• 検索結果がありません。

effective. The need to activate the core muscles was recognize and began from the second half of the 1980s. Bergmark

N/A
N/A
Protected

Academic year: 2021

シェア "effective. The need to activate the core muscles was recognize and began from the second half of the 1980s. Bergmark"

Copied!
8
0
0

読み込み中.... (全文を見る)

全文

(1)

INTRODUCTION

 In recent years, core strength has become widely recognized as being important to an athletes’ performance in most sports. We have even realized that an increase in core strength has improved movement and prevented people from falling down, which is important in the health care environment. Despite all of the knowledge on this subject, there still lacks evidence about which specific exercise is most

effective. The need to activate the core muscles was recognize and began from the second half of the 1980s. Bergmark

1)

classified Local muscles to work in the stability of spine and classified Global muscles to work to be involved in the general body movement. His results showed that the movement of the transverse abdominis, which is one of the Local muscles, contracts first before movement of extremities in healthy people, but starts late or does not work with on people who

The influence that the small angle tilt exercise

in the sitting position gives to the Local muscle activation

1 Satoshi MIYASHITA, Ph.D  1 Takeo HASHIGUCHI, Ph.D

2 Haro OGAWA  3 Hideki KATO  4 Tomoya MIYASHITA

5 Masanori SUZUKI  1 Katsuhiro MANABE, MS  1 Soichiro IWANUMA, Ph.D

1

Teikyo University of Science. 

2

San Francisco Giants 

3

Miyota Central Memorial Hospital

4

Shinshu University 

5

Tokyo Eisei Gakuen College 要旨

 近年、スポーツ分野をはじめ医療・介護現場で、体幹トレーニング(コアトレーニング)はパフォーマンスの向上のため盛ん に行われている。本研究では四肢の運動に先だって収縮する腹横筋に着目し、従来観察できなかった低負荷運動時における効果 的な Local muscle の収縮を得ることを目的とした。対象者は健常男子 22 名。平均年齢 16.2 ± 0.7 歳であった。運動課題は、安 静椅座位、10cm 前方傾斜、最大前方傾斜の 3 つとした。それぞれの課題時、超音波エコーにて外腹斜筋、内腹斜筋、腹横筋厚 を測定し、3 筋の合計筋厚を 100%とし、各筋の筋厚比率を求めた。その結果、測定した 3 筋の要因において有意な主効果が認め られたが(F(2,42)= 520.37, p<0.01)、軸側、利き側の要因においては主効果が認められなかった。また運動課題と 3 筋厚比較 において有意な交互作用が認められた(F(4,84)=5.43, p<0.01)。特に Local muscle である腹横筋の筋厚比率は静止座位および 最大前方傾斜に比べ、10cm 前方傾斜において有意に高かった(静止座位 <10cm 前方傾斜(p<0.05)、10cm 前方傾斜 > 最大前方 傾斜(p<0.05))。以上のことから、体幹トレーニングで Local muscle を活性化させるには、従来の高負荷運動ではなく、低負荷 運動によって選択的に腹横筋を活性化させることが示された。このことはスポーツ選手に限らず、運動範囲が減少する高齢者に おいても応用可能であることが期待できる。

Abstract

 In recent years, core training has become a very popular method to improve performance among athletes and has also contributed to people within different clinical settings throughout our health care system. In this research, we focus on the transverse abdominis contraction that is the prior muscle to upper and lower extremities exercises. The aim was to also obtain effective local muscle contraction during lower resistance exercises, which could not be conventionally observed. The subjects are 22 healthy males who are an average of 16.2 ± 0.7 years in age. Exercises consist of three types of postures; rest sitting position, 10cm front tilt position, and maximum front tilt position. Each muscle (external oblique, internal oblique and transverse abdominis) thickness ratio was measured by ultrasound during each exercise. The total muscle thickness of the three muscles are used as 100% , and used to determine the muscle thickness ratio of each muscle. The results were very successful (F (2.42)=520.37, p<0.01).However, a significant effect was observed in the 3 muscles measured (F (2,42) = 520.37, p <0.01) but significant effect of the dominant side or was not observed. In addition, significant interaction in motor tasks and the three muscles was observed (F (4,84) = 5.43, p <0.01).The most significant result was the muscle thickness ratio of the transverse abdominis, which is also known as the local muscle. When comparing sitting in rest position or sitting with maximum forward tilt with sitting with 10cm front tilt, sitting with 10cm front tilt showed higher significance (rest sitting < 10cm front tilt (p<0.05), 10cm front tilt > maximum front tilt (p<0.05)).As shown above, in order to activate the Local muscle in core trainings, rather than doing the conventional high resistance exercise, selectively activating the transverse abdominis with low resistance exercise is more effective. It would be possible to modify the activity so that the exercise can be applied not only to athletes but also to elderly patients.

