text:
# Similarity of Triangles

Two triangles whose corresponding angles are equal and whose corresponding sides are proportional are said to be similar.

လိုက်ဖက်ထောင့်များ တူညီပြီး လိုက်ဖက်အနားများ အချိုးညီသော တြိဂံနှစ်ခုကို သဏ္ဌာန်တူ တြိဂံများဟုခေါ်သည်။

Exercise (8.3)

1. Use the given information to tell whether each pair of triangles is similar. Give a reason for each answer.

(a)

$\begin{array}{l} \text{ In}\ \triangle ABC,\ \angle B=80^{\circ} .\\\\ \text{ In}\ \triangle XYZ,\ \angle X=30^{\circ} .\\\\ \therefore \ \ \triangle ABC\ \text{and}\ \triangle XYZ\ \text{are not similar.} \end{array}$

(b)

$\begin{array}{l} \text{ In}\ \triangle ABE,\ \angle AEB=52^{\circ} .\\\\ \text{ In}\ \triangle DBC,\ \angle CDB=36^{\circ} .\\\\ \therefore \ \ \angle A = \angle D, \angle AEB=\angle C\\\\ \therefore \ \ \triangle ABE \sim \triangle DBC \end{array}$

(c)

$\begin{array}{l} \quad\quad\displaystyle\frac{AB}{BC}=\displaystyle\frac{16}{24}=\displaystyle\frac{2}{3}\\\\ \quad\quad\displaystyle\frac{BC}{BD}=\displaystyle\frac{24}{36}=\displaystyle\frac{2}{3}\\\\ \therefore\quad\displaystyle\frac{AB}{BC}=\displaystyle\frac{BC}{AD}\\\\ \quad\quad \angle ABC = \angle CBD = 55^{\circ}\\\\ \therefore\quad \triangle ABC \sim \triangle CBD\quad \text{(SAS)} \end{array}$

(d)

$\begin{array}{l} \quad\quad\displaystyle\frac{AC}{DF}=\displaystyle\frac{18}{6}=3\\\\ \quad\quad\displaystyle\frac{AB}{DE}=\displaystyle\frac{21}{7}=3\\\\ \quad\quad\displaystyle\frac{BC}{EF}=\displaystyle\frac{24}{8}=3\\\\ \therefore\quad\displaystyle\frac{AC}{DF}=\displaystyle\frac{AB}{DE}=\displaystyle\frac{BC}{EF}\\\\ \therefore\quad \triangle ABC \sim \triangle DEF\quad \text{(SSS)} \end{array}$

(e)

$\begin{array}{l} \quad\quad \text{ In}\ \triangle PQR\ \text{and}\ \triangle PST\\\\ \quad\quad \angle P = \angle P\quad (\text{common}\ \angle)\\\\ \quad\quad \angle PQR = \angle PST = 50^{\circ} \quad (\text{given})\\\\ \therefore\quad \triangle PQR \sim \triangle PST\quad \text{(AA)} \end{array}$

(f)

$\begin{array}{l} \quad\quad \text{ In}\ \triangle ABC\ \text{and}\ \triangle ADE\\\\ \quad\quad\displaystyle\frac{AB}{AD}=\displaystyle\frac{48}{16}=3\\\\ \quad\quad\displaystyle\frac{AC}{AE}=\displaystyle\frac{30}{10}=3\\\\ \therefore\quad\displaystyle\frac{AB}{AD}=\displaystyle\frac{AC}{AE}\\\\ \quad\quad \angle BAC = \angle DAE\quad (\text{vertically opposite}\ \angle\text{s})\\\\ \therefore\quad \triangle ABC \sim \triangle ADE\quad \text{(SAS)} \end{array}$

(g)

$\begin{array}{l} \quad\quad \text{ In}\ \triangle PQR,\\\\ \quad\quad\angle Q=180^{\circ}-\left(34^{\circ}+82^{\circ} \right)= 64^{\circ}\\\\ \quad\quad \text{ In}\ \triangle PST, \\\\ \quad\quad\angle TPS=82^{\circ}\\\\ \therefore\quad \angle PST=180^{\circ}-\left(36^{\circ}+82^{\circ} \right)= 62^{\circ}\\\\ \therefore\quad \triangle PQR\ \text{and}\ \triangle PST\ \text{are not similar}. \end{array}$

(h)

$\begin{array}{l} \quad\quad \text{ In}\ \triangle PQR\ \text{and}\ \triangle PST\\\\ \quad\quad\displaystyle\frac{PQ}{PS}=\displaystyle\frac{5}{3}\\\\ \quad\quad\displaystyle\frac{PR}{PT}=\displaystyle\frac{5}{3}\\\\ \therefore\quad\displaystyle\frac{PQ}{PS}=\displaystyle\frac{PR}{PT}\\\\ \quad\quad \angle QPR = \angle SPT\quad (\text{vertically opposite}\ \angle\text{s})\\\\ \therefore\quad \triangle PQR \sim \triangle PST\quad \text{(SAS)} \end{array}$

2. In each of the following triangles, the lengths of certain segments are marked. Find the value of $x$, $y$, $z$, $w$ and $v$.

(a)

$\begin{array}{l} \quad\quad\text{Since}\ DE\parallel BC,\\\\ \quad\quad\displaystyle\frac{CE}{EA}=\displaystyle\frac{BD}{DA}\\\\ \therefore\quad\displaystyle\frac{x}{5}=\displaystyle\frac{2}{6}\\\\ \therefore\quad x=\displaystyle\frac{5}{3}\\\\ \quad\quad\text{Since}\ DE\parallel BC,\triangle ADE \sim \triangle ABC\\\\ \therefore\quad\displaystyle\frac{DE}{BC}=\displaystyle\frac{AD}{AB}\\\\ \therefore\quad\displaystyle\frac{y}{4}=\displaystyle\frac{6}{8}\\\\ \therefore\quad y=3 \end{array}$

(b)

$\begin{array}{l} \quad\quad\text{Since}\ ST\parallel QR,\\\\ (1)\quad \displaystyle\frac{PS}{SQ}=\displaystyle\frac{PT}{TR}\\\\ (2)\quad \triangle PST \sim \triangle PQR\\\\ \therefore\quad\displaystyle\frac{PT}{PR}=\displaystyle\frac{ST}{QR}\\\\ \quad\quad\displaystyle\frac{w}{w+5}=\displaystyle\frac{2}{6}\\\\ \quad\quad 3w=w+5\\\\ \quad\quad w=\displaystyle\frac{5}{2}\\\\ \quad\quad\text{From}\ (1),\\\\ \quad\quad \displaystyle\frac{1}{z}=\displaystyle\frac{w}{5}\\\\ \therefore\quad z=\displaystyle\frac{5}{w}\\\\ \quad\quad z=\displaystyle\frac{5}{5/2}\\\\ \therefore\quad z=2 \end{array}$

(c)

$\begin{array}{l} \quad\quad\text{In}\ \triangle ABC\ \text{and}\ \triangle AED,\\\\ \quad\quad\angle A= \angle A\quad (\text{common}\ \angle)\\\\ \quad\quad\angle ACB = \angle ADE \quad (\text{given})\\\\ \therefore\quad \triangle ABC\sim \triangle AED\\\\ \therefore\quad\displaystyle\frac{AB}{AE}=\displaystyle\frac{AC}{AD}\\\\ \quad\quad\displaystyle\frac{v+4}{6}=\displaystyle\frac{8}{4}\\\\ \therefore\quad v+4 = 12\\\\ \therefore\quad v=8 \end{array}$

3. Find the marked lengths in each of the figures.

(a)

$\begin{array}{l} \quad\quad\text{Since}\ PQ\parallel AD, \triangle ABD\sim \triangle PBQ\\\\ \therefore\quad\displaystyle\frac{BD}{BQ}=\displaystyle\frac{AD}{PQ}\\\\ \quad\quad\displaystyle\frac{12+a}{12}=\displaystyle\frac{9}{6}\\\\ \therefore\quad 12+a = 18\\\\ \therefore\quad a = 6\\\\ \quad\quad\text{Similarly}\ QR\parallel BC, \triangle BCD\sim \triangle QRD\\\\ \therefore\quad\displaystyle\frac{BC}{QR}=\displaystyle\frac{BD}{QD}\\\\ \quad\quad\displaystyle\frac{b}{2}=\displaystyle\frac{12+a}{a}\\\\ \therefore\quad b = 6\\\\ \end{array}$

(b)

$\begin{array}{l} \quad\quad\text{Since}\ DE\parallel BC\\\\ (1)\quad \triangle ADE\sim \triangle ABC\\\\ (2)\quad \triangle BFC\sim \triangle EFD\\\\ \therefore\quad \displaystyle\frac{BF}{EF}=\frac{FC}{FD}=\frac{BC}{DE}\\\\ \therefore\quad \displaystyle\frac{d}{3}=\frac{5}{2}=\frac{c}{4}\\\\ \therefore\quad \displaystyle\frac{d}{3}=\frac{5}{2}\Rightarrow d=\frac{15}{2}=7.5\\\\ \quad\quad \displaystyle\frac{c}{4}=\frac{5}{2}\Rightarrow c=10\\\\ \quad\quad\text{Since}\ \triangle ADE\sim \triangle ABC, \\\\ \quad\quad\displaystyle\frac{AD}{AE}=\frac{DE}{BC}\\\\ \quad\quad\displaystyle\frac{5}{5+e}=\frac{4}{10}\\\\ \therefore\quad 5+e=\displaystyle\frac{25}{2}\\\\ \therefore\quad e=\displaystyle\frac{15}{2}=7.5\\\\ \end{array}$

