropoids, including tool-users (e.g., chimpanzees), and species not considered tool-users in free-ranging conditions (e.g., spider monkeys). Additional selective pressures besides simply tool use appear responsible for the relative reduction in primate forelimb forces.

Effects of support size and orientation on symmetric gaits in free-ranging tamarins of Amazonian Peru: implications for the functional significance of primate gait sequence patterns

The adoption of a specific gait sequence pattern during symmetrical locomotion has been proposed to have been a key advantage for the exploitation of the fine branch niche in early primates. Diverse aspects of primate locomotion have been extensively studied in technically equipped laboratory settings, but evolutionary conclusions derived from these investigations have rarely been verified in wild primates. Bridging the gap from the lab to the field, we conducted an actual performance determination of symmetrical gaits in two free-ranging tamarin species ( Saguinus mystax and Saguinus fuscicollis) of Amazonian Peru by analyzing high-speed video recordings of naturally occurring locomotor bouts. Tamarins arguably represent viable models for aspects of early primate locomotion. We tested three specific hypotheses derived from laboratory studies to test for the influence of support size and orientation and to gain further insight into the functional significance of primate gait sequence patterns: (1) The tamarins utilize symmetrical gaits at a higher rate on small supports than on larger ones. (2) During symmetrical locomotion on small supports, diagonal sequences are utilized at a higher rate than on larger supports. (3) On inclines, diagonal sequences are predominantly used and on declines, lateral sequences are predominantly used. Our results corroborated hypotheses 1 and 3. We found no clear support for hypothesis 2. In conclusion, our results add to the notion that primate gait plasticity, rather than uniform adoption of diagonal sequence gaits, enabled early primates to accommodate different support types and effectively exploit the small branch niche.

Electron-impact ionization of ground and metastable neon

Electron-impact ionization of neutral neon continues to be of interest both from a fundamental and an applied perspective. Non-perturbative calculations are required for the electron-impact ionization of ground and excited states of neon. R-matrix with pseudostates (RMPS) calculations for the electron-impact ionization of the 1s22s22p6 ground state of Ne at higher incident electron energies are now made possible by the further partitioning of each individual LS partial wave across a greater number of processors. Furthermore, instead of a long sequence of parallel matrix diagonalizations, every partial wave diagonalization is now carried out concurrently. The ground-state cross section is found to be in excellent agreement with recent experimental measurements. The previous work of Ballance et al (2004 J. Phys. B: At. Mol. Opt. Phys. 37 4779), for the ionization of the 1s22s22p53s excited state of Ne, has been extended to examine the ionization of the 1s22s22p54s and 1s22s22p55s excited states. Very good agreement is found between R-matrix with pseudostates and time-dependent close-coupling calculations for the 1s22s22p54s excited state. The non-perturbative cross sections for low nl are used to scale semi-classical impact parameter cross sections to high nl. Ionization cross sections for the ground and all excited states will enable the accurate determination of the generalized collisional-radiative ionization rate coefficient for neutral Ne in finite plasmas.

More than multiplication in a 12 × 12 multiplication table

This note presents demonstrations of mathematics that emerges when problems are posed with respect to a combined 12 × 12 multiplication table showing multiplier and multiplicand. Through processes such as recognizing and extending patterns, specializing and generalizing particular functional relationships between the diagonal and row sequences are compressed. Insights obtained from the various methods can be used to deepen teachers’ and students’ understandings of possible ways to bridge arithmetic and algebra.

Kinematics of level terrestrial and underwater walking in the California newt, Taricha torosa

Salamanders are acknowledged to be the closest postural model of early tetrapods and are capable of walking both in a terrestrial environment and while submerged under water. Nonetheless, locomotion in this group is poorly understood, as is underwater pedestrian locomotion in general. We, therefore, quantified the movements of the body axis and limbs of the California newt, Taricha torosa, during steady-speed walking in two environments, both of which presented a level surface: a treadmill and a trackway that was submerged in an aquarium. For treadmill walking at a relative speed of 0.63 snout-vent lengths (SVL)/sec, newts used a diagonal couplets lateral sequence walk with a duty factor of 77%. In contrast, submerged speeds were nearly twice as fast, with a mean of 1.19 SVL/sec. The submerged gait pattern was closer to a trot, with a duty factor of only 41%, including periods of suspension. Environment appears to play a critical role in determining gait differences, with reduction of drag being one of the most important determinants in increasing duration of the swing phase. Quantitative analysis of limb kinematics showed that underwater strides were more variable than terrestrial ones, but overall were strikingly similar between the two environments, with joint movement reversals occurring at similar points in the step cycle. It is suggested that the fundamental walking pattern appears to function well under multiple conditions, with only minor changes in motor control necessary.

Padé Unitarizations: A Critical Look

We use the Linear Sigma Model to test the convergence of different sequences of Pade Approximants when used to unitarize the low energy pi-pi scattering amplitude. We find that, in this particular case, diagonal sequences reproduce the sigma pole with high accuracy, as opposed to other sequences that have been discussed extensively.

Forelimb proportions and kinematics: how are small primates different from other small mammals?

