Consecutive Elements Research Articles

Skeletal landmarks for TKR implantations: Evaluation of their accuracy using EOS imaging acquisition system

Lower extremity alignment remains one essential objective during total knee replacement. Implants positioning analysis requires selecting reliable skeletal landmarks. Our objective was to in vivo evaluate the precision of the implemented skeletal landmarks. This evaluation was based on multiple three-dimensional (3D) computer reconstructions of the lower extremity derived from an EOS biplanar low-dose X-ray system acquisition. A 3D angle measurement protocol was used. Currently defined landmarks carry a tolerable uncertainty margin, which can still probably be further improved. Nine lower extremity 3D computer reconstructions were obtained from an EOS protocol based on seven simultaneous A-P and lateral views performed in standing position. A database was established by four operators; finally, building up a total of 99 in vivo 3D reconstructions of these nine lower extremities. Specific algorithms were used for such 3D reconstructions of lower extremities based on bone points and pre-identified contours on X-ray. Four femoral landmarks and four tibial landmarks were thus defined. For each bone and each landmark studied, a mean landmark for the 11 consecutive series elements was established. The deviation from each constructed landmark to the corresponding mean landmark was calculated based on the anteroposterior (x), longitudinal (y) and mediolateral axes (z), in translation (Tx, Ty, Tz) and in rotation (Rx, Ry, Rz). Uncertainty was estimated by the 95% confidence interval (95% CI). The landmarks located at the middle of the segment joining the center of each posterior condyle and at the barycenter of the plateaux showed a greater reliability; these landmarks uncertainty (95% CI) of Tx, Ty, Tz was less than 1, 0.5, 1.5 mm for the femur and 1.5, 0.6, 0.6 mm for the tibia, respectively. The femoral landmarks using the center or posterior edge of the posterior condyles to define the mediolateral axis were retained; for rotations Rx, Ry, and Rz, uncertainty remained less than 0.3, 4, and 0.5 degrees. All of the tibial landmarks had a comparable reliability in rotation, 95% of the Rx and Rz deviations were under 0.5 and 1.3 degrees, respectively, with a mean error less than 1 degrees . For the tibial rotation Ry, the mean error was greater (4 degrees), with uncertainty (95% CI) at 11.2 degrees. All tibial translations showed a mean error of 1 mm. The 3D implantation angles were measured on two patients using preoperative 3D skeletal reconstructions and 3D geometric models of the implants repositioned on postoperative EOS knee X-rays. The posterior condyles are rarely involved in the arthritic wear process, making them an anatomic landmark of choice in the analysis of the femoral component positioning. The femoral landmarks using the posterior condyles were sufficiently reliable for clinical use. However, the posterior contours of the tibial plateaux were less precise. The knees should be staggered from an anteroposterior perspective on the EOS lateral images so that they can be visualized separately. The anatomic zones on which the skeletal landmarks are based are usually removed by the bone cuts, making it preferable to save the preoperative computer reconstructions to analyze the postimplantation 3D reconstruction. The lower extremity skeletal landmarks precision relates to the quality of the corresponding 3D reconstructions. Except for tibial rotation, all the translation and rotation parameters were estimated within a mean error margin inferior to 1.2 mm and 1.3 degrees, respectively. Making the reconstruction algorithms more robust would render certain anatomic zones even more precise. Biplanar low-dose EOS X-ray system is a tool of the future to generate 3D knee X-rays that can improve the evaluation and follow-up of total knee arthroplasty patients.

Une remarque sur le spectre des nombres de Pisot

Let θ be a Pisot number less than 2, m a positive rational integer, and A m the set of the polynomials with coefficients in { 0 , 1 , … , m } evaluated at θ. We give a lower bound for the greatest limit point of common differences of consecutive elements of A m . To cite this article: T. Zaïmi, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The Three Gap Theorem and Riemannian geometry

The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.

Asymptotic Behavior of Beta-Integers

Beta-integers (“β-integers”) are those numbers which are the counterparts of integers when real numbers are expressed in an irrational base β > 1. In quasicrystalline studies, β-integers supersede the “crystallographic” ordinary integers. When the number β is a Parry number, the corresponding β-integers realize only a finite number of distances between consecutive elements and are in this sense the most comparable to ordinary integers. In this paper, we point out the similarity of β-integers and ordinary integers in the asymptotic sense, in particular for a subclass of Parry numbers – Pisot numbers for which their Parry and minimal polynomial coincide.

