One-dimensional Mapping Research Articles

О связи свойств одномерных отображений управляющих функций с хаосом в уравнении специального вида с запаздывающим аргументом

A differential equation of a special form, which contains two control functions f and g and one delayed argument, is analyzed. This equation has a wide application in biology for the description of dynamic processes in population, physiological, metabolic, molecular-genetic, and other applications. Specific numerical examples show the correlation between the properties of the one-dimensional mapping, which is generated by the ratio f /g, and the presence of chaotic dynamics for such equation. An empirical criterion is formulated that allows one to predict the presence of a chaotic potential for a given equation by the properties of the one-dimensional mapping f /g.

Forts of quadratic polynomials under iteration

Since some dynamical behaviors of a one-dimensional mapping are influenced by the number of forts, attention is paid to the change of the number under iteration. For simple computation we work on quadratic polynomials. We use the theory of polynomial complete discrimination system to give a symbolic algorithm for the number of forts of iterated polynomials and apply the algorithm to quadratic functions, which proves an alternative result that the number either persists to be 1 or tends to infinity under iteration. We further compute the number for iterates of order 2,3,…,7 in the above infinity case and obtain critical values of the parameter at which the number changes. Those changes with finitely many data display a conjectured Fibonacci rule.

The effects of conventional and oval chainrings on patellofemoral loading during road cycling: an exploration using musculoskeletal simulation

The aim of the current investigation was to utilize a musculoskeletal simulation approach to resolve muscle forces during the pedal cycle, to specifically examine the effects of chainring geometry on patellofemoral loading during cycling. 15 healthy male recreational cyclists rode a stationary cycle ergometer at a fixed cadence of 70 RPM in two chainring conditions (round and oval). Patellofemoral loading was explored using a musculoskeletal simulation and mathematical modeling approach. Differences between chainring conditions across the entire pedal cycle were examined using one-dimensional statistical parametric mapping, and patellofemoral force experienced per 20 km was explored using a paired samples t test. No significant (P > 0.05) differences in patellofemoral force or stress were found throughout the pedal cycle between chainring conditions. It was also shown that no significant (P > 0.05) differences in patellofemoral force per 20-km joint were evident (round 38,576.40 N/kg s and oval = 35,637.00 N/kg s). The current analysis found no effects of chainring geometry, on the forces experienced by the patellofemoral joint during the pedal cycle.

Biomechanical analysis of gait waveform data: exploring differences between shod and barefoot running in habitually shod runners

The aim of this study was to utilise one-dimensional statistical parametric mapping to compare differences between biomechanical and electromyographical waveforms in runners when running in barefoot or shod conditions.Fifty habitually shod runners were assessed during overground running at their current 10-km race running speed. Electromyography, kinematics and ground reaction forces were collected during these running trials. Joint kinetics were calculated using inverse dynamics. One-dimensional statistical parametric mapping one sample t-test was conducted to assess differences over an entire gait cycle on the variables of interest when barefoot or shod (p<0.05).Only sagittal plane differences were found between barefoot and shod conditions at the knee during late stance (18–23% of the gait cycle) and swing phase (74–90%); at the ankle early stance (0–6%), mid-stance (28–38%) and swing phase (81–100%). Differences in sagittal plane moments were also found at the ankle during early stance (2, 4–5%) and knee during early stance (5–11%). Condition differences were also found in vertical ground reaction force during early stance between (3–10%).An acute bout of barefoot running in habitual shod runners invokes temporal differences throughout the gait cycle. Specifically, a co-ordinative responses between the knee and ankle joint in the sagittal plane with a delay in the impact transient peak; onset of the knee extension and ankle plantarflexion moment in the shod compared to barefoot condition was found. This appears to affect the delay in knee extension and ankle plantarflexion during late stance. This study provides a glimpse into the co-ordination of the lower limb when running in differing footwear.

Effect of taping on multi-segmental foot kinematic patterns during walking in persons with chronic ankle instability.

