Drinfeld-Twisted Supersymmetry and Non-Anticommutative Superspace

索末菲原子模型著名理论物理学家阿诺德·索末菲（Arnold Sommerfeld ，1868～1951），是德国慕尼黑大学理论物理研究院院长，他对玻尔原子理论的扩充和他所著的《原子结构和光谱线》这部深具影响的教科书，被他的学生誉为“原子物理学的圣经”。

在量子力学史上，他赢得量子力学三大重要学派领袖之一的声誉，他在培养人才方面是无与伦比的，他有能把像海森伯、泡利这样的毛头小伙子精雕细琢成杰出科学家的神奇本领。

爱因斯坦1922年很赞赏地说道：“我特别欣赏您培养出了如此众多的青年才俊。

”3、1 索末菲的生平1868年12月5日，索末菲生于东普鲁士的柯尼斯堡（Königsberg ）（今俄罗斯的加里宁格勒），是中欧理论物理的发源地，德国成立的第一个数学和物理研究班就诞生在这里。

中学时代索未菲和德国实验物理学家维恩是同学，1886年进入柯尼斯堡大学数学教授林德曼（C ．Lindemann ）指导的数学——物理研究班主修数学，同当时许多别的数学家一样，索未菲运用开尔文勋爵的数学物理理论对麦克斯韦电磁场方程的进行了概述，并对应用数学产生了浓厚的兴趣。

于是，他从林德曼的数论领域转变到开尔文勋爵的数学对物理学的应用研究，他研究过电子波的物理特性和关于旋转陀螺的理论，对于应用复变函数理论解决边界问题颇有造诣。

1891年，他在康尼斯堡的数学物理教授沃尔克曼（P ．V olkmann ）的指导下，完成了数学物理方面的博士学位论文。

1893～1894年在哥廷根的矿物研究所担任数学家克莱因（F ．Klein ）的助手。

1897年任克劳斯塔尔矿业学校的数学教授。

1900年由克莱因推荐，在亚琛工业大学任工程力学教授。

在此期间，他致力于把数学和工程力学联系起来，使工程力学有坚实的数学基础；这是克莱因一贯的主张。

1906年起任慕尼黑大学理论物理学教授，不久主持建立了理论物理研究院并任院长。

1905年爱因斯坦（A ．Einstein ）的关于狭义相对论的论著发表以后，在1907年德国自然研究者大会上，索末菲曾为爱因斯坦的理论辩护，而且他在这个领域所做的工作，为后来的轫致辐射理论提供了理论基础。

人工智能是一门新兴的具有挑战力的学科。

自人工智能诞生以来,发展迅速,产生了许多分支。

诸如强化学习、模拟环境、智能硬件、机器学习等。

但是,在当前人工智能技术迅猛发展,为人们的生活带来许多便利。

下面是搜索整理的人工智能英文参考文献的分享，供大家借鉴参考。

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richard feynman英文介绍Richard Feynman, a name synonymous with brilliance and innovation in the realm of physics, left an indelible mark on the scientific community with his groundbreaking contributions and charismatic personality. Born on May 11, 1918, in Queens, New York, Feynman exhibited an early aptitude for mathematics and science, foreshadowing his future as one of the most influential physicists of the 20th century.Feynman's journey into the world of physics began at the Massachusetts Institute of Technology (MIT), where he obtained his Bachelor's degree in 1939. His academic pursuits then led him to Princeton University, where he earned his Ph.D. in physics in 1942. It was during his time at Princeton that Feynman's genius began to shine, particularly in the field of quantum mechanics.One of Feynman's most notable contributions to physics came in the form of his diagrams, now famously known as Feynman diagrams. These graphical representations revolutionized the way physicists approached quantum electrodynamics (QED) by providing a visual framework for understanding the behavior of subatomic particles. Feynman diagrams allowed for the visualization of complex interactions between particles, leading to significant advancements in the field.In addition to his theoretical contributions, Feynman was also a gifted teacher and communicator of science. His lectures at the California Institute of Technology (Caltech) became legendary for their clarity, wit, and insight. Feynman had a unique ability to convey complex ideas in simple terms, making physics accessible to students and enthusiasts alike.Feynman's insatiable curiosity and unconventional approach to problem-solving set him apart from his peers. He had a knack for questioning conventional wisdom and was never afraid to challenge the status quo. This fearless attitude not only fueled his own research but also inspired future generations of physicists to think outside the box.Beyond his scientific achievements, Feynman was also known for his colorful personality and adventuresome spirit. He had a passion for playing the bongo drums, cracking safes, and exploring the mysteries of the natural world. Feynman's zest for life was infectious, and he approached both his work and his hobbies with boundless enthusiasm.Throughout his illustrious career, Feynman received numerous accolades and honors, including the Nobel Prize in Physics in 1965 for his contributions to the development of quantum electrodynamics. He was also awarded the Albert Einstein Award and the Oersted Medal, among others, cementing his legacy as one of the preeminent physicists of his time.In his later years, Feynman continued to inspire and educate through his writings and lectures. His books, including "Surely You're Joking, Mr. Feynman!" and "What Do You Care What Other People Think?", offer glimpses into his brilliant mind and irreverent sense of humor.Richard Feynman passed away on February 15, 1988, but his legacy lives on in the hearts and minds of those who continue to be inspired by his work. His contributions to physics not only expanded our understanding of the universe but also served as a testament to the power of curiosity, creativity, and perseverance in the pursuit of knowledge. As we reflect on Feynman's life and achievements, we are reminded of the profound impact that one individual can have on the world through dedication, passion, and a relentless quest for truth.。