Poisson statistics via the Chinese Remainder Theorem

We consider the distribution of spacings between consecutive elements in subsets of Z / q Z , where q is highly composite and the subsets are defined via the Chinese Remainder Theorem. We give a sufficient criterion for the spacing distribution to be Poissonian as the number of prime factors of q tends to infinity, and as an application we show that the value set of a generic polynomial modulo q has Poisson spacings. We also study the spacings of subsets of Z / q 1 q 2 Z that are created via the Chinese Remainder Theorem from subsets of Z / q 1 Z and Z / q 2 Z (for q 1 , q 2 coprime), and give criteria for when the spacings modulo q 1 q 2 are Poisson. Moreover, we also give some examples when the spacings modulo q 1 q 2 are not Poisson, even though the spacings modulo q 1 and modulo q 2 are both Poisson.

The cross algorithm for phase-aberration correction in medical ultrasound images formed with two-dimensional arrays

Common-midpoint signals in the near-field signal-redundancy (NFSR) algorithm for one-dimensional arrays are acquired using three consecutive transducer elements. An all-row-plus-two-column algorithm has been proposed to implement the one-dimensional NFSR algorithm on two dimensional arrays. The disadvantage of this method is that its ambiguity profile is not linear and a timeconsuming iterative method has to be used to linearize the ambiguity profile. An all-row-plus-two-column-and-a-diagonal algorithm has also been proposed. Its ambiguity profile is linear, but it is very sensitive to noise and cannot be used. In this paper, a novel cross algorithm is proposed to implement the NFSR algorithm on two-dimensional arrays. In this algorithm, common-midpoint signals are acquired using four adjacent transducer elements, which is not available in one-dimensional arrays. Its advantage includes a linear ambiguity profile and a higher measurement signal-to-noise ratio. The performance of the cross algorithm is evaluated theoretically. The region of redundancy is analyzed. The procedure for deriving the phaseaberration profile from peak positions of cross-correlation functions between common-midpoint signals is discussed. This algorithm is tested with a simulated data set acquired with a two-dimensional array, and the result shows that the cross algorithm performs better than the all-row plus-twocolumn NFSR algorithm.

Modified sign method for testing the fractality of Gaussian noise

Fractal Gaussian noise is a stationary Gaussian sequence of zero-mean random variables whose sums possess the stochastic self-similarity property. If the random variables are independent, the self-similarity coefficient equals 1/2. The sign criterion for testing the hypothesis that the parameter equals 1/2 against the alternative H � 1/2 is based on counting the sign change rate for elements of the sequence. We propose a modification of the criterion: we count sign change indicators not only for the original random variables but also for random variables formed as sums of consecutive elements. The proof of the asymptotic normality of our statistics under the alternative hypothesis is based on the theorem on the asymptotics of the covariance of sign change indicators for a zero-mean stationary Gaussian sequence with a slowly decaying correlation function.

The bandpass problem: combinatorial optimization and library of problems

A combinatorial optimization problem, called the Bandpass Problem, is introduced. Given a rectangular matrix A of binary elements {0,1} and a positive integer B called the Bandpass Number, a set of B consecutive non-zero elements in any column is called a Bandpass. No two bandpasses in the same column can have common rows.

Incremental Maintenance of Online Summaries Over Multiple Streams

We propose a novel approach based on predictive quantization (PQ) for online summarization of multiple time-varying data streams. A synopsis over a sliding window of most recent entries is computed in one pass and dynamically updated in constant time. The correlation between consecutive data elements is effectively taken into account without the need for preprocessing. We extend PQ to multiple streams and propose structures for real-time summarization and querying of a massive number of streams. Queries on any subsequence of a sliding window over multiple streams are processed in real time. We examine each component of the proposed approach, prediction, and quantization separately and investigate the space-accuracy trade-off for synopsis generation. Complementing the theoretical optimality of PQ-based approaches, we show that the proposed technique, even for very short prediction windows, significantly outperforms the current techniques for a wide variety of query types on both synthetic and real data sets.

Simple relationships between the ionization energies of one-, two- and three-electron ions of consecutive elements

The ionization energy of one-electron ions can be calculated from a well-known equation that is based on quantum mechanics and on the Bohr model, but no theoretically justified equation is available for the calculation of the ionization energies of multi-electron ions. I report here simple empirical relationships between the ionization energies of one-, two- and three-electron ions of elements whose atomic numbers are Z, Z + 1 and Z + 2. On the basis of these relationships, an equation was constructed for the calculation of the ionization energies of two- and three-electron ions (IE(2el)(Z) and IE(3el)(Z), respectively) as a function of Z only: [equation: see text] where N = the number of electrons, i.e. 2 or 3. For N = 3, this equation is only valid when Z > N, being inaccurate for the neutral Li atom. Graphs of the difference between calculated and experimental values of the ionization energies as a function of Z reveal inaccurate experimental results that are impossible to detect by inspection of the ionization energy itself. On the basis of the present results, more accurate values can be predicted for these ionization energies. A striking example is the inaccuracy of the traditional handbook value of IE(3el)(Fe).