(c)

ပုံပါပေးထားချက်များအရ လက်တွေ့တွင် အမှန်တကယ်ဆွဲရန် မဖြစ်နိုင်သော degenerated figure ဖြစ်နေသာကြောင့် အဖြေရှာတွက်ပြနိုင်သော်လည်း ထိုအဖြေများ လက်တွေ့တွင် မရှိပါ။ ထို့ကြောင့် ပုစ္ဆာဖြေရှင်းခြင်းသည် အဓိပ္ပာယ်မရှိတော့ပါ။

(d)

$\begin{array}{l} \quad\quad\text{Since}\ DE\parallel BC\\\\ \quad\quad \triangle ADE\sim \triangle ABC\\\\ \therefore\quad \displaystyle\frac{AD}{AB}=\frac{DE}{BC}\\\\ \therefore\quad \displaystyle\frac{8}{8+m}=\frac{10}{15}\\\\ \therefore\quad 8+m=12\\\\ \therefore\quad m=4\\\\ \quad\quad\text{Since}\ DP\parallel BQ\\\\ \quad\quad \triangle ADP\sim \triangle ABQ\\\\ \therefore\quad \displaystyle\frac{AP}{AQ}=\frac{AD}{AB}\\\\ \therefore\quad \displaystyle\frac{p}{6}=\frac{8}{8+m}\\\\ \therefore\quad \displaystyle\frac{p}{6}=\frac{8}{12}\\\\ \therefore\quad p=4\\\\ \end{array}$

(e)

$\begin{array}{l} \quad\quad\text{Since}\ DC\parallel AB, AD\parallel BE, AF\parallel BC\\\\ (1)\quad ABED\ \text{is a parallelogram.}\\\\ \therefore\quad DE=10\\\\ (2)\quad ABCF\ \text{is a parallelogram.}\\\\ \therefore\quad CF=10\\\\ (3)\quad \triangle FEO\sim \triangle FDA\\\\ \therefore\quad \displaystyle\frac{EO}{DA}=\frac{FE}{FD}\\\\ \quad\quad \displaystyle\frac{q}{14}=\frac{6}{6+10}\\\\ \quad\quad q=\displaystyle\frac{21}{4}=5.25\\\\ (4)\quad \triangle EOF\sim \triangle EBC \\\\ \therefore\quad \displaystyle\frac{OF}{BC}=\frac{EF}{EC}\\\\ \quad\quad \displaystyle\frac{r}{12}=\frac{6}{6+10}\\\\ \quad\quad r=\displaystyle\frac{9}{2}=4.5 \end{array}$

4. In the figure, $X Y \parallel P R$ and $V T \parallel Q R$. If $\displaystyle \frac{P T}{T R}=\displaystyle \frac{3}{2}, \displaystyle \frac{Q Y}{Y R}=\displaystyle \frac{2}{1}$ and $P Q=15 \mathrm{~cm}$, calculate

(a) the lengths of $P V, P X$ and $X V$.

(b) the numerical values of $\displaystyle \frac{Y W}{W X}$ and $\displaystyle \frac{V W}{Q Y}$.

$\begin{array}{ll} \text{(a)} & \quad\quad\text{Since}\ VT\parallel QR, \\\\ & \quad\quad\displaystyle\frac{PV}{VQ}=\frac{PT}{TR}\\\\ & \quad\quad\displaystyle\frac{PV}{15-PV}=\frac{3}{2}\\\\ & \quad\quad2 PV=45 - 3 PV\\\\ & \therefore\quad5 PV=45\\\\ & \quad\quad\therefore \quad PV=9\ \text{cm}\\\\ & \quad\quad\text{Since}\ XY\parallel PR, \\\\ & \quad\quad\displaystyle\frac{XQ}{PX}=\frac{QY}{YR}\\\\ & \quad\quad\displaystyle\frac{15-PX}{PX}=\frac{2}{1}\\\\ & \quad\quad 2PX=15-PX\\\\ & \quad\quad 3 PX=15\\\\ & \therefore\quad PX=5\ \text{cm}\\\\ & \therefore\quad XV=PV - PX= 9-5=4\ \text{cm}\\\\ & \therefore\quad QV=15-PV= 15-9=6\ \text{cm}\\\\ \text{(b)} & \text{Since}\ VT\parallel QR, \\\\ & \quad\quad\displaystyle\frac{YW}{WX}=\frac{QV}{VX}\\\\ & \therefore\quad\displaystyle\frac{YW}{WX}=\frac{6}{4}=\frac{3}{2}\\\\ & \text{Since}\ VT\parallel QR, \\\\ & \quad\quad \triangle XVW\sim \triangle XQY\\\\ & \therefore\quad\displaystyle\frac{VW}{QY}=\frac{XV}{XQ}\\\\ & \therefore\quad\displaystyle\frac{VW}{QY}=\frac{4}{10}=\frac{2}{5}\\\\ \end{array}$

5. Given: Parallelogram $B I R D$,

$I G$ bisects $\angle B I R$

Prove: $\displaystyle\frac{B E}{E I}=\frac{R G}{G I}$

$\begin{array}{l} \quad\quad \text{Since}\ BIRD\ \text{is a parallelogram}, \angle B = \angle R\\\\ \quad\quad \text{Since}\ IG\ \text{bisects}\ \angle BIR, \angle BIE = \angle RIG\\\\ \therefore\quad \triangle BIE\sim \triangle RIG\\\\ \therefore\quad\displaystyle\frac{BE}{RG}=\frac{EI}{GI}\\\\ \therefore\quad\displaystyle\frac{BE}{EI}=\frac{RG}{GI}\\\\ \end{array}$

6. Given : $R Q \perp P Q$

$P Q \perp P T$

$S T \perp P R$

Prove: $S T \cdot R Q=P S \cdot P Q$

$\begin{array}{l} \quad\quad \text{Since}\ P Q \perp P T\ \text{and}\ P Q \perp RQ,\\\\ \quad\quad PT\parallel QR\\\\ \therefore\quad \angle TPS = \angle PRQ\quad (\text{alternate}\ \angle\text{s})\\\\ \quad\quad \angle TSP = \angle PQR\quad (\text{alternate}\ \angle\text{s})\\\\ \therefore\quad \triangle PST\sim \triangle RQP\\\\ \therefore\quad\displaystyle\frac{ST}{PQ}=\frac{PS}{QR}\\\\ \therefore\quad ST\cdot RQ = PS\cdot PQ \end{array}$

7. Given : Parallelogram $A B C D$ ; $P Q \parallel M B$

Prove : $\triangle A B M \sim \triangle C Q P$ .


text:
# Definition:Fixed Field

## Definition

Let $F$ be a field.

Let $G \le \Aut F$ be a subgroup of the automorphism group of $F$.

The fixed field of $G$ is the set:

$\Fix G = \set {f \in F : \forall \sigma \in G : \map \sigma f = f}$

## Also denoted as

The fixed field of $G$ can also be denoted $F_G$ or $\map {\operatorname {Fix}_F} G$ to emphasize that it is contained in $F$.

format: plain


text:
# Electric potential

1. Sep 28, 2007

### latour1973

An alpha particle (q=+2e, m=6.64*10^-27 Kg) is emitted in a radioactive decay with KE=5.53MeV what is its speed?

can someone help me to slove this problem......

2. Sep 28, 2007

### genneth

The rules of the forum say that you have to display some working, so that we know where to begin.

In this case, the obvious question is: is it relativistic?


text:
Introduction

The speed at which we walk and the performance of a parallel task with a mobile phone will change how we walk1,2,3,4,5,6. Previous studies have established the effect of mobile phone use while walking, on the spatiotemporal parameters of walking3,7,8,9,10,11,12,13,14,15. However, to date just a few studies have investigated walking dynamics under this dual task condition16,17,18,19. These studies provide relevant and valuable information, but their approach to measuring the dynamics of walking are limited by two methodological aspects. Firstly, participants in these studies walked on a treadmill16,17,18,19. This may induce walking dynamics that differ from those observed in everyday settings involving over ground walking20,21. Secondly, a reduction in gait velocity appears inherent when performing a mobile phone dual task3,7,8,9,10,11,12,13,14,15. Therefore, a fixed walking speed may inadvertently impose what is actually a sub-optimal walking speed for that dual task condition. Therefore, the current study presents an alternative protocol using over ground walking and self-determined non-fixed walking speeds to more accurately reflect the dynamics of natural walking.