キーワード:ローカルマッスル、腹横筋収縮、低負荷運動、体幹トレーニング・サスペンション・エクササイズ

Keywords:Local Muscle, Transverse abdominis, Low resistance exercise, Core training, Suspension exercise.

(2)

have suffered from chronic lower back pain

2-3)

. This report shows therapists and researchers the importance of recognition to enhance the activity of the transverse abdominis in chronic lower back pain patients. After body stability is achieved, and the mechanism of the limb movement is revealed, athletes aiming to improve competitiveness came to reconsider conventional high resistance exercise which was thought to be involved in spinal stabilization using surface layer muscles such as rectus abdominis and elector spinae

4)

.Promoting the contraction of the Local muscles, which located at the deep layer, is difficult. Sapsford et al.

5-6)

,reports that the contraction of the basic pelvic muscles has induced transverse abdominis contraction. They recommend, "pulling the body in" method

7-9)

, which is to pull the umbilicus toward the spine, and verbal instruction like "putting force on the abdomen to stop urination" is tried and has been promoted as a way to encourage the contraction of the transverse abdominis. However, as for core training, it becomes a muscle contraction in the static condition. Therefore, the effectiveness of making the Local muscles active during exercise remains unknown.

 Based on Sapsford's research, we considered a need for a method to obtain a valid contraction of the transverse abdominis during exercises.

We researched movement of the upper and lower extremities with no gravity by using two- suspension equipment (manufactured by Redcord co., Ltd.).The results indicate that it is possible to promote the contraction of the transverse abdominis as a Local muscle with suppressing the activity of the Global muscles

10)

.Top athlete teams use this method, but we think that this high resistance exercise is not appropriate for people who suffer from decreased physical strength, pain, and old age. By considering both exercises in supine position in static conditions developed by Sapsford, and the training we reported for unstable environments in prone position, we were successful in making each Local muscle activate. However, it still needs to consider the training method with a valid intermediate load amount to train for the

athletes in the preparation stage and elderly.

Conventional low resistance exercise is usually done in static condition; therefore development of low resistance training during exercise needs to be considered.

 In this study, the purpose is to validate the possibility of the transverse abdominis contracting effectively by restraining the activity of the Global muscles during forward tilt exercise by measuring the thickness of external oblique, internal oblique, and transverse abdominis in a sitting position which is the position that has been incorporated into the training in a number of facilities for the elderly.

SUBJECTS

 Subjects participating in this study are 22 healthy males, age 16.2 ± 0.7, average height 168.3

± 5.8cm (179.0cm 〜 157.6cm),average weight 63.5 ± 8.8kg (83.6kg 〜 47.5kg).They have signed an agreement of compliance with the code of ethics based on the Declaration of Helsinki.

METHODS

 The tasks (Figure 1) are the following three conditions, and the thickness of both right and left of three different muscles (external oblique, internal oblique, and transverse abdominis) were measured by an ultrasonic measuring device.

Task1:RestSittingPosition

 Subjects are placed under the suspension equipment (Redcord Co. Redcord trainer),in a shallow sitting position on a height adjustable chair with both knee joints 90 degrees and ankle joints 90 degrees. Upper extremity is positioned so that the rope is hung down from the suspension device and is held in the subject's hand perpendicularly with both shoulders flexed 90 degrees with elbows extended.

Task2:10cmFrontTiltPosition

 The rope is hung down vertically 10cm

forward from the original position. The subject

is instructed to forward tilt to touch the rope and

stop the movement at the time when the finger

(3)

touches the guide rope and this is termed as the 10cm front tilt task position.

Task3:MaximumFontTiltPosition

 The subject is forward tilted to maximum to a position where the angle of bilateral knee joints and ankle joints do not change, and this is termed as the maximum forward tilted position.

 The subject is asked to sit shallow in the seat so it makes it easier to carry out the movement of each limb position. It is explained verbally that only the hip joint can be move and it is corrected if an inappropriate of posture was observed during the task. Execution order of the exercise is the following; the measurement of Task 2 is obtained after Task 1. After Task 2, the subject is instructed to sit back in normal sitting position at rest and then performed and obtained the measurement of Task 3.