To evaluate multi-segmental foot kinematic patterns in chronic ankle instability (CAI) participants during walking, and to investigate the influence of high-Dye and low-Dye taping on these kinematic patterns. Cross-sectional study. Kinematic data of 12 non-injured controls and 15 CAI participants were measured with a three-dimensional motion analysis system during barefoot walking. In addition, the CAI participants walked with high-Dye and low-Dye taping. A rigid Plug-in gait model and the Rizzoli 3D Multi-Segment Foot Model were used to measure multi-segmental foot kinematic patterns. One-dimensional statistical parametric mapping was used to compare barefoot walking of the control and CAI group, and to evaluate differences between walking barefoot and walking with high-Dye and low-Dye taping within the CAI group. Compared to the control group, CAI participants showed a decreased ankle dorsiflexion during loading response (p=0.025) and a more inverted calcaneus in relation to the shank during the initial swing phase (p=0.024). A more inverted position of the metatarsus in relation to the midfoot was observed after low-Dye taping during almost the entire stance phase (p=0.017). No significant differences were found for high-Dye taping. Significant differences in kinematic patterns were found in the ankle joint and rearfoot, but not in the mid- and forefoot in CAI participants. The application of low-Dye taping resulted in a significantly increased inverted position of the forefoot, which can be considered as a less desirable effect for patients with CAI. No other effects of high-Dye and low-Dye taping on kinematic patterns were revealed.

Are two-dimensional measured frontal plane angles related to three-dimensional measured kinematic profiles during running?

ObjectivesTo investigate the temporal relationship between two-dimensional measured frontal plane joint angles and three-dimensional measured kinematic profiles during the stance phase of running, and to assess the intra- and intertester reliability of the two-dimensional angles. DesignObservational study. SettingResearch laboratory. ParticipantsFifteen injury-free elite athletes. Main outcome measuresContralateral pelvic drop (CPD), femoral adduction (FA), hip adduction (HA) and knee valgus (KV) were measured at the deepest landing position during midstance with two-dimensional video analysis during running. CPD, HA and knee abduction were measured continuously during the entire stance phase through three-dimensional motion analysis. One-dimensional statistical parametric mapping was used to examine the temporal relationships between the two-dimensional angles and three-dimensional kinematic profiles. In addition, intra-class correlation coefficients (ICC) were calculated to assess the intra- and intertester reliability of the two-dimensional angles. ResultsTwo-dimensional CPD, FA and HA were significantly related to the three-dimensional HA kinematic profile. Two-dimensional CPD was significantly related to the three-dimensional CPD kinematic profile. No significant relationship was found between two-dimensional KV and three-dimensional knee abduction. Excellent intra- and intertester reliability was found for the two-dimensional angles (ICC 0.90–0.99). ConclusionsThese findings support implementing two-dimensional video analysis to evaluate CPD and HA during running.

基于一维混沌映射和类标准映射的彩色图像加密新算法

提出基于一维混沌映射和类标准映射的彩色图像加密新算法。通过改进的一维混沌映射构造出一种新的类标准映射；通过空间相位图、Lyapunov指数曲线和时间序列测试等对其进行了混沌特性分析，分析结果表明新类标准映射具有混沌参数空间大、良好的随机性等混沌特性。基于一维混沌映射和新类标准映射设计了一种彩色图像加密算法。不同于传统的置乱–扩散机制，该算法采用预先扩散–置乱–扩散结构，在预先扩散阶段采用一维混沌映射产生初始向量对明文图像进行预处理，然后利用本文构造的类标准映射产生随机数对预处理后的图像进行加密。本文算法采用一维混沌系统和多维混沌系统对图像按照依行依列的方式对明文图像进行加密，不但安全性大大提高，而且加快了加密的速度。事实上，本文提出算法的安全性能分析包括密钥空间分析、密钥敏感性分析和统计分析等均表明本文提出的加密算法具有密钥空间大、密钥敏感性强、可抵抗统计分析攻击、蛮力攻击、差分攻击、已知明文和选择明文攻击等优良特性。 In this paper, a new color image encryption algorithm based on one-dimensional chaotic mapping and quasi-standard mapping is proposed. A new quasi-standard mapping is constructed by an improved one-dimensional chaotic map. The chaotic characteristics of the new quasi-standard mapping are analyzed by spatial phase diagram, Lyapunov exponential curve and time series test. The results show that the new quasi-standard mapping has large chaotic parameter space, good randomness and other chaotic characteristics. Then, a color image encryption algorithm is designed based on one-dimensional mapping and new quasi-standard mapping. It is different from the traditional scrambling-diffusion mechanism. The algorithm uses a pre-diffusion-scrambling-diffusion structure, which uses one-dimensional chaotic mapping creating the initial vector to preprocess the plain image in the pre-diffusion stage, and then the pre-processed image is encrypted by generating the random number and using the new quasi-standard mapping. Furthermore, we encrypt the plain image in row by row and column by column, as a result, the security is greatly improved as well as the speed of encryption is accelerated. In fact, the security performance analysis of the proposed algorithm, including key space analysis, key sensitivity analysis and statistical analysis etc., which shows that the encryption algorithm proposed has large key space, strong key sensitivity and can resist various known attacks, such as statistical analysis attack, brute-force attack, differential attack, known plaintext and chosen plaintext attack, etc.