Eleven Euclidean distances are enough

The well-known three-distance theorem states that there are at most three distinct gaps between consecutive elements in the set of fractional parts of the first n multiples of any real number. We generalise this theorem to higher dimensions under a suitable formulation. The three-distance theorem can be thought of as a statement about champions in a tournament. The players in the tournament are geodesics between pairs of multiples of the given real number (modulo 1), two edges play each other if and only if they overlap, and an edge loses only against edges of shorter length that it plays against. According to the three-distance theorem, there are at most three distinct values for the lengths of undefeated edges. In the plane and in higher dimensions, we consider fractional parts of multiples of a vector of real numbers, and two edges play if their projections along any axis overlap. In the plane, there are at most 11 values for the lengths of undefeated edges.

Extended set partitions with successions

We enumerate set partitions by strings of consecutive elements, or successions, and obtain a formula for the number of partitions with successions of arbitrary length. Our approach involves direct operations on the objects within the blocks of partitions. The succession concept is extended to m -regular partitions by means of two algorithms for transforming partitions. We also present a succession-based connection between integer partitions and set partitions, and obtain an application to the enumeration of partitions of arbitrary subsets of { 1 , 2 , … , n } by successions.

An Analogue of the Thue-Morse Sequence

We consider the finite binary words $Z(n)$, $n \in {\Bbb N}$, defined by the following self-similar process: $Z(0):=0$, $Z(1):=01$, and $Z(n+1):=Z(n)\cdot\overline{Z(n-1)}$, where the dot $\cdot$ denotes word concatenation, and $\overline{w}$ the word obtained from $w$ by exchanging the zeros and the ones. Denote by $Z(\infty)=01110100 \dots$ the limiting word of this process, and by $z(n)$ the $n$'th bit of this word. This sequence $z$ is an analogue of the Thue-Morse sequence. We show that a theorem of Bacher and Chapman relating the latter to a "Sierpiński matrix" has a natural analogue involving $z$. The semi-infinite self-similar matrix which plays the role of the Sierpiński matrix here is the zeta matrix of the poset of finite subsets of ${\Bbb N}$ without two consecutive elements, ordered by inclusion. We observe that this zeta matrix is nothing but the exponential of the incidence matrix of the Hasse diagram of this poset. We prove that the corresponding Möbius matrix has a simple expression in terms of the zeta matrix and the sequence $z$.

On uniquely [formula omitted]-determined permutations

Motivated by a new point of view to study occurrences of consecutive patterns in permutations, we introduce the notion of uniquely k -determined permutations. We give two criteria for a permutation to be uniquely k -determined: one in terms of the distance between two consecutive elements in a permutation, and the other one in terms of directed hamiltonian paths in the certain graphs called path-schemes. Moreover, we describe a finite set of prohibitions that gives the set of uniquely k -determined permutations. Those prohibitions make the application of the transfer matrix method possible for determining the number of uniquely k -determined permutations.

Ranking k maximum sums

Given a sequence of n real numbers and an integer parameter k , the problem studied in this paper is to compute k subsequences of consecutive elements with the sums of their elements being the largest, the second largest, …, and the k th largest among all possible range sums of the input sequence. For any value of k , 1 ≤ k ≤ n ( n + 1 ) / 2 , we design a fast algorithm that takes O ( n + k log n ) time in the worst case to compute and rank all such subsequences. We also prove that our algorithm is optimal for k = O ( n ) by providing a matching lower bound. Moreover, our algorithm is an improvement over the previous results on the maximum sum subsequences problem (where only the subsequences are requested and no ordering with respect to their relative sums will be determined). Furthermore, given the fact that we have computed the ℓ th largest sums, our algorithm retrieves the ( ℓ + 1 ) th largest sum in O ( log n ) time, after O ( n ) time of preprocessing.

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \(\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert\), as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form \(A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert\), where \(h:[0,1] \rightarrow {\Bbb C}\) is a continuous function with \(\int_0^1 h(t) \, {\rm d} t \ne 0\), \({\frac{a}{q}}\) runs over \({\cal F}_{\!Q}\), the set of Farey fractions of order Q in the unit interval [0,1] and \({\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}}\) are consecutive elements of \({\cal F}_{\!Q}\). We show that the limit limQ→∞Ah(Q) exists and is independent of h.