 A diagnostic ultrasound device by MicroMax

(SonoSite. Companies, Ltd.) is used for all three tasks. We also used an 8MHz linear probe that can measure the surface image of each muscle;

external oblique, internal oblique, and transversus abdominis. The pointing of the marking equipment

of the diagnostic ultrasound device is used to obtain our measurement of the thickness of the muscles based on the fascia. For the position to apply the probe, the probe is place at 10cm lateral to the umbilicus across the horizontal line level of the umbilicus, and perpendicular to the axillary line. The point of measurement for Task 1 are marked and are used for all tasks’ measurement.

Also the person who is measuring the muscles is pre-train and the same person is assigned to all tasks’ measuring to minimize the measurement error as much as possible.

 Due to the fact that different people have different muscle size and body shapes, we compared the muscle thickness ratio of each subject by dividing each muscle thickness by the sum of the thickness of the 3 muscles. We also compared 6 categories including; exercise and position (resting sitting, 10cm front tilt, maximum forward tilt),each side (dominant leg side and non-dominant leg side).

 For statistical processing, exercise task by (rest

sitting, 10cm front tilt, maximum front tilt),right

and left leg (dominant side and non-dominant

side),muscles (external oblique, internal oblique,

Fig1:Tasksconsistedofthreeconditions;

(4)

transverse abdominis) are analyzed. The analysis of variance (ANOVA) is based on each category of 3 × 2 × 3;which was the main factor. For those which produced significant main effects and interactions, we examined by multiple comparison test (Bonferoni).The significance average level was 5% .

RESULT

1. Each number is the average and standard deviation of thickness of each muscleThickness of muscle 3 positions.

1-1Restsitting.

 External oblique (dominant side) 31.7 ± 5.9% ,

(non-dominant side) 30.4 ± 4.0% , Internal oblique

(dominant side) 50.0 ± 5.7% , (non-dominant side)

50.2 ± 3.7 % , Transverse abdominis (dominant side) 18.3 ± 2.8 % , (non-dominant side) 19.4 ± 2.6% .

1-210cmfronttilting

 External oblique (dominant side) 29.9 ± 4.1%,

(non-dominant side) 28.6±4.2%,Internal oblique

(dominant side) 49.3±4.4%, (non-dominant side)

50.9±4.6%, Transverse abdominis (dominant side)

20.8 ± 2.5%, (non-dominant side) 20.5 ± 2.5%.

1-3Maximumfronttilting

 External oblique (dominant side) 20.9 ± 3.6%,

(non-dominant side) 27.0 ± 4.9%, Internal oblique

(dominant side) 51.5 ± 5.4%, (non-dominant side)

53.6 ± 4.6%,Transverse abdominis (dominant side) 19.4 ± 2.5 (non-dominant side) 19.4 ± 2.7%.

 The ratio of the internal oblique muscle is about 50% each for posture and side and was the highest percentage. Subsequently,the external oblique is 2nd highest with about 30%,and transverse abdominal was the third with about 20%.

2. Based on each exercise (rest sitting, 10cm front tilt, maximum front tilt),the differences between left and right (dominant side and non- dominant side),muscles (external abdominal oblique, internal oblique, transverse abdominis),

the analysis of each main factor ANOVA (3 × 2 × 3) shows significant main effect observed in the muscle (F (2,42) = 520.37, p <0.01).

No significant main effect was observed with comparison in the left and right. In addition,

Fig2:TasksandTheratioofthicknessofthemuscle

OE(externaloblique),OI(internaloblique),TrA(transverseabdominis)

(5)

a significant interaction was observed in the exercise and three muscles (F (4,84) = 5.43, p

<0.01).Figure 2 shows the results of multiple comparison tests. To compare 3muscles in exercise tasks the result of thickness of the muscle was 1st. internal oblique, 2nd. external oblique,3rd.(Internal oblique > external abdominal oblique > transverse abdominis,p

<0.05).

Comparisonbetween3musclesandexercisetask, 3-1 The ratio of thickness of the external abdominis was significantly higher than maximum forward tilt (rest sitting = 10cm front tilt (ns),

10cm front tilt = maximum front tilt (ns),rest sitting>maximum front tilt (p <0.05)).Therefore, because the result of the ratio of rest sitting is the highest, it shows we work with this posture in our normal life.

3-2 The ratio of thickness of the internal oblique with maximum tilt sitting position was 10cm higher than others, (rest sitting = 10cm front tilt (ns),10cm front tilt <maximum front tilt

(p < 0.05),rest sitting > maximum front tilt (p

<0.05)).Internal oblique will tend to significantly move rather than others during the exercise.