3D Multi-segment foot kinematics in children: A developmental study in typically developing boys

BackgroundThe relationship between age and 3D rotations objectivized with multisegment foot models has not been quantified until now. The purpose of this study was therefore to investigate the relationship between age and multi-segment foot kinematics in a cross-sectional database. MethodsBarefoot multi-segment foot kinematics of thirty two typically developing boys, aged 6–20 years, were captured with the Rizzoli Multi-segment Foot Model. One-dimensional statistical parametric mapping linear regression was used to examine the relationship between age and 3D inter-segment rotations of the dominant leg during the full gait cycle. ResultsAge was significantly correlated with sagittal plane kinematics of the midfoot and the calcaneus-metatarsus inter-segment angle (p<0.0125). Age was also correlated with the transverse plane kinematics of the calcaneus-metatarsus angle (p<0.0001). ConclusionGait labs should consider age related differences and variability if optimal decision making is pursued. It remains unclear if this is of interest for all foot models, however, the current study highlights that this is of particular relevance for foot models which incorporate a separate midfoot segment.

Efficacy of ACL injury risk screening methods in identifying high-risk landing patterns during a sport-specific task.

Screening methods sensitive to movement strategies that increase anterior cruciate ligament (ACL) loads are likely to be effective in identifying athletes at-risk of ACL injury. Current ACL injury risk screening methods are yet to be evaluated for their ability to identify athletes' who exhibit high-risk lower limb mechanics during sport-specific maneuvers associated with ACL injury occurrences. The purpose of this study was to examine the efficacy of two ACL injury risk screening methods in identifying high-risk lower limb mechanics during a sport-specific landing task. Thirty-two female athletes were screened using the Landing Error Scoring System (LESS) and Tuck Jump Assessment. Participants' also completed a sport-specific landing task, during which three-dimensional kinematic and kinetic data were collected. One-dimensional statistical parametric mapping was used to examine the relationships between screening method scores, and the three-dimensional hip and knee joint rotation and moment data from the sport-specific landing. Higher LESS scores were associated with reduced knee flexion from 30 to 57ms after initial contact (P=0.003) during the sport-specific landing; however, no additional relationships were found. These findings suggest the LESS and Tuck Jump Assessment may have minimal applicability in identifying athletes' who exhibit high-risk landing postures in the sport-specific task examined.

Dynamics of the simplest piecewise linear discontinuous mappings

The dynamics of one-dimensional piecewise-linear mapping with one rupture is analyzed. In the parameter space, regions with different dynamic effects are defined.

How reliable are knee kinematics and kinetics during side-cutting manoeuvres?

IntroductionSide-cutting tasks are commonly used in dynamic assessment of ACL injury risk, but only limited information is available concerning the reliability of knee loading parameters. The aim of this study was to investigate the reliability of side-cutting data with additional focus on modelling approaches and task execution variables. MethodsEach subject (n=8) attended six testing sessions conducted by two observers. Kinematic and kinetic data of 45° side-cutting tasks was collected. Inter-trial, inter-session, inter-observer variability and observer/trial ratios were calculated at every time-point of normalised stance, for data derived from two modelling approaches. Variation in task execution variables was regressed against that of temporal profiles of relevant knee data using one-dimensional statistical parametric mapping. ResultsVariability in knee kinematics was consistently low across the time-series waveform (≤5°), but knee kinetic variability was high (31.8, 24.1 and 16.9Nm for sagittal, frontal and transverse planes, respectively) in the weight acceptance phase of the side-cutting task. Calculations conveyed consistently moderate-to-good measurement reliability. Inverse kinematic modelling reduced the variability in sagittal (∼6Nm) and frontal planes (∼10Nm) compared to direct kinematic modelling. Variation in task execution variables did not explain any knee data variability. ConclusionSide-cutting data appears to be reliably measured, however high knee moment variability exhibited in all planes, particularly in the early stance phase, suggests cautious interpretation towards ACL injury mechanics. Such variability may be inherent to the dynamic nature of the side-cutting task or experimental issues not yet known.

Dynamics of the Simplest Piecewise Linear Discontinuous Mappings

The dynamics of one-dimensional piecewise linear mapping with one rupture is investigated. In the space of parameters we defined regions with different dynamical effects.