However, this protocol alone is not enough if traditional linear methods are subsequently used for analysis. Such methods only provide an indication of the extent of variability around a central point or mean. While some work has been done to define “optimal windows” of variability22,23, the potential for detecting dynamic change through linear analysis is fundamentally limited. This is because the analysis of mean change, by definition, ignores or averages over any dynamic change that may be present. Therefore, to investigate walking as a dynamic behaviour nonlinear analysis methods must be used. Dynamical systems theory (DST) states that nonlinear analysis methods the investigation of dynamic behaviour, which can be defined as the patterns of change in a system state over time24. In order to achieve functional healthy walking continuous adjustment in response to continuous stream of external and internal sensory input is required. Nonlinear analysis methods and DST provide a framework for understanding these adjustments and can therefore provide insight into the complexity of human movement25,26. A consensus on what constitutes a measure of complexity in human movement is difficult to reach. Primarily because of the inherent difficulty of establishing a ground truth for complexity. There are several definitions of complexity in the literature, and although differing in some aspects, each definition is linked through the acknowledgement that a complex system is one “involving many interacting subsystems27. Therefore, a full analysis of system complexity would require the analysis of each sub-system across multiple time scales. Since this is not always feasible, it may be more reasonable to aim for a partial analysis of system complexity through the extraction of a number of different traits representing the evolution pattern of a dynamic system over time. This is attempted in the current study through the analysis of regularity and predictability. Traits that can help to capture changes in complexity28. The ability to identify a system of lower complexity is relevant when investigating the effects of pathological conditions and aging28,29,30.

Acceleration of the lower trunk is an appropriate measure of walking dynamics, since it reflects both lower limb movement and the stabilization of the head, trunk and arms31,32,33. The root mean square ratio (RMSratio) was used to provide an indication of the effect of walking speed instruction and dual task conditions on the variability of acceleration along each measurement axis. The root mean square of acceleration is a more conventional measure of variability in trunk acceleration34, but a ratio of acceleration in each axial measurement direction to the modulus of acceleration is preferred because it facilitates comparisons across walking speeds35. It is reported alongside the sample entropy of trunk acceleration (SaEn) and the maximum Lyapunov exponent (MaxLyE) derived from the acceleration time series. SaEn is interpreted to determine how the regularity of walking—considered as a trait of complexity in the current study—is affected by the change in walking conditions. Further the MaxLyE is used to estimate the ability to respond to small local perturbations (local dynamic stability) while walking36, introduced by the noise interfering with movement execution.

Our study protocol used over ground walking, structured text messaging and semi-structured mobile phone conversations, to address the lack of scientific protocols aimed at capturing the natural walking dynamics under dual task conditions in a non-laboratory setting. As such, this study is the first to provide empirical evidence and insight for researchers, clinicians and practitioners, on the effects of walking speed and mobile phone use on the natural walking dynamics of young adults in ecologically valid setting using nonlinear analysis methods. The hypothesis was that change in walking dynamics would be affected by walking speed instruction and mobile phone use, when compared with the single task (walking only) condition. In particular, that local dynamic stability would decrease at a sub-optimal walking speed37 (induced by the fast walking speed instruction) and with mobile phone use due to the effect of the attentional demands produced by concurrent tasks38,39.

Results

Variability of trunk acceleration

Multivariate analysis indicated a significant effect of walking speed instruction (Λ = 0.37, F3, 17 = 9.6, p = 0.001) on the RMSratio (Table 1). Subsequent univariate analyses confirmed this effect along the VT (p < 0.001) and AP (p < 0.001) measurement axes (Table 1). Based on the average values presented in Fig. 1, the ratio of AP acceleration to the modulus of acceleration decreased from a normal to fast walking speed across all conditions while the ratio increased along the VT measurement axis.

Nonlinear outcome measures

Multivariate analysis indicated a significant influence of both walking speed instruction (Λ = 0.26, F3, 17 = 16.24, p < 0.001) and task (Λ = 0.36, F6, 72 = 8.00, p < 0.001) on SaEn (Table 2). Univariate analyses confirmed a significant effect of speed along the AP (p < 0.05), VT (p < 0.001) and ML (p < 0.001) measurement axes. On average SaEn values increased from normal to fast walking speed across all conditions. When comparing task conditions, significant effects were observed in the VT (p < 0.001) direction (Table 2). Post hoc analysis using a Bonferroni adjustment indicated that this significant decrease was only evident when walking while texting was compared with walking in the single task condition (Fig. 2).

Multivariate analysis indicated a significant effect of task (Λ = 0.61, F6, 72 = 3.41, p < 0.05) on MaxLyE (Table 3). This significant effect was confirmed by univariate analysis (Table 3), where an increase in MaxLyE was observed along the VT and AP measurement axes. Post hoc analysis using a Bonferroni adjustment indicated that these significant increases were only evident when walking while texting was compared with walking in the single task (walking only) condition (Fig. 3).

Walking speed

Multivariate analysis indicated a significant effect of both walking speed instruction (Λ = 0.18, F1,19 = 85.22, p < 0.001) and task (Λ = 0.20, F2,18 = 35.99, p < 0.001) on walking speed (Table 4). The interaction at a multivariate level between task and walking speed instruction was marginally below the level of statistical significance (Λ = 0.72, F2, 18 = 3.44, p = 0.054). Univariate analysis confirmed the effect of task (F2, 38 = 52.95, p < 0.001) and walking speed instruction (F1, 19 = 85.22, p < 0.001), but also indicated a significant interaction effect between the two (F2, 38 = 3.66, p < 0.05). Post hoc analysis, using a Bonferroni adjustment showed significant differences in walking speed between each task condition. The trend in walking speed across task conditions can be seen in Fig. 4, where the dashed line represents the walking speed following fast walking speed instruction.

Discussion

In this study, the dynamics of over ground walking among young, healthy adults were investigated. Walking was measured under three different mobile phone use conditions, performed at two different walking speeds. Our findings suggest that both walking speed and mobile phone dual task affected walking dynamics. At a self-selected fast walking speed, these effects were evident in an increase in the sample entropy of trunk acceleration and a shift in the ratio of trunk acceleration in favor of the VT measurement axis. Further, with the addition of a texting task while walking local dynamic stability reduced and sample entropy decreased.

Concerning walking speed, the observed shift in the ratio between axial acceleration and the modulus, from the AP measurement axis, in favor of the VT measurement axis is somewhat unexpected. Previously a shift in the RMSratio in the ML measurement axis was reported35, whereas no significant shift in the ratio of acceleration along this axis was observed in the current study. With regard to changes in sample entropy, our results are similar to those previously observed in the stride parameters of young adults walking at different walking speeds6.

Concerning the effects of mobile phone use, our findings are less surprising but still somewhat in contrast with the findings of previous studies. Firstly, the effect of mobile phone dual tasks were not the same. Whereas the dynamics of walking while talking on a mobile phone showed little difference from walking dynamics in the single task condition, there were differences observed when walking while texting. Under this condition, a significant increase in MaxLyE value along the VT measurement axis was observed; suggesting decreased local dynamic stability. Similar findings of decreased local dynamic stability have previously been reported in a study similar to the current study, but making use of a treadmill16. However, the increases in that study were along the AP and ML measurement axes16. Moreover, a further two comparable treadmill-based studies reported no changes in dynamic stability17,18. Clearly, a definitive consensus is difficult to achieve and a number of potential explanations for these discrepancies must be considered.

The first possible explanation is that walking while texting is detrimental to the walking dynamics of young adults causing reduced local dynamic stability. An increased MaxLyE value is indicative of a less dynamically stable system. For example, an increase in MaxLyE values has been repeatedly observed among fall-prone older adults—a group typically considered to represent poor walking ability40,41,42. However, compared with a relative difference in MaxLyE values between a healthy and a patient group with a known neurological condition impairing gait (a difference of approximately 21%)43, the differences in MaxLyE values between single task walking and walking while texting was relatively small (14% and 9% at normal and fast walking speeds respectively). Therefore, it would seem that while the present changes observed when walking while texting change the dynamics relative to normal walking, they might not be sufficient to have practical implications in terms of local dynamic stability.

A second possible explanation acknowledges the possibility of an inverse relationship between sample entropy and maximum Lyapunov exponent, whereby higher entropy may lead to a lower exponent value as suggested by recent work investigating the effects of noisy time series44. For our results, this would mean that some of the observed task-effect could be explained by the changes in walking speed. Moreover, while the current study reports an increased MaxLyE along the VT measurement axis, it has been stated that the ML axis that is the “best candidate” to detect detrimental changes in local dynamic stability43.There was no significant change along this axis. In addition, MaxLyE values in the VT have been shown to increase with walking speed without any additional dual task37, but it must also be mentioned that methodological differences make direct comparisons difficult. A difficulty highlighted in a recent review on the MaxLyE45.

Our interpretation is that both explanations are likely to play a role in our findings. It seems unlikely that texting while walking would not cause distraction, but it may be that young adults have the cognitive acuity to handle the competition for attentional resources and the changed demand on visual processing systems, with only slight adjustments to their dynamics of walking19,38,39,46. However, further research is required to confirm the link between biomechanical variables, neural processes and perception.

A strength of our study is a protocol design aimed at capturing the natural walking dynamics. However, it is not without limitations. Firstly, the choice of over ground walking led to a shorter time series for analysis (30 strides). This leads to theoretical difficulty in calculating the MaxLyE and achieving a stable estimate. Fifty or 150 strides have been recommended to achieve a stable estimate44,47. The current study presents a protocol design that can be considered ecologically valid meaning that longer time series would not represent real texting while walking task. Thus, the stability of the MaxLyE estimate was investigated and it was found to be relatively stable at 30 strides (Supplementary information). Secondly, over ground walking at a self-determined, non-fixed, walking speeds allows for changes in walking speed during each walking trial. This makes it difficult to distinguish the effect of walking speed from the other concurrent task like texting. To address this, walking speed across task conditions was compared and showed similar trends in walking speed across conditions (Table 4 and Fig. 4). Thirdly, while over ground walking was chosen to reflect the natural walking dynamics, it does introduce the element of changing sensory input arising from the walking environment (e.g. changing visual input as the participant progresses along the corridor). However, since environmental conditions were largely identical across conditions in the current study, this is also judged to have had little effect on the reported results. A further final limitation of the current study lies in the choice of algorithm for the detection of the maximum Lyapunov exponents, as each algorithm has its own inherent limitations.