3-3 The ratio of thickness of transverse abdominis with 10cm front tilt sitting position is higher than others, (rest sitting <10cm front tilt (p <0.05),

10cm front tilt = maximum front tilt (ns),10cm front tilt > maximum front tilt (p <0.05)).The ratio of thickness of transverse abdominis with 10cm forward tilt that is small movement is significantly gradually increased. That means that the exercise with less tilt posture is the most effective exercise for stabilizing the core and trunk.

DISCUSSION

 This study started with the purpose of finding an effective method for activating Local muscles during lower resistance exercise.

The mass weight ratio of the trunk is 47.9%

male 48.7% female calculated with the ratio of each body part

11)

.This result shows that it is important to consider the function of the trunk

movement. In this study,we focus on the movement of the transverse abdominis,located at a deep layer instead of the muscle located at a superficial layer,with electromyogram (EMS).

EMS analyzes core muscles, which are the basic measurements. Transverse abdominis starts the initial contractions for the each movement, and can move with "Feed-Forward" during fast speed, but not seen during slow speed. In the case of upper extremity exercises, the transverse abdominis contracts 30ms before any other body parts move. During the lower extremities exercises, the transverse abdominis starts contraction 110 ms faster than any other body part

12)

. Therefore, we consider that it is necessary to actively work out the Local muscle during exercise rather than in the static position. Hodges reports that the transverse abdominis will contract first before each extremities move, but it does not indicate the strength of the contraction force.

Also in this research, we measured the timing of the contractions of the muscle with embedded electrode needles in the body. It is difficult to do so during exercise. Therefore, measurement of thickness of the muscle was taken by using an ultrasound device instead of electrode needles.

Ultrasounds device can measure each position and posture during exercise. We conclude this method of observation is most effective. Moreover, it can also observe the thickness of the muscle changing during the exercise. McMeeken

13)

has reported the relevance between using an ultrasonic diagnostic for measuring changes thickness in the transverse abdominis devise, apposed to the EMG. The results showed that there is a strong correlation between the myoelectric activity changing and increasing the thickness of the muscle.

 Ito

14)

,considered the structure of the muscle

fibers of the side of the abdominal muscle, the

thickest muscle layer that can be seen in the

internal oblique muscle. The external oblique

muscle is the 2nd, and the transverse abdominis is

the 3rd thickness. Males have thicker layers of all

three muscles compared to females.Males have

more muscle fibers in 1mm

2

of external oblique

and the internal oblique. However, no difference is

(6)

seen in transverse abdominis and rectus abdominis.

Therefore, we want to consider the Global muscles, which are internal and external oblique, and Local muscle, which is the transverse muscle, separately.

At the initial check, the abdominal MRIs of 2 people showed that the area of the non-dominant side of muscles are thicker than the dominant side of each muscles (external oblique, internal oblique, and transverse abdominis).The active ratio of each muscle can be calculated based on the total ratio 100% of thickness of three abdominal muscles, excluding the size of body and muscle.

 In this research, the differences of active ratio of each right and left side cannot be observed. In other words, if the muscle sizes are different, the active ratio does not show any differences. We would like to build small muscle mass based on observing the shape, but it is very common for one side of the trunk to be built more due to excessive workouts.

 Conventional Local muscle exercise give indication to "pulling the body in" to athletes and patients during rehabilitation, to prompt the transverse abdominis contraction in the standing position. However, using an ultrasonic diagnostic device it has been observed that in many subjects, they do not work well for muscle contraction of the transverse abdominis because the subject does not fully comprehend how to do the proper exercise.

 Also, using the Sapsford methods, the placebo indication was made to the subject

"to stop urination", which concluded in better understanding opposed to the indication "pulling the body in". On the spine position, it enables to contract the transverse abdominis under the curbing of activating the Global muscles. 

However, subjects have an effective contraction of transverse abdominis on the supine position, when performing the same instructions in the standing position, Global muscles contract strongly, and have less effect on the transverse abdominis.

This shows that the internal oblique and external oblique muscles work well for maintaining the posture, and provides evidence that the external oblique and the internal oblique have similar

affects to the transverse abdominis. Change is not seen when verbal instructions are given rather than with spine position. The "pulling the body in" and "to stop the urination” two way exercises should be done in static position, and the loading can be classified into those of the ultra-low load.