Dynamical modeling and multi-periodic behavior analysis on pulse train controlled DCM-DCM BIFRED converter

According to four different operating modes of a BIFRED converter, a piecewise smooth linear time-varying model for pulse train (PT)-controlled BIFRED converter operating in dual discontinuous conduction modes (DCM-DCM) is established, from which time-domain waveforms and phase portraits under different loads are obtained by numerical simulations. By analyzing the variation of the charge of an output capacitor in a switching period, a one-dimensional approximate discrete mapping model for PT-controlled DCM-DCM BIFRED converter is derived, in which the multi-periodic behaviors with the variations of the load resistance and the input voltage are studied, and the stability analysis and the estimations of the load resistance ranges are performed. Research findings indicate that the dynamical behaviors depicted by the one-dimensional approximate discrete mapping model are in good agreement with those described by the piecewise smooth linear time-varying model, well revealing the complicated multi-periodic behaviors existing in the PT-controlled DCM-DCM BIFRED converter. By PSIM circuit simulations and physical circuit experimental measurements, the waveforms obtained are consistent with the results of theoretical analysis, effectively validating the feasibility of the two dynamical models.

Can two-dimensional measured peak sagittal plane excursions during drop vertical jumps help identify three-dimensional measured joint moments?

BackgroundLess optimal sagittal plane movement patterns are believed to increase knee injury risk in female athletes. To facilitate clinical screening with a user-friendly method, the purpose of the present study was to examine the temporal relationships between two-dimensional measured sagittal plane kinematics and three-dimensional joint moments during the double-leg drop vertical jump (DVJ) and single-leg DVJ (SLDVJ). MethodsFifty injury-free female athletes were tested. Maximal excursions of hip flexion, knee flexion and ankle dorsiflexion were measured through two-dimensional video analysis. Three-dimensional motion and ground reaction forces were recorded to calculate external hip flexion, knee flexion and knee abduction moments during the entire stance phase of DVJ and SLDVJ. One-dimensional statistical parametric mapping was used to examine relationships between peak two-dimensional kinematic variables and three-dimensional moment profiles. ResultsHip flexion was significantly related to the hip and knee flexion moment for both tests and knee abduction moment for DVJ during the time frames corresponding with highest three-dimensional moments, while knee flexion was significantly related to the hip flexion moment during these time frames. No significant relationships were found for ankle dorsiflexion with any of the joint moments. ConclusionsTwo-dimensional measured sagittal plane hip flexion angles at the deepest landing position were associated with peak joint moments of the hip and knee during DVJ and SLDVJ, while the amount of knee flexion was only associated with the hip flexion moment. Assessment of knee injury risk with two-dimensional video analysis could benefit from measuring maximal hip flexion, more so than knee flexion.

Analysis of the Applicability of the Lattice Boltzmann Method in Targeting a Chaotic Flame Front Model

In this paper, we present utilization of the lattice Boltzmann method (LBM) to analyze spatiotemporal chaotic fields. To this end, the one-dimensional Kuramoto–Sivashinsky equation, which is a proper model for unstable flame front flutter, is rewritten in terms of lattice Boltzmann equation components and is then solved. After accuracy checks, we alter the computational domain size and also the control parameter of the problem, which is representative of the kinematic viscosity in real flows, to capture the symmetry-breaking phenomenon. In the next step, since the method is in fact a one-dimensional explicit mapping for the probability density functions, we adopt the targeting algorithm of chaotic fields using LBM. The results show that the method can be applied to chaotic fields with confidence.

Nonlinear mixed solitary—Shear waves and pulse equi-partition in a granular network

We study primary pulse transmission in a two-dimensional granular network composed of two ordered chains that are nonlinearly coupled through Hertzian interactions. Impulsive excitation is applied to one of the chains (designated as ‘excited chain’), and the resulting transmitted primary pulses in both chains are considered, especially in the non-directly excited chain (the ‘absorbing chain’). A new type of mixed nonlinear solitary pulses–shear waves is predicted for this system, leading to primary pulse equi-partition between chains. An analytical reduced model for primary pulse transmission is derived to study the strongly nonlinear acoustics in the small-amplitude approximation. The model is re-scalable with energy and parameter-free, and is asymptotically solved by extending the one-dimensional nonlinear mapping technique of Starosvetsky (2012). The resulting nonlinear maps governing the amplitudes of the mixed-type waves accurately capture the primary pulse propagation in this system and predict the first occurrence of energy equipartition in the network. To confirm, in part, the theoretical results we experimentally test a series of two-dimensional granular networks, and prove the occurrence of strong energy exchanges leading to eventual pulse equi-partition between the excited and absorbing chains, provided that the number of beads is sufficiently large.