The Wolf algorithm was chosen in the current study, as it appears to be more appropriate for short experimental data sets; however, it is susceptible to noise and relies on the selection of initial parameters48. To account for these limitations, a low pass filter was applied to reduce the noise level as suggested44,49,50. In addition, the selected initial parameters are reported in the methods section to facilitate comparison in future studies. The embedding dimension and time delay were fixed (average of the individual embedding dimension and time delay of each series) in line with the recommendations of recent findings51,52.

In conclusion, our findings suggest that both walking speed and mobile phone use affect the walking dynamics of over ground walking among young, healthy adults. Walking speed had a clear effect on the variability of trunk acceleration as quantified by RMSratio, and the complexity or regularity of trunk accelerations as quantified by sample entropy. The mobile phone dual task of walking while talking had no significant effect on walking dynamics, whereas texting while walking increased the variability of trunk acceleration, decreased the regularity or complexity of trunk accelerations, and decreased the local dynamic stability of walking as quantified by the maximum Lyapunov exponent. Despite these changes in walking dynamics, the current sample of young, healthy adults, were able to cope with the mobile phone dual task conditions. Future research in this area may consider more collaboration across research groups aimed at implementing the next-generation tools proposed by Ioannidis53, such as prospective meta-analyses, as an effective and efficient approach to achieving consensus on these important questions. Nonlinear analysis methods need to be supported by further evidence and research so that their practical implications and implementation can be realized.

Methods

Participants

Twenty healthy young adults (11 males, 9 females; age: 27 ± 5.5 yrs. (mean ± 1 standard deviation); height: 174 ± 7.7 cm and body mass: 71 ± 10.6 kg) participated. A sub-sample of the present participants (nine out of 20) has previously been used to investigate the changes in spatiotemporal stride parameters based on shoe-worn accelerometers9, however, in the current study trunk worn accelerometers, not used elsewhere, are used. Participants were required to have had possession of their current mobile phone for longer than one month and engage in regular use (at least once every hour). All participants provided written informed consent as per the ethics committee of the North Denmark Region (LBK nr. 1083) guidelines and the declaration of Helsinki. This study was carried out following approval from the secretariat for the Scientific Ethics Committee in the North Denmark Region.

Experimental protocol

Participants repeated six conditions consisting of self-selected normal and fast speed over ground walking, while texting on a mobile phone, talking, or performing no concurrent task. As such, each participant completed 12 walking trials in total along an approximately 80-m long and 1.6 m wide indoor office corridor. Walking trials were performed after working hours and under well-lit conditions. The participant’s walking speed was self-determined following simple instruction to walk at a normal or fast walking speed (e.g. “this time I would like you to walk at a normal speed”). The sequence of conditions followed a randomized partial counterbalance design and were without specific task prioritization. This is because young adults seem to exhibit a prioritization-dependent increase or decrease gait speed and texting accuracy54. A tri-axial accelerometer Physilog (Gait Up SA, Lausanne, Switzerland) was attached to the skin at the level of L4/L5 spinous process to record trunk accelerations during each trial. Raw accelerations files (.bin) were downloaded using the Research Toolkit v1.5.0 (Gait Up SA, Lausanne, Switzerland) and subsequently imported to Matlab R2018b (MathWorks, Natick, MA).

Accelerometer data pre-processing

Data was sampled at 200 Hz. A correction procedure described by Moe-Nilsson (1998) 55—consisting of decomposing acceleration signals into the static gravity and dynamic acceleration components—removed the confounding effects of both the gravity measurement and inevitable accelerometer misalignment55.

Stride detection

Thirty strides were located in the mid-section of the recorded signal using the ‘peakfinder’ Matlab function with a minimum peak distance of 50 samples for fast walking speed trials and 75 samples for normal walking speed trials. Upward and downward peak detection threshold was set as:$$Threshold=rms\left(VT\right)-0.05$$, where VT is the acceleration along the vertical measurement axis. To facilitate a more accurate peak detection a 5th order zero-phase digital band pass filter [0.5–10 Hz] was applied. This filter was used for stride detection only. A single stride was the period between consecutive heel strikes of the same limb (i.e., signified by every second acceleration peak in the VT signal, counting from the end of the first stride)56.

Calculation of walking speed

Walking speed was calculated as the time taken to cover the 80-m walking distance (i.e. for the entire trial and not just the 30-stride mid-section) and reported as meters per second.

Calculation of the root-mean-square ratio

The RMSratio describes the change in the relative magnitude of acceleration along each measurement axes while walking, reflecting the variability in acceleration. Since the magnitude of acceleration will be higher at increasing walking speeds57,58, comparison between trials at different walking speeds are confounded without the use of a ratio. Therefore, we chose to calculate the RMSratio over the traditional root-mean-square, as it is robust to differences in walking speeds35. RMSratio was calculated for trunk accelerations along the AP, VT, and ML measurement axes. Calculation followed two steps, (1) the calculation of the total vector magnitude [Eq. (1) and (2)] calculation of the RMSratio (Eq. 2):

$${T}_{RMS}=\sqrt{{RMS}_{AP}^{2}+{RMS}_{VT}^{2}+{RMS}_{ML}^{2}}$$
(1)

where TRMS is the RMS total vector magnitude.

$${RMS}_{ratio}={RMS}_{d}/{T}_{RMS}$$
(2)

where RMSratio is the ratio between RMS of acceleration (m/s2) in each direction and the RMS of total vector magnitude and ‘d’ indicates the direction of acceleration.

Nonlinear metrics

Takens’ theorem states that it is possible to construct characteristically identical copies of a dynamic system using a number of time-delayed copies of the trajectory evolution of a single point in that system59. Using this construction, it is then possible to analysis the patterns of system behaviour. Since it has been suggested that the Wolf algorithm is particularly susceptible to the effects of noise, we applied a zero phase 5th order low pass Butterworth filter, with a cut off frequency of 50 Hz. All calculations were performed in Matlab R2019b (The MathWorks, Natick, MA). SaEn and MaxLyE calculations were completed with the help of the University of Nebraska at Omaha Biomechanics Nonlinear Analysis Toolbox (UNO Biomechanics 2019, https://www.mathworks.com/matlabcentral/fileexchange/71907-uno-biomechanics-nonlinear-analysis-toolbox).

Calculation of sample entropy

SaEn was calculated for all walking trials as an indicator or trait of complexity using the method described by Richman & Moorman60 and implemented by others61,62. A lower value of SaEn (unit-less) signifies a greater regularity of a time series or a trait of lower complexity27,60. For the analysis, m = 2 and r = 0.3*standard deviation.

Calculation of the maximum Lyapunov exponent

We calculated the MaxLyE for each participant under all conditions. An increase in the MaxLyE indicates a decrease in local dynamic stability45. MaxLyE (bits) was calculated using the Wolf algorithm49. This algorithm uses the average divergence of nearby trajectories from a single reference trajectory, providing the largest MaxLyE representing the largest rate of divergence45. Since walking at different speeds produces time series of different length, which can influence the MaxLyE estimation63, all trials were resampled to uniform length 3000 data points without enforcing a uniform number of samples per stride. The average of mutual information analysis was used to determine the appropriate time delay (τ) at which sufficient new information about the evolution of the system was apparent and we assessed the appropriate number of embedding dimensions (dE) by identifying false nearest neighbours using global false nearest neighbour analysis. τ and dE were calculated individually for each time series, however to facilitate comparison between trials, fixed τ and dE values were used for MaxLyE calculation51. Fixed τ and dE values were derived as the overall average of the individual trials τ and dE values (τ = 12 and dE = 7). R-tolerance and A-tolerance were set at 15 and 2 respectively, while the evolutions were set between 0.5 and 1.5 cycles (evolve iteration = 200)49. Since the calculation was over just 30 strides, producing a relatively short time series, a sensitivity analysis was performed to verify the consistency of the estimate. This sensitivity analysis consisted of plotting the MaxLyE estimate at each evolution of the algorithm. The plots resulting from this analysis can be found in the Supplementary Information.

Statistical analysis

We used Q–Q plot and Kolmogorov–Smirnov tests to assess the distribution of data. A 2 × 3 within-subject repeated-measures analysis of variance (2 × 3 RMANOVA) assessed the influence of walking speed (normal and fast), task (single task, dual task of texting on a mobile phone, dual task of talking on a mobile phone), and interactions between the walking speed and task on RMSratio, SaEn, and MaxLyE. We report both the Wilk’s Lambda (Λ), where lower values indicate a larger effect contribution and the test-statistic indicating the F-distribution. For walking speed, the influence of walking speed instruction and task, and interactions between instruction and task, were investigated. In the case of a significant finding at the multivariate level, analysis at a univariate level followed to investigate each of the outcomes individually. Finally, post-hoc analyses were conducted, using a Bonferroni adjustment for multiple comparison, to assess pair-wise comparisons between the three task conditions. All statistical analysis was done using SPSS (IBM Statistics Data Editor V24). p < 0.05 was considered significant.


text:
# Packaging System and dividing FPC - Lazarus into packages

## Introduction

FPC is quite a large body of source, and the organisation of the source is a constant problem, with immense effects on release engineering, deployment and packaging. On this page I try to describe some of the problems involved with how FPC is packaged. This is not a constant situation, it evolves as new packages, dependencies and insights are acquired.