The crouched limb posture of small mammals enables them to react to unexpected irregularities in the support. Small arboreal primates would benefit from these kinematics in their arboreal habitat but it has been demonstrated that primates display certain differences in forelimb kinematics to other mammals. The objective of this paper is to find out whether these changes in forelimb kinematics are related to changes in body size and limb proportions. As primates descended from small ancestors, a comparison between living small primates and other small mammals makes it possible to determine the polarity of character transformations for kinematic and morphometric features proposed to be unique to primates. Walking kinematics of mouse lemurs, brown lemurs, cotton-top tamarins and squirrel monkeys was investigated using cineradiography. Morphometry was conducted on a sample of 110 mammals comprising of primates, marsupials, rodents and carnivores. It has been shown that forelimb kinematics change with increasing body size in such a way that limb protraction increases but retraction decreases. Total forelimb excursion, therefore, is almost independent of body size. Kinematic changes are linked to changes in forelimb proportions towards greater asymmetry between scapula and radius. Due to the spatial restriction inherent in the diagonal footfall sequence of primates, forelimb excursion is influenced by the excursion of the elongated hind limb. Hindlimb geometry, however, is highly conserved, as has been previously shown. The initial changes in forelimb kinematics might, therefore, be explained as solutions to a constraint rather than as adaptations to the particular demands of arboreal locomotion.

Gaits of Japanese macaques (Macaca fuscata) on a horizontal ladder and arboreal stability

Most primates use diagonal sequence (DS), diagonal couplets (DC) gaits when they walk or run quadrupedally, and it has been suggested that DSDC gaits contribute to stability in their natural arboreal habitats compared to other symmetrical gaits. However, this postulate is based solely on studies of primate gaits using continuous terrestrial and arboreal substrates. A particular species may select suitable gaits according to the substrate properties. Here, we analyzed the gaits of Japanese macaques moving on a horizontal ladder with rung intervals ranging from 0.40 to 0.80 m to elucidate the relative advantages of each observed form of gait. The rung arrangement forced our macaques to choose either diagonal coupling or DS gaits. One macaque consistently used diagonal coupling (i.e., DSDC and LSDC gaits) across narrow and intermediate rung intervals, whereas the other macaque used DS gaits (i.e., DSDC and DSLC gaits). At wider rung intervals, both macaques shifted to a two-one sequence (TOS), which is characterized by two nearly simultaneous touchdowns of both forelimbs and one touchdown of each hind limb in a stride. The transition to the TOS sequence increased the duration of support on multiple limbs, but always included periods of a whole-body aerial phase. These results suggest that Japanese macaques prefer DSDC gaits, because the diagonal coupling and DS contribute separately to stability on complex supports compared to the lateral coupling and lateral sequence. We also postulate that stability triggers the transition from symmetrical gaits to the TOS sequence.

Diagonal sequence property in Banach spaces with weaker topologies

We investigate the diagonal sequence property in Banach spaces with weaker topologies. In particular, we present examples of Banach spaces with weaker locally convex topologies which have the diagonal sequence property but are not Fréchet–Urysohn. The examples answer negatively a question of Averbukh and Smolyanov. We give also a very simple proof of the fact that each Banach space contains a subset A whose weak closure includes 0, but 0 is not contained in the weak closure of any bounded subset of A.

Morphology and function of the forelimb in arboreal frogs: specializations for grasping ability?

Frogs are characterized by a unique morphology associated with their saltatory lifestyle. Although variation in the form and function of the pelvic girdle and associated appendicular system related to specialized locomotor modes such as swimming or burrowing has been documented, the forelimbs have typically been viewed as relatively unspecialized. Yet, previous authors have noted versatility in forelimb function among arboreal frogs associated with feeding. Here we study the morphology and function of the forelimb and hand during locomotion in two species of arboreal frogs (Litoria caerulea and Phyllomedusa bicolor). Our data show a complex arrangement of the distal forelimb and hand musculature with some notable differences between species. Analyses of high-speed video and video fluoroscopy recordings show that forelimbs are used in alternating fashion in a diagonal sequence footfall pattern and that the position of the hand is adjusted when walking on substrates of different diameters. Electromyographic recordings show that the flexors of the hand are active during substrate contact, suggesting the use of gripping to generate a stabilizing torque. Measurements of grasping forces in vivo and during stimulation experiments show that both species, are capable of executing a so-called power grip but also indicates marked differences between species, in the magnitude of forces generated. Stimulation experiments showed an increased control of digit flexion in the more specialized of the two species, allowing it to execute a precision grip paralleled only by that seen in primates.

Hallucal grasping behavior in Caluromys (Didelphimorphia: Didelphidae): Implications for primate pedal grasping

A key feature in primate evolution is a foot with a divergent opposable hallucal metatarsal bearing a large peroneal process. Extant primates are characterized by a powerful hallucal grasp—an either “euprimate” or “plesiadapoid-euprimate” ancestor acquisition—that facilitates the exploitation of fine branches, an ability that increased the fitness of ancestral euprimates. In this context, the didelphid marsupial Caluromys has been used as the extant analog to this primate morphotype stage due to some morphological, ecological, and behavioral features. However, the extent to which and the positional and support contexts in which Caluromys uses powerful hallucal grasping are not known. This renders analogies to any mode of “euprimate” or “stem primate” grasping poorly substantiated. The present paper quantifies locomotor and postural behavior, support use, and associated frequencies of hallucal grasping in captive Caluromys philander via analysis of video recordings. During locomotion, Caluromys primarily used diagonal sequence walk, clamber, and climb, whereas stand, foot-hang, and bipedal stand were the dominant postures. Small, fine, horizontal, and moderately inclined branches were frequently used. Overall rates of “apparently powerful hallucal grasps” were high, but were exceptionally high during clamber, climb, foot-hang, and bipedal stand. Additionally, an “apparently powerful hallucal grasp” was very common upon fine, small, steep, and vertical branches. The extensive use of such powerful hallucal grasping provided stability and safety that enabled Caluromys to proficiently utilize fine branches of various orientations. The ability to negotiate such unstable supports, further reflected in foot anatomy, provides evidence that the morphobehavioral complex of Caluromys can serve as an extant analog to the plesiadapoid-euprimate ancestor, represented as a terminal branch feeder with effective hallucal grasping.