 When you think about load of the exercise, we would like to mention what the range of each high load and low load is. Richardson et al.

15)

, recommends the "3 steps work out menus" for transverse abdominis. Step1, no weight bearing, no additional load, anti-gravity. Step 2, on weight- bearing, gradually increase the load on the Local muscles, and integrate both load and gravity. Step 3 is to maintain the intermediate lumbar pelvis position, gradually increase the load to control the joints. According to this 3 steps work out, "pulling the body in" and "the way to stop urination with placebo" should be done in step 1.Although the risk is low, it is very difficult to adapt to the elderly who do not receive verbal instructions well. For athletes who need to increase the load for warming up, finding the contraction of the transverse abdominal muscle through exercise is a very important issue .

 We found the 10cm forward tilting exercise is the most effective for increasing the size of transverse abdominis, which is the Local muscle during the CKC (Closed Kinetic Chain) using the suspension equipment. CKC is a similar exercise to the motion of human activities. This exercise is considered to be directly linked to performance.

This explains the proximal part starting to move

after the distal part is stable using his own weight

for the load. On the other hand, OKC can be

described as the load is trained utilizing its own

weight. Typical exercises are bench press using

weight plates. Unlike CKC, this explains the distal

part starting to move after the proximal part is

stable. In the case of OKC, the load becomes

the weight (example: barbell),it can damage

the body if it exceeds the limits. On the other

hand, CKC is very popular recently because it

is close to human activities because it uses its

own weight, has low risk and is possible to easily

create an environment for the CKC. Using this

(7)

equipment, automatic movement from passive exercise, resistance exercise and a wide range of motion exercises can be combined such as dexterity exercise using a specific muscle, which can be enabled in various scenes such as dexterity exercises. This functional training equipment has been applied to the medical clinic, health care facilities, and athletic gyms. We have been researching this topic because many of facilities use this suspension device in order to research the front tilt exercise.

 Many of the facilities educate people to maintain the posture from static sitting position to maximum forward tilt to increase ROM of shoulder joint and spine. During the observation in many cases at starting position, the pelvic with static sitting is posterior tilt (Fig.1).At the end, as the subject moves to the maximum front tilt with sitting, the pelvic is forward tilt. They say that the pelvic muscle and diaphragm work to stabilize the trunk with Local and Global muscles.

The instructor wants to confirm the position of both pelvic angle and spinal position. We observed the pelvic is able to maintain horizontal position with front tilt on the chair during the exercise.

The result shows that the 10cm forward tilt was the best position to achieving and maintaining the pelvic horizontal position. Therefore, this leads to determining the three different exercises.

 As we discuss each exercise, the thickness of external and internal oblique muscles are higher at the maximum front tilt in sitting position, compared to the transverse abdominis, which is the highest at the 10cm front tilt position and to be lower at the maximum front tilt position.

This indicates that when requesting the exercise to the subject, it is very important to set up the range of tilt during the exercise. Conventionally, more muscle activity has been see during the wide range of motion. But the result shows that the Local muscle makes the spine stable and is most active at 10cm forward tilt. Therefore, it has been suggested to change the range of motion during exercise to find the best position where the muscle is more active. At the point of the 10cm forward tilt, the size of the muscle decreases and

the pressure of contraction on the transverse abdominis through the end at the maximum forward tilt position increases the load of one's own body weight. Higher gravity makes Global muscles activate for posture stability. It also becomes easy training the Local muscle.

 The importance of contraction of transeverse abdominis for the athletes and patients has become recognized for treating lower back pain. The trunk stabilization of patients has been decreasing in the clinic. There are several reasons of this cause; the muscle itself is developing pain and damage; lower muscle power due to limited activity after the damage; fear of pain causing subconscious or conscious restriction of movement.

The purpose is not to remove the pain but rather to strengthen the spinal column stability, reduce substitution of movement and to have a system of core stabilization in order to promote the prevention of recurrence. Because of this, it is very important that the effective Local muscle activation with a lower load should be done at the middle position between the static (stationary)

position and the middle of the dynamic position.

 In the future, if it is possible to create exercises collaborating the Global muscle and the Local muscle, which have different purposes and features, it would be valuable in helping improve athlete performance, prevent injuries, and would assist anti-aging. We believe that this is going to be the new method of exercise to control small loads and possibly be applicable for rehabilitation.