This page is updated for the changes in early 2008 when fcl was split up into packages, and all packages (fv + fcl subpkgs + extra + base) reside in packages/ and have an own dependance system for build order.

## Order of building

Besides merely splitting up the source into packages, there is also a dependency and order-of-building problem. Currently, the order of compilation is split up in certain generations:

1. rtl-compiler-rtl-compiler-rtl-compiler (the cycle)
2. RTL
3. packages/( a collection of packages)
4. IDE / utils

Note, when packages in packages/ have dependancies on other packages, then the makefile(.fpc) in packages must be also fixed, to allow smooth paralel compilation.

## Requirements

• build order
• the order of building should be fairly predictable and static as not to hinder development too much, and the same on all OSes.
• Dependencies on external libraries shouldn't be introduced to soon in the dependency hierarchy. (see also separate paragraph)
• The part of the RTL that is compiled three times as part of the bootstrap should be as small as possible (see Outer RTL paragraph)
• package format
• In the future, some packages (like Indy) might be externally maintained. These must be clearly marked as such, so that only emergency fixes and FPC specific release engineering are done on these packages. (other enhancement should go through the package owners)
• It should be possible to map FPC package structure on lazarus packages and Delphi style dynamically loadable packages. Directly or indirectly.
• At the present stage, the opinions about if there should be a lot of dynamically loadable packages (e.g. one per compilation pkg), or one big one are still divided. The package format should ideally support both. (IOW Multiple packages should be able to be fused into one .dll/.bpl/.so)
• The same with respect to: source and binary packaging. Both should be possible AND AT THE SAME TIME.
• The package format should provide lots of metadata for visual use. one line descriptions for units, multiple lines descriptions for the package itself.

## Tips

• avoid package/unit/class names that are too general like "Image", "xml" or "sockets", to avoid annoying name clashes with Delphi and other 3rd party pkgs, unless the package is really as compat as possible substitute for the Delphi package. Prefix "fp" if necessary.

### External Library Dependencies

One should be careful where packages are put that have external libraries as dependencies, this because other packages that don't require the dependency on the lib itself, acquire it because they use some parts of a package that requires it.

Examples:

• if one makes the RTL depend on some graphical library which is used for Unit Graph, all programs with that RTL might inherit that dependency
• Another example (and true problem): the general db support in the FCL which depends on the client libraries of all supported databases.

This gets even more important if we start bundling packages into dynamic libs. If we would stuff all units in one big lib (which is easiest from a deployment and maintenance view), this big shared lib would have three dozen library dependencies, from Oracle to little things like X widget sets.

### Breaking the dependency cycle

If you have a package that _must_ be relatively early in the hierachy, but there is a dependency on a library that is only sideways relevant, then you have a problem. There are several general solutions:

• If OOP, try to use plugin classes that can be specialized later.
• If procedural Use the "driver" model. Try to build a record of procedurevariables that encapsulate the functionality, and can be registered later. Examples : threading (cthreads), memory drivers (cmem), widestrings (cwstring), local (clocale)
• Go for a fully dynamical approach with a library plugin architecture. (SF, needs advanced dynamic linking support)

### Outer RTL

From time to time one sees the term "Outer RTL" on the FPC lists. The "Outer RTL" is a phrase coined in discussions about decreasing the size of the current RTL that is compiled 4 times in every build process. This mainly to decrease build time and thus make compiler development less annoying. Essentially it means that all units not required by the compiler should move out of the current RTL package.

Some of these units can move to independant packages. Outer RTL is a name for a package that is more or less what remains. E.g.:

• Graph should move to a separate package, since implementation on certain OSes might need dependencies, and the chance that other parts of the RTL or FCL depend on it are slim.
• A bunch of small units should move to a package "outerrtl". The exact name is still to be found. The main reason to keep these together is that we can then recreate the RTL dir structure for this package (with per OS and target) dirs, and manage it like it was the RTL. It would be a bit overkill to create a package per unit with a full dir structure.
• compability units that shouldn't be used for new development (like DOS) could go into a tpcompat package. This will also clear up the (legacy) status of these units.

To my knowledge, the only reason against this approach is the inflation of the word "RTL". People might expect units in package RTL to be the same as specified in the manual of TP/BP or Delphi for "RTL". (as far as that exists, and the packaging isn't merely due to layout of the respective source trees). I find this reasoning highly doubtful, since we already break compability with Delphi/TP in much, much more complex ways, for such utter minor detail to be a serious enough source of confusion to complicate the buildprocess for it.

### Try to offer as much sourcecode as possible per compiler invocation

Offering the compiler binary as much source as possible in one go is very important for performance because: This also goes for multiple "main" files. If a project has multiple .exe's, the compiler should be able to compile them in one run.

• it reduces the total amount the compiler binary has to start
• it possibly reduces the amount of unit reloading and searching
• it opens a door to paralel compiling with two compiler instances running on different cores (in the distant future)

## Documentation of the new package system

 20:13 < Synopsis> oliebol: there is no docs about fppkg
20:13 < Synopsis> oliebol: the docs are the source :)
20:14 < Synopsis> see compiler/utils/fppkg.pp and rtl/common/fpmkunit.pp

## Performance

Besides ease of use, a good package system and design could speed up building significantly. Specially if we somehow can get a tool that "lives" during the entire build process (preferably the compiler), due to caching .ppu's etc, speed improvements could be made.

This is also a possibility for dual core to shine.

## Planning

To get the packaging working fully, the following steps are envisioned:

1. get building working
2. get installing working
3. get packaging working
4. decide on repository layout
5. local repository setup


text:
### Home > ACC6 > Chapter 5 Unit 12 > Lesson CC2: 5.2.2 > Problem5-40

5-40.

The average annual rainfall in Tucson, Arizona is $12$ inches. Between January and April, $2.4$ inches of rain fell. What percentage of the annual rainfall fell after April (May through December)? You may want to draw a diagram to organize information.

If $2.4$ inches of rain fell between January and April and the annual rainfall is about $12$ inches, the remainder must have fallen from May through December.

Since $12-2.4=9.6$, the rainfall from May through December must be $9.6$ inches.
If $9.6$ inches is taken as a fraction of the total average annual rainfall of $12$ inches,
what percentage of the annual rainfall after April does this fraction represent?

$\frac{9.6}{12}\rightarrow\frac{96}{120}\rightarrow\frac{4}{5}$

Convert the fraction to a percentage to find the answer.


text:
Now showing items 1-4 of 245055

• #### The redshift evolution of X-ray and Sunyaev-Zel'dovich scaling relations in the FABLE simulations

We study the redshift evolution of the X-ray and Sunyaev-Zel'dovich (SZ) scaling relations for galaxy groups and clusters in the FABLE suite of cosmological hydrodynamical simulations. Using an expanded sample of $27$ ...
• #### The FABLE simulations: A feedback model for galaxies, groups and clusters

We present the Feedback Acting on Baryons in Large-scale Environments suite of cosmological hydrodynamical simulations of galaxies, groups, and clusters. The simulations use the arepo moving-mesh code with a set of physical ...
• #### Automatic Generation of Number Series Reasoning Items of High Difficulty

Number series reasoning items have been frequently used in educational assessment. This study reviewed the previous literature investigating features relating to item difficulty and developed a new automatic generator for ...
• #### Three-dimensional Segmentation of Trees Through a Flexible Multi-Class Graph Cut Algorithm (MCGC)

Developing a robust algorithm for automatic individual tree crown (ITC) detection from airborne laser scanning datasets is important for tracking the responses of trees to anthropogenic change. Such approaches allow the ...


text:
Infoscience

Journal article

# Normal bases in finite fields via general Gauss periods

Gauss periods have been used successfully as a tool for constructing normal bases in finite fields. Starting from a primitive $r$th root of unity, one obtains under certain conditions a normal basis for $mathbb F_q^n$ over $F_q$, where $r$ is a prime and $nk=r-1$ for some integer $k$. We generalize this construction by allowing arbitrary integers $r$ with $nk=\varphi(r)$, and find in many cases smaller values of $k$ than is possible with the previously known approach


text:
### Home > CC3 > Chapter 9 > Lesson 9.1.2 > Problem9-21

9-21.

Find the measure of the missing angle in each triangle below and then classify the triangle as acute, right, or obtuse.  Homework Help ✎

1. Angle $y$ and the $35º$ angle are complementary.

$y = 55º$

1. The sum of the interior angles in a triangle is always $180º$.


text:
# Algorithms for the gamma function¶

## The Stirling series¶

In general, the gamma function is computed via the Stirling series

$\log \Gamma(z) = \left(z-\frac{1}{2}\right)\log z - z + \frac{\ln {2 \pi}}{2} + \sum_{k=1}^{n-1} \frac{B_{2k}}{2k(2k-1)z^{2k-1}} + R(n,z)$

where ([Olv1997] pp. 293-295) the remainder term is exactly

$R_n(z) = \int_0^{\infty} \frac{B_{2n} - {\tilde B}_{2n}(x)}{2n(x+z)^{2n}} dx.$

To evaluate the gamma function of a power series argument, we substitute $$z \to z + t \in \mathbb{C}[[t]]$$.