Structural collapse analysis of framed structures under impact loads using ASI-Gauss finite element method

The main objective of this study is to devise a technique, which, when implemented into finite-element codes, is efficiently applicable to impact collapse analyses of framed structures. In this study, the formerly developed adaptively shifted integration (ASI) technique for the linear Timoshenko beam element is modified into the ASI-Gauss technique by placing the numerical integration points of the two consecutive elements forming an elastically deformed member in such a way that stresses and strains are evaluated at the Gaussian integration points of the two-element member. On comparison with the ASI technique, the ASI-Gauss technique proves its higher accuracy and efficiency in elastic range. Moreover, instead of applying impact loads in the form of nodal forces, we consider the impact phenomenon by means of contacts between the elements involved and the elemental contact algorithm is verified from the point of conservation of energy. Impact analyses considering member fracture with different sets of parameters are performed using a high-rise framed structure and a small aircraft. From the results obtained, we can observe propagation phenomena of impact loads and shock waves. Also, a proper difference in impact damage is obtained by different sets of parameters. The results also indicate that the mass of the aircraft has a stronger influence on impact damage than its velocity. Moreover, soon after impact, tensile stresses are observed in the columns that were compressed by dead loads before impact.

Optimal leaf ordering of complete binary trees

Ordering a set of items so as to minimize the sum of distances between consecutive elements is a fundamental optimization problem occurring in many settings. While it is NP -hard in general, it becomes polynomially solvable if the set of feasible permutations is restricted to be compatible with a tree of bounded degree. We present a new algorithm for the elementary case of ordering the n leaves of a binary tree with height log n + O ( 1 ) . Our algorithm requires O ( n 2 log n ) time and O ( n ) space. While the running time is a log-factor away from being asymptotically optimal, the algorithm is conceptually simple, easy to implement, and highly practical. Its implementation requires little more than a few bit-manipulations.

Overlaps help: Improved bounds for group testing with interval queries

Given a finite ordered set of items and an unknown distinguished subset P of up to p positive elements, identify the items in P by asking the least number of queries of the type ‘‘does the subset Q intersect P?”, where Q is a subset of consecutive elements of { 1 , 2 , … , n } . This problem arises, e.g., in computational biology, in a particular method for determining splice sites in genes. We consider time-efficient algorithms where queries are arranged in a fixed number s of stages: In each stage, queries are performed in parallel. In a recent bioinformatics paper, we proved optimality (subject to lower-order terms) with respect to the number of queries, of some strategies for the special cases p = 1 or s = 2 . Exploiting new ideas, we are now able to provide improved lower bounds for any p ⩾ 2 and s ⩾ 3 and improved upper bounds for larger s. Most notably, our new bounds converge as s grows. Our new query scheme uses overlapping query intervals within a stage, which is effective for large enough s. This contrasts with our previous results for s ⩽ 2 where optimal strategies were implemented by disjoint queries. The main open problem is whether overlaps help already in the case of small s ⩾ 3 . Anyway, the remaining gaps between the current upper and lower bounds for any fixed s ⩾ 3 amount to small constant factors in the main term. The paper ends with a discussion of practical implications in the case that the positive elements are well separated.

Turbulent channel flow with either transverse or longitudinal roughness elements on one wall

Direct numerical simulation results are presented for turbulent channel flows with two-dimensional roughness elements of different shapes. The focus is mainly on a geometry where the separation between consecutive roughness elements is small and for which the rate of change of the roughness function with respect to the separation between consecutive elements is large. Roughness elements are placed either along the flow direction or orthogonally to it. In the latter case, the drag is increased. For the former case, the possibility of drag reduction reflects the different relative contributions from viscous and Reynolds shear stresses. The Reynolds shear stress depends on the shape of the surface more than the viscous stress and is closely related to the near-wall structures. For orthogonal elements, there is no satisfactory correlation between the roughness function and parameters describing the roughness geometry. On the other hand, a satisfactory collapse of the data is achieved when the roughness function is plotted against the root mean square wall-normal velocity averaged over the plane of the roughness crests. Relative to a smooth wall surface, the Reynolds stress tensor near the wall tends to become more isotropic when the elements are orthogonal to the flow and less isotropic when the elements are aligned with the flow. The interdependencies between the departure from isotropy in the wall region, the organization of the wall structures, and the magnitude of the drag are assessed by examining the rotational component of the turbulent kinetic energy production and the probability density function of the helicity density.