CONCLUSIONS

 In conventional trunk exercises, it requires

subjects to be in stable and unstable positions with

heavy resistance in order to activate the Local

muscle with small resistance. In our research

findings, 10 cm forward tilt methodology of the

lower resistance exercises is able to create

activation of transverse abdominis. As shown

above, in order to activate the Local muscle in

core trainings, rather than doing the conventional

high resistance exercise, selectively activating the

transverse abdominis with low resistance exercise

is more effective. It would be possible to modify

(8)

the activity so that the exercise can be applied not only to athletes but also to elderly patients.

Reference

1) Bergmark A:Stability of the lumbar spine. A study in mechanical engineering. Acta Orthop Scand Suppl 230:1-54, 1989

2) Hides JA, Richardson CA, Jull GA : Multifidus Muscle recovery is not automatic after resolution of acute, first-episode low back pain.

Spine21:2763-2769, 1996.

3) Hodges PW, Richardson CA : Feedforward contraction of transversus abdominis is not influenced by the direction of arm movement.

Exp Brain Res 114 :362-370, 1997.

4) Syunichi Ito:Lumbago treatment and recurrence prevention by lumber spine stabilization exercise, Orthopaedics21, No6, 2008

5) Sapsford, R.R., Hodges, P.W., Richardson, C.A., et al.:Co-activation of the abdominal and pelvic floor muscles during voluntary exercise, Neurourology and Urodynamics, 20, 31-42, 2001.

6) Sapsford, R.R. and Hodges, P.W.:Contraction of the pelvic floor muscles during abdominal maneuvers, Arch Phys Med Rehabilitation, 82, 1081-1088, 2001.

7) Strohl K.P, Mead J, Banzett R.B, Loring S.H, Kosch P.C : Regional differences in abdominal muscle activity during various manoeuvres in humans. Journal of Applied Physiology51, 1471- 1476, 1981.

8) Locote M, Clevalier A.M, Mirander A, Bleton J.P, Stevenin P : Clinical evaluation of muscle function. Churchill Livingstone, Edinburgh, 1987.

9) D e T r o y e r A , E s t e n n e M , N i n a m e V , VanGansbeke D, Gorini M:Transversus abdminis muscle function in humans. Journal of Applied Physiology 68, 1010-1016, 1990.

10) Satoshi Miyashita, Yoshihiro Wada, Masanori Suzuki: Examination of effective trunk muscle training method –The transversus abdominis muscle and the internal pblique muscle measure musclular thickness of different movement- , Nihonbashi gakkan university bulletin No. 11, 41-51, 2012.

11) Shuji Matsui:Physical exercise science guide basic course, Journal of health, physical education and recreation112, kyourinsyoin.

1967.

12) Hodges P.W., Richardson C.A.:Contraction of the abdominal muscles associated with movement of the lower limb. Physical Therapy, 77, 132- 144, 1997.

13) McMeeken JM, Beith ID, Newham DJ:The relationship between EMG and change in thickness of transversus abdminis, Clinical Biomechanics, 19, (4), 337-342, 2004.

14) Jyunji Ito:Myofibrous organization in human abdominal muscles, Journal of The Showa Medical Association, 48(4), 471-483, 1988.

15) Richardson C., Jull G., Hodges P., Hides J.:

Therapeutic Exercise for spinal segmental

stabilization in low back pain, Churchill

Livingstone, United kingdom, 103-123, 1999.

参照

関連したドキュメント

Key words: multitime maximum principle, curvilinear integral cost, variational PDEs, adjoint PDEs, m-needle variations.. 1 Multitime

We present sufficient conditions for the existence of solutions to Neu- mann and periodic boundary-value problems for some class of quasilinear ordinary differential equations.. We

For further analysis of the effects of seasonality, three chaotic attractors as well as a Poincar´e section the Poincar´e section is a classical technique for analyzing dynamic

Tanaka , An isoperimetric problem for infinitely connected complete open surfaces, Geometry of Manifolds (K. Shiohama, ed.), Perspec- tives in Math. Shioya , On asymptotic behaviour

It leads to simple purely geometric criteria of boundary maximality which bear hyperbolic nature and allow us to identify the Poisson boundary with natural topological boundaries

Correspondingly, the limiting sequence of metric spaces has a surpris- ingly simple description as a collection of random real trees (given below) in which certain pairs of

We formalize and extend this remark in Theorem 7.4 below which shows that the spectral flow of the odd signature operator coupled to a path of flat connections on a manifold

While conducting an experiment regarding fetal move- ments as a result of Pulsed Wave Doppler (PWD) ultrasound, [8] we encountered the severe artifacts in the acquired image2.