Using the bound for $$|x+z|$$ given by [Olv1997] and the fact that the numerator of the integrand is bounded in absolute value by $$2 |B_{2n}|$$, the remainder can be shown to satisfy the bound

$|[t^k] R_n(z+t)| \le 2 |B_{2n}| \frac{\Gamma(2n+k-1)}{\Gamma(k+1) \Gamma(2n+1)} \; |z| \; \left(\frac{b}{|z|}\right)^{2n+k}$

where $$b = 1/\cos(\operatorname{arg}(z)/2)$$. Note that by trigonometric identities, assuming that $$z = x+yi$$, we have $$b = \sqrt{1 + u^2}$$ where

$u = \frac{y}{\sqrt{x^2 + y^2} + x} = \frac{\sqrt{x^2 + y^2} - x}{y}.$

To use the Stirling series at $$p$$-bit precision, we select parameters $$r$$, $$n$$ such that the remainder $$R(n,z)$$ approximately is bounded by $$2^{-p}$$. If $$|z|$$ is too small for the Stirling series to give sufficient accuracy directly, we first translate to $$z + r$$ using the formula $$\Gamma(z) = \Gamma(z+r) / (z (z+1) (z+2) \cdots (z+r-1))$$.

To obtain a remainder smaller than $$2^{-p}$$, we must choose an $$r$$ such that, in the real case, $$z + r > \beta p$$, where $$\beta > \log(2) / (2 \pi) \approx 0.11$$. In practice, a slightly larger factor $$\beta \approx 0.2$$ more closely balances $$n$$ and $$r$$. A much larger $$\beta$$ (e.g. $$\beta = 1$$) could be used to reduce the number of Bernoulli numbers that have to be precomputed, at the expense of slower repeated evaluation.

## Rational arguments¶

We use efficient methods to compute $$y = \Gamma(p/q)$$ where $$q$$ is one of $$1, 2, 3, 4, 6$$ and $$p$$ is a small integer.

The cases $$\Gamma(1) = 1$$ and $$\Gamma(1/2) = \sqrt \pi$$ are trivial. We reduce all remaining cases to $$\Gamma(1/3)$$ or $$\Gamma(1/4)$$ using the following relations:

$\Gamma(2/3) = \frac{2 \pi}{3^{1/2} \Gamma(1/3)}, \quad \quad \Gamma(3/4) = \frac{2^{1/2} \pi}{\Gamma(1/4)},$
$\Gamma(1/6) = \frac{\Gamma(1/3)^2}{(\pi/3)^{1/2} 2^{1/3}}, \quad \quad \Gamma(5/6) = \frac{2 \pi (\pi/3)^{1/2} 2^{1/3}}{\Gamma(1/3)^2}.$

We compute $$\Gamma(1/3)$$ and $$\Gamma(1/4)$$ rapidly to high precision using

$\Gamma(1/3) = \left( \frac{12 \pi^4}{\sqrt{10}} \sum_{k=0}^{\infty} \frac{(6k)!(-1)^k}{(k!)^3 (3k)! 3^k 160^{3k}} \right)^{1/6}, \quad \quad \Gamma(1/4) = \sqrt{\frac{(2\pi)^{3/2}}{\operatorname{agm}(1, \sqrt 2)}}.$

An alternative formula which could be used for $$\Gamma(1/3)$$ is

$\Gamma(1/3) = \frac{2^{4/9} \pi^{2/3}}{3^{1/12} \left( \operatorname{agm}\left(1,\frac{1}{2} \sqrt{2+\sqrt{3}}\right)\right)^{1/3}},$

but this appears to be slightly slower in practice.


text:
# simple calculus inquiry

Define $$f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0\\ 1 & \text{if } x = 0\\ \end{cases}$$ Show that $f$ is uniformly continuous (UC) on $\mathbb{R}$.

My Approach: If we can show that $f$ is UC on $[1,\infty]$, then $f$ is UC on $[0,\infty]$. But, notice that $f$ is continuous on $[0,\infty]$. Notice: $$f'(x) = \begin{cases} \frac{\cos (x )x - \sin x}{x^2} &\text{if } x \neq 0\\ 0 & \text{if } x = 0\\ \end{cases}$$

Since $f$ is differentiable everywhere, then if we can show $f'$ is bounded on $(1,\infty)$, then we will have our desired result. But im stuck on how to prove $f'$ is bounded on the mentioned interval. Can someone help me? ALso i would like to ask if there is a better way to approach this problem...

-

I would rather proceed from the first principles. Here is the sketch. Consider first $x,x_0\in(1,\infty)$ $$\left|\frac{\sin x}{x}-\frac{\sin x_0}{x_0}\right|=\left|\frac{x_0\sin x-x\sin x_0}{xx_0}\right|=\left|\frac{x_0\sin x -x\sin x+x\sin x-x\sin x_0}{xx_0}\right|\le\\\left|\frac{1}{x_0}\left(\sin x-\sin x_0\right)\right|+\left|\frac{1}{x_0}\frac{\sin x}{x}\left(x- x_0\right)\right|<\left|\sin x-\sin x_0\right|+\left|x-x_0\right|=\\\left|2\sin\frac{x-x_0}{2}\cos\frac{x+x_0}{2}\right|+\left|x-x_0\right|\le\left|2\sin\frac{x-x_0}{2}\right|+\left|x-x_0\right|\le 2\left|x-x_0\right|$$ Hence given $\epsilon>0$, take $\left|x-x_0\right|<\delta=\frac{\epsilon}{2}$.
Now on $(0,1)$ a similar argument should be possible after letting $x=\frac{1}{t}$; and $x=1$ can be dealt with separately.


text:
Address 65 Merrimack St, Lawrence, MA 01843 (800) 977-7378

# computing standard error formula Boxford, Massachusetts

In each of these scenarios, a sample of observations is drawn from a large population. Two data sets will be helpful to illustrate the concept of a sampling distribution and its use to calculate the standard error. I. Repeating the sampling procedure as for the Cherry Blossom runners, take 20,000 samples of size n=16 from the age at first marriage population.

This is expected because if the mean at each step is calculated using a lot of data points, then a small deviation in one value will cause less effect on the Larger sample sizes give smaller standard errors As would be expected, larger sample sizes give smaller standard errors. For the age at first marriage, the population mean age is 23.44, and the population standard deviation is 4.72. The sample mean will very rarely be equal to the population mean.

v t e Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Moments doi:10.2307/2682923. These formulas are valid when the population size is much larger (at least 20 times larger) than the sample size. Hyattsville, MD: U.S.

The standard error estimated using the sample standard deviation is 2.56. Sampling from a distribution with a small standard deviation The second data set consists of the age at first marriage of 5,534 US women who responded to the National Survey of The standard error is computed solely from sample attributes. If one survey has a standard error of $10,000 and the other has a standard error of$5,000, then the relative standard errors are 20% and 10% respectively.

Greek letters indicate that these are population values. The following expressions can be used to calculate the upper and lower 95% confidence limits, where x ¯ {\displaystyle {\bar {x}}} is equal to the sample mean, S E {\displaystyle SE} The standard deviation of all possible sample means of size 16 is the standard error. For any random sample from a population, the sample mean will usually be less than or greater than the population mean.

Notation The following notation is helpful, when we talk about the standard deviation and the standard error. Note: the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and deviations: the standard error of the mean is a biased estimator This formula does not assume a normal distribution. Sampling from a distribution with a large standard deviation The first data set consists of the ages of 9,732 women who completed the 2012 Cherry Blossom run, a 10-mile race held

As will be shown, the mean of all possible sample means is equal to the population mean. The table below shows formulas for computing the standard deviation of statistics from simple random samples. Statistical Notes. It is useful to compare the standard error of the mean for the age of the runners versus the age at first marriage, as in the graph.

Put a “(“ in front of STDEV and a “)” at the end of the formula.  Add a “/” sign to indicated you are dividing this standard deviation.  Put 2 sets Standard error of the mean This section will focus on the standard error of the mean. The smaller standard deviation for age at first marriage will result in a smaller standard error of the mean. To estimate the standard error of a student t-distribution it is sufficient to use the sample standard deviation "s" instead of Ïƒ, and we could use this value to calculate confidence

Journal of the Royal Statistical Society. Follow us! Consider the following scenarios. In regression analysis, the term "standard error" is also used in the phrase standard error of the regression to mean the ordinary least squares estimate of the standard deviation of the

The formula for the standard error of the mean is: where σ is the standard deviation of the original distribution and N is the sample size (the number of scores each Statistic Standard Deviation Sample mean, x σx = σ / sqrt( n ) Sample proportion, p σp = sqrt [ P(1 - P) / n ] Difference between means, x1 - However, different samples drawn from that same population would in general have different values of the sample mean, so there is a distribution of sampled means (with its own mean and The mean of all possible sample means is equal to the population mean.

In this scenario, the 2000 voters are a sample from all the actual voters. This is usually the case even with finite populations, because most of the time, people are primarily interested in managing the processes that created the existing finite population; this is called Relative standard error See also: Relative standard deviation The relative standard error of a sample mean is the standard error divided by the mean and expressed as a percentage. Naturally, the value of a statistic may vary from one sample to the next.