Impact of Knee Modeling Approach on Indicators and Classification of Anterior Cruciate Ligament Injury Risk

The aim of this study was to determine whether using a direct kinematic (DK) or inverse kinematic (IK) modeling approach could influence the estimation of knee joint kinematics, kinetics, and ACL injury risk classification during unanticipated side cutting. The three-dimensional motion and force data of 34 amateur Australian rules footballers conducting unanticipated side-cutting maneuvers were collected. The model used during the DK modeling approach was an eight-segment lower body model with the hip, knee, and ankle free to move in six degrees of freedom. During the IK modeling approach, the same eight-segment model was used; however, translational constraints were imposed on the hip, knee, and ankle joints. The similarity between kinematic and kinetic waveforms was evaluated using the root mean square difference (RMSD) and the one-dimensional statistical parametric mapping (SPM1D). The classification of an athlete's ACL injury risk was determined by correlating their peak knee moments with a predefined injury risk threshold. The greatest RMSD occurred in the frontal plane joint angles (RMSD = 10.86°) and moments (RMSD = 0.67 ± 0.18 N·m·kg(-1)), which were also shown to be significantly different throughout the stance phase in the SPM1D analysis. Both DK and IK modeling approaches classified the same athletes as being at risk of ACL injury. The choice of a DK or an IK modeling approach affected frontal plane estimates of knee joint angles and peak knee moments during the weight acceptance phase of unanticipated side cutting. However, both modeling approaches were similar in their classification of an athlete's ACL injury risk.

ARE TWO-DIMENSIONAL MEASURED SAGITTAL PLANE KINEMATICS RELATED TO THREE-DIMENSIONAL JOINT LOADING DURING THE DROP VERTICAL JUMP TEST?

BackgroundAcute and overuse knee injuries are common in sports involving jump-landing movements, especially in female athletes. More erect sagittal plane movement patterns are believed to increase injury risk. Whilst three-dimensional...

Interface evaluation of experimental dental adhesives with nanostructured hydroxyapatite incorporation

Abstract The aim of this study was to evaluate the adhesive interface with dentin of an experimental adhesive resin with nanostructured hydroxyapatite addition. The organic phase of the adhesive resin was produced by mixing 50 wt.% Bis-GMA, 25 wt.% TEGDMA and 25 wt.% HEMA. CQ and EDAB were added at 1 mol% to all groups, according to the monomer moles. HAnano was added at the following two concentrations: 0 and 2 wt%. One commercial adhesive system was used as control. Nine lower incisor bovine teeth were used to produce interfaces of adhesive resin and dentin. Tooth slices were analysed using the following micro-Raman parameters: a 100 mW diode laser with 785 nm wavelength and spectral resolution of ~ 3–5 cm-1. One-dimensional mapping was performed over a 150 μm line across the adhesive-dentine interface at 1 μm intervals using a computerized XYZ stage. These areas covered the composite resin, adhesive layer, hybrid layer, partially demineralised and un-affected dentine and were visualised and focused at x500 magnification. Raman analysis showed the penetration of experimental and commercial adhesive systems into dentin. HAnano was observed into the hybrid layer. Based on results of the present study, is possible to observe resin and nanostructured hydroxyapatite penetration at the hybrid layer.

一种图像DCT域的混沌加密方法研究

文中针对JPEG图像压缩的特点，提出了一种图像DCT域的混沌加密方法。该算法通过一维Logistic映射迭代产生的混沌矩阵来置乱经过量化的DCT系数矩阵，根据JPEG图像压缩标准，以8 × 8像素块为单位对DCT系数矩阵进行位置置乱，从而使DCT低频成分仍然保留在矩阵的左上角。在混沌序列与置乱后的DCT系数矩阵进行异或操作时，采用只对感兴趣DCT系数加密的方案。利用以上两种方式，不仅提高了图像的加密速度，而且避免了在图像DCT域置乱和异或所导致的图像压缩率降低这一现象。最后利用混沌符号矩阵来扰乱DCT系数的符号位。仿真实验表明该算法具有良好的加密效果，速度快，安全性高，易于硬件实现。In this paper, we suggest one chaos image encryption method in DCT domain according to the characteristics of JPEG image compression. The chaotic models used in this algorithm are Logistic mapping and Chebyshev mapping. The one-dimensional Logistic mapping is used to generate a chaotic sequence which is considered as control matrix for scrambling DCT coefficient matrix. Ac-cording to the JPEG image compression standard, scrambling the DCT coefficient matrix by 8 × 8 pixels in blocks can make the low-frequency composition remain in the upper-left corner of the DCT matrix. When we implement XOR operation between chaotic sequence and DCT coefficient matrix, we only encrypt interested DCT coefficients. Using the above two methods, we not only improve the encryption speed of the image, but also avoid low image compression rates due to scrambling the image encryption method. At last, we use the two chaotic models to generate sign matrix. The simulation results show that the algorithm has good encryption effects, fast encryption speed, and high security.