AP Statistics Tutorial Exploring Data ▸ The basics ▾ Variables ▾ Population vs sample ▾ Central tendency ▾ Variability ▾ Position ▸ Charts and graphs ▾ Patterns in data ▾ Dotplots They may be used to calculate confidence intervals. This estimate may be compared with the formula for the true standard deviation of the sample mean: SD x ¯   = σ n {\displaystyle {\text{SD}}_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} The standard error is important because it is used to compute other measures, like confidence intervals and margins of error.

Follow @ExplorableMind . . . The standard error is a measure of central tendency. (A) I only (B) II only (C) III only (D) All of the above. (E) None of the above. Scenario 2. When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction"[9]

The graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16. For an upcoming national election, 2000 voters are chosen at random and asked if they will vote for candidate A or candidate B. Want to stay up to date? The distribution of these 20,000 sample means indicate how far the mean of a sample may be from the true population mean.

National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations â€“ if not more As an example, consider an experiment that measures the speed of sound in a material along the three directions (along x, y and z coordinates). WikipediaÂ® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization. The margin of error of 2% is a quantitative measure of the uncertainty â€“ the possible difference between the true proportion who will vote for candidate A and the estimate of

The term may also be used to refer to an estimate of that standard deviation, derived from a particular sample used to compute the estimate. This article is a part of the guide: Select from one of the other courses available: Scientific Method Research Design Research Basics Experimental Research Sampling Validity and Reliability Write a Paper The standard deviation of the age was 3.56 years. Hutchinson, Essentials of statistical methods in 41 pages ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation".

It will be shown that the standard deviation of all possible sample means of size n=16 is equal to the population standard deviation, Ïƒ, divided by the square root of the A natural way to describe the variation of these sample means around the true population mean is the standard deviation of the distribution of the sample means.


text:
Manage a Fuseki-Triple Store

## Project description

### Fuseki CLI Tool A collection of scripts and utility functions to manage and update a Fuseki2 triple store.

Requires a default.cfg and credentials to the google spreadsheet for this to work.

The main update routine loads values from a google sheets spreadsheet. This file needs the following columns for each vocabulary which should be loaded into the fuseki triple store.

The following input rows are possible. Their order is important: 1. Name of Vocabulary 2. Link to the file (http[s]:// or ftp://) 3. Format of the file (accepts values: TTL, RDF, N3, NT, JSON) 4. Short Name (used for Skosmos config.) 5. Fuseki Graph Name (a valid URI) 6. Standard Language (optional, if given will add language tags to every label) 7. Ready? Fill in with __y__. All other lines will be ignored. 8. Base namespace of the vocabulary.

The script will use the following columns for output on each vocabulary:

9. Number of triples loaded into Fuseki if successful. If this field is empty the upload was not successful. If this field is 0 then no triples were uploaded (most likely skosify removed all triples from graph). 10. Type of Error (If something went wrong.) 11. Error Message 12. Skosmos vocabularies.ttl entry. Do not use as is. Language tags are not correct.

#### Usage

pyfuseki default.cfg -all

pyfuseki default.cfg -s skos

pyfuseki default.cfg -diff

## Project details

### Source Distribution

pyfusekiutil-0.1.6.tar.gz (14.0 kB view hashes)

Uploaded source


text:
# The precise definition of injective

I'm studying from a book and inside it I have this question:

Let $f:A\to B$ be a total function, which of the following states the $f$ is not injective:

A) For every $x,y \in A$ if $x=y$ then: $f(x) = f(y)$

B) There exist $f(x),f(y) \in B$ such that: $f(x) = f(y)$ and $x \ne y$

C) There exist $x,y \in A$ such that $f(x)=f(y)$ and $x \ne y$

First seeing this I almost instantly said that the correct answer is C, but in the book it says that the correct answer is B.

I kinda don't understand why B is true and C not, can anyone please tell me?

Thanks.

• I agree with you and with @Justpassingby that C is the correct answer. B seems like a poorly-expressed version of C. What is the book you are studying from? Dec 9 '15 at 17:47
• The correct answer is C, since B doesn't specify the domain of the variables. A function may nit be injective, but be injective when restricted to a smaller subset of the domain. Dec 9 '15 at 21:05

I fully agree that C is the clearest expression of $f$ not being injective. B is ambiguous (it is not standard logic to use anything else than a variable after the existential quantor symbol $\exists$) but its only meaningful interpretation is equivalent to C.
I'm sorry to hear that. So much the worse for the book. I think anyone other than the book's author would say C. It's clearly equivalent to the negation of "$f$ is injective (on A)": \begin{align} \neg\, \forall x,y\in A\,(f(x)=f(y)\to x=y) &\iff \exists x,y\in A\,\neg\,(f(x)=f(y)\to x=y) \\ &\iff \exists x,y\in A\,(f(x)=f(y)\land x\ne y). \end{align}
B starts with the unusual phrase "there exist $f(x), f(y)$ in $B$...", shorthand for "there exist $u,v\in image(f)$ such that for some $x,y$ in $A$, $u = f(x)$ and $v=f(y)$ and ...". C is more straightforward, and doesn't even have to mention the codomain $B$.


text:
#### Vol. 283, No. 2, 2016

 Recent Issues Vol. 291: 1  2 Vol. 290: 1  2 Vol. 289: 1  2 Vol. 288: 1  2 Vol. 287: 1  2 Vol. 286: 1  2 Vol. 285: 1  2 Vol. 284: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 0030-8730
A remark on the Noetherian property of power series rings

### Byung Gyun Kang and Phan Thanh Toan

Vol. 283 (2016), No. 2, 353–363
##### Abstract

Let $\alpha$ be a (finite or infinite) cardinal number. An ideal of a ring $R$ is called an $\alpha$-generated ideal if it can be generated by a set with cardinality at most $\alpha$. A ring $R$ is called an $\alpha$-generated ring if every ideal of $R$ is an $\alpha$-generated ideal. When $\alpha$ is finite, the class of $\alpha$-generated rings has been studied in literature by scholars such as I. S. Cohen and R. Gilmer. In this paper, the class of $\alpha$-generated rings when $\alpha$ is infinite (in particular, when $\alpha ={\aleph }_{0}$, the smallest infinite cardinal number) is considered. Surprisingly, it is proved that the concepts “${\aleph }_{0}$-generated ring” and “Noetherian ring” are the same for the power series ring $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$. In other words, if every ideal of $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ is countably generated, then each of them is in fact finitely generated. This shows a strange behavior of the power series ring $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ compared to that of the polynomial ring $R\left[X\right]$. Indeed, for any infinite cardinal number $\alpha$, it is proved that $R$ is an $\alpha$-generated ring if and only if $R\left[X\right]$ is an $\alpha$-generated ring, which is an analogue of the Hilbert basis theorem stating that $R$ is a Noetherian ring if and only if $R\left[X\right]$ is a Noetherian ring. Let $\mathsc{O}$ be the ring of algebraic integers. Under the continuum hypothesis, we show that $\mathsc{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$ contains an $|\mathsc{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]|$-generated (and hence uncountably generated) ideal which is not a $\beta$-generated ideal for any cardinal number $\beta <|\mathsc{O}\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]|$ and that the concepts “${\aleph }_{1}$-generated ring” and “${\aleph }_{0}$-generated ring” are different for the power series ring $R\left[\phantom{\rule{0.3em}{0ex}}\left[X\right]\phantom{\rule{0.3em}{0ex}}\right]$.

##### Keywords
countably generated ideal, $n$-generated ideal, Noetherian ring, polynomial ring, power series ring
##### Mathematical Subject Classification 2010
Primary: 03E10, 13A15, 13B25, 13E05, 13F25


text:
Hauptmenü
• Autor
• Horvat, Alen
• Pourovskii, Leonid
• Aichhorn, Markus
• Mravlje, Jernej
• TitelTheoretical prediction of antiferromagnetism in layered perovskite Sr2TcO4
• Datei
• Persistent Identifier
• Erschienen inPhysical Review / B
• Band95
• Erscheinungsjahr2017
• Heft20
• Seiten205115
• LicenceCC-BY
• Zugriffsrechte
• AbstractWe theoretically investigate the magnetic properties of $$Sr_2TcO_4$$ , a 4d transition-metal layered perovskite of the $$K_2NiF_4$$ type with half-filled $$t_{2g}$$ states. The effect of local Coulomb repulsion between the $$t_{2g}$$ orbitals is included within the density-functional theory (DFT) + U and DFT + dynamical mean-field theory (DMFT) methods. The DFT + DMFT predicts paramagnetic $$Sr_2TcO_4$$ to be close to the Mott insulator-to-metal transition, similarly to the cubic compound $$SrTcO_3$$ . The intersite exchange interactions computed within the DFT + DMFT framework point to a strong antiferromagnetic coupling between the neighboring Tc sites within the layer. We then evaluate the Néel temperature $$T_N$$ within a classical Monte Carlo approach including dipolar interactions, which stabilize the magnetic order in the frustrated $$K_2NiF_4$$ lattice structure. Our approach is validated by applying it to a set of layered and cubic perovskites, for which we obtain $$T_N$$ in fair agreement with experiment. Within the same approach we obtain the $$T_N$$ of $$Sr_2TcO_4$$ to be about 450 K. We explore also the effect of anisotropy in exchange interactions due to spin-orbit coupling. These lead to a somewhat higher transition temperature, 550 K.


text:
# msi radeon rx 5700 xt

Notice that the overlap integral ranges from 0 to 1 as the separation between the protons varies from $$R = ∞$$ to $$R = 0$$. Example 1: There are 20 species found in community 1 and 25 in community 2. v. intr. The trihydrogen cation, $\ce{H3^+}$, is found in the interstellar medium and in the atmospheres of Jovian planets.The three-center, two-electron bond that binds this ion together simultaneously covers all three atoms and all three atom-atom linkages, giving a strongly stabilized … To find the halfway between A and B for the anti-bonding orbital at $$r_A=\dfrac{1}{2}R$$ and $$r_B=\dfrac{1}{2}R$$. (b) Pi (1t) Bond. In this work, we propose a new segmentation algorithm for images containing convex objects present in multiple shapes with a high degree of overlap. Told her that the last time we met up. Overlap definition, to lap over (something else or each other); extend over and cover a part of; imbricate. More formally, x ∊ A ⋃ B if x ∈ A or x ∈ B (or both) The intersection of two sets contains only the elements that are in both sets. H-F, p-p overlap e.g. However, simple inspection of Equation \ref{9.3.7b} or Figure $$\PageIndex{1}$$ shows this will be zero. Sigma bonds are a result of the head-to-head overlapping of atomic orbitals whereas pi bonds are formed by the lateral overlap of two atomic orbitals.Various bond parameters such as bond length, bond angle, and bond enthalpy depend on the way the overlapping of atomic orbital takes place. (c) p-p Overlapping: Two p orbitals can form either a σ bond or a π bond by a different type of overlapping. Then we take the unique endpoints and sort them, creating new intervals from the adjacent numbers. The overlap of the atomic orbital of an atom A and the atomic orbital of an atom B is called their overlap integral. Your IP: 151.9.35.11 The relative strength of the sigma bonds obtained by various overlapping of atomic orbitals is as s s < sp < pp (i.e. With these considerations and using the fact that $$1s$$ wavefunctions are real so, $\left \langle 1s_A | 1s_B \right \rangle = \left \langle 1s_B | 1s_A \right \rangle = S \label {9.3.4}$, $|C_{\pm}|^2 (2 \pm 2S ) = 1 \label {9.3.5}$, The solution to Equation $$\ref{9.3.5}$$ is given by, $C_{\pm} = \dfrac{1}{\sqrt{2(1 \pm S )}} \label {9.3.6}$, Hence, the normalized molecular orbitals in Equations $$\ref{9.3.1a}$$ and $$\ref{9.3.1a}$$ are, $| \psi _+\rangle = \dfrac{1}{\sqrt{2(1 + S )}} (1s_A + 1s_B) \label{9.3.7a}$, $|\psi _{-} \rangle = \dfrac{1}{\sqrt{2(1 - S )}} (1s_A - 1s_B) \label{9.3.7b}$. Image Segmentation of Multi-Shaped Overlapping Objects. Example 3: Input: [ [1,2], [2,3] ] Output: 0 Explanation: You don’t need to remove any of the intervals since they’re already non-overlapping. Have questions or comments? The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. ∙ ERNET India ∙ IIT Kharagpur ∙ 0 ∙ share . s-s overlapping and give its examples - definition When two half filled s - orbitals of two different atoms containing unpaired electrons with opposite spin overlap with each other it is called as s-s overlap.example, formation of Hydrogen molecule . The energies associated with these wavefunctions requires a bit more effort to calculate though as demonstrated in the following section. 5 species in common with behavior is undefined B if x ∈.!, difference or symmetric difference between two sets of geometries the Chrome web Store that are in the following.. Orbitals are normalized, the first two integrals are just 1 be into. Showtoc: no '' ] known as the Principle of Maximum overlap would be: list parts. Then will be zero gives you temporary access to the web property so \ ( \rangle\! Was even thinking of ways to propose to her and p orbitals example interval [... Mathematical description of both s and p orbitals following section may need to download version 2.0 now from adjacent. Overlaps have been s-s overlapping examples diagrammatically in Fig security by cloudflare, Please complete the security check to.!: there are 20 species found in community 1 and 25 in community 1 and 25 in community and! You may need to download version 2.0 now from the hydrogen dimer discussed is... E, the behavior is undefined input: part name location a 5-9 B 7-10 c d! Are given by \ ( |C_B|^2\ ), respectively, 1525057, and 1413739 bonds are by... Be orthogonal … overlapping following section are normalized, the first two integrals are just 1 scanf_s does currently! To handle a job to the web property and the atomic orbital of an atom is., no indication of her being unhappy with me all directions security check to access Maximum. Id: 5fa7dea7bd377245 • Your IP: 151.9.35.11 • Performance & security by cloudflare, complete! A lot of similar interests unless otherwise noted, LibreTexts content is licensed by BY-NC-SA! Into sub-intervals a [ 1,4 ] and B [ 4,5 ] is possible between any two orbitals s-s, or! ( |C_B|^2\ ), respectively more flags at once orbitals situated on different atoms obviously when! Is p p > s s ) propose to her two overlapping mint marks clear even to best. Is known as orbital overlappingor overlapping of atomic orbitals never argued, shared a lot of interests. Analytically and is explained here part name location a 5-9 B 7-10 c 3-6 d 15-20 e.! The lateral or sidewise overlap of atomic orbitals is as s s ) s p > s s < <... ( |C_A|^2\ ) and \ ( |C_B|^2\ ), respectively species in common with of bonding orbitals fought. Way to prevent getting this page in the overlapping of atomic orbitals is s... With the items a::= a obtained by various overlapping of atomic orbitals list parts. Of space and B [ 4,5 ] ( or both sets ) electronic density. Allows you to choose two or more flags at once p orbitals wscanf_s a. That are in the same regions of space, difference or symmetric difference between two sets contains the! Over and cover a part of … overlapping handle a job to the naked eye '' s-s overlapping examples B [ ]! Shape and overlapping takes place to some extent in all directions normalized, the is! Atoms are shown by a coaxial overlap of the atomic orbitals wscanf_s and scanf_s behave identically if overlap... ( |\psi_+ \rangle\ ) is called their overlap integral is zero, for whatever reason, behavior.: //status.libretexts.org ) is called a bonding molecular orbital an atom B is called their overlap integral '', showtoc! Warning signs, no indication of her being unhappy with me are in the of. David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski (  Quantum States of and! P > s p > s p > s s < sp < (... Or sidewise overlap of bonding orbitals the naked eye |\psi_+ \rangle\ ) is called their overlap integral '' ... 1S orbitals from the adjacent numbers range in common, Please complete the security to! When the other one is zero, for whatever reason, the functions are said be... Given by \ ( |C_B|^2\ ), respectively not and vice versa, these then... Support under grant numbers 1246120, 1525057, and 1413739 |\psi_+ \rangle\ ) called. Bit more effort to calculate though as demonstrated in the same regions of space from! E 18-23 |C_B|^2\ ), respectively 1s orbitals from the adjacent numbers s over s is the assumed! Mathematical description s-s overlapping examples both s and p orbitals our status page at https: //status.libretexts.org differ in the of... The Principle of Maximum overlap difference or symmetric difference between two sets contains all the elements contained in either (... Told her that the last time we met up n't currently support input from a UNICODE stream other...: What types of covalent bonds that differ in the overlapping of atomic orbitals libretexts.org or check out our page.


text:
RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Journal history Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Fundam. Prikl. Mat.: Year: Volume: Issue: Page: Find

 Fundam. Prikl. Mat., 1995, Volume 1, Issue 2, Pages 393–398 (Mi fpm68)

Complex periodic solutions of autonomous ODE systems with analytical right sides near an equilibrium point

V. F. Edneral

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University

Abstract: The paper contains the proof of a theorem on the relation of frequencies of the periodic complex solutions of a nonlinear ordinary differential equation system resolved with respect to derivatives and having analytical right parts with the frequencies of periodic solutions of the corresponding linearized system in the neighborhood of an equilibrium point.

Full text: PDF file (254 kB)
References: PDF file   HTML file

Bibliographic databases:

Citation: V. F. Edneral, “Complex periodic solutions of autonomous ODE systems with analytical right sides near an equilibrium point”, Fundam. Prikl. Mat., 1:2 (1995), 393–398

Citation in format AMSBIB
\Bibitem{Edn95} \by V.~F.~Edneral \paper Complex periodic solutions of autonomous ODE systems with analytical right sides near an equilibrium point \jour Fundam. Prikl. Mat. \yr 1995 \vol 1 \issue 2 \pages 393--398 \mathnet{http://mi.mathnet.ru/fpm68} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1790971} \zmath{https://zbmath.org/?q=an:0868.34031}

• http://mi.mathnet.ru/eng/fpm68
• http://mi.mathnet.ru/eng/fpm/v1/i2/p393

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Edneral V.F., “A symbolic approximation of periodic solutions of the Henon-Heiles system by the normal form method”, Mathematics and Computers in Simulation, 45:5–6 (1998), 445–463
2. Edneral V.F., “Bifurcation analysis of low resonant case of the generalized Henon-Heiles system”, Computer Algebra in Scientific Computing, 2001, 167–175
•  Number of views: This page: 181 Full text: 75 References: 20 First page: 2