The brilliance of passion is a force that can light up the world,igniting the darkest corners of our lives with a glow that is both inspiring and transformative.When passion is allowed to flourish,it can lead to incredible achievements and personal growth.In the realm of arts,passion is the driving force behind the creation of masterpieces that have stood the test of time.Artists who pour their heart and soul into their work,like Vincent Van Gogh with his swirling Starry Night or Beethoven with his powerful Symphony No.9,have left an indelible mark on history.Their passion not only fueled their creative process but also resonated with audiences,inspiring generations to appreciate and create art.In sports,athletes who are fueled by passion often push their bodies to the limits, achieving feats that seem impossible.Take,for instance,the marathon runners who, despite exhaustion,find the strength to cross the finish line,or the gymnasts who defy gravity with their acrobatic routines.Their dedication and love for their sport are evident in every move they make,and it is this passion that propels them to greatness.In the field of science and innovation,passion is the catalyst for groundbreaking discoveries and inventions.Think of figures like Albert Einstein,whose passion for understanding the universe led to the development of the theory of relativity,or Marie Curie,who was driven by her curiosity to study radioactivity,ultimately winning two Nobel Prizes.Their unwavering commitment to their work has shaped our understanding of the world and opened up new frontiers of knowledge.Passion also plays a vital role in social and political movements.Leaders like Martin Luther King Jr.and Mahatma Gandhi were propelled by their fervent belief in justice and equality,leading to significant changes in society.Their passion was infectious,rallying people to join their cause and fight for a better world.On a personal level,allowing passion to bloom can lead to a more fulfilling life.When we pursue our interests and dreams with fervor,we experience a sense of purpose and joy that is unmatched.This passion can also help us overcome obstacles and challenges,as it provides the motivation to keep going even when the going gets tough.In conclusion,the radiance of passion is a beacon that illuminates the path to success, both for individuals and society as a whole.It is a powerful force that can transform lives, inspire greatness,and bring about positive change.By nurturing and embracing our passions,we can unlock our full potential and contribute to a brighter,more vibrant world.。

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alexander fleming英语介绍Alexander Fleming: An IntroductionAlexander Fleming, a pioneer in the field of medical science, is widely known for his discovery of penicillin, a groundbreaking antibiotic that revolutionized the world of medicine. Born on August 6, 1881, in Lochfield, Scotland, Fleming grew up to make significant contributions to the field of bacteriology and earned his place in history as one of the greatest scientists of all time. This article will delve into the life, work, and legacy of Alexander Fleming.Early Years and EducationAlexander Fleming was born into a farming family to Hugh Fleming and Grace Stirling Morton. His upbringing in the countryside cultivated a strong work ethic and a deep fascination with nature. At the age of 13, he moved to London to live with his older brother and pursue his education.Fleming's academic journey led him to attend the Royal Polytechnic Institution, where he studied biology, physics, and chemistry. His passion for research and experimentation flourished during this time, and he graduated with distinction in 1902. Fleming's thirst for knowledge prompted him to further his studies at St. Mary's Hospital Medical School, where he eventually obtained a Bachelor of Medicine degree.Groundbreaking Discovery of PenicillinIn 1928, Alexander Fleming stumbled upon one of his most notable discoveries entirely by accident. While working at St. Mary's Hospital,Fleming noticed that a petri dish containing Staphylococcus bacteria had been contaminated with mold. Surprisingly, the mold inhibited the growth of the bacteria and created a clear zone around it.Intrigued by this phenomenon, Fleming isolated the mold and identified it as a strain of the Penicillium genus. He named the substance produced by the mold "penicillin" and conducted further experiments to determine its potential medical applications. Fleming's research revealed that penicillin possessed remarkable antibacterial properties, capable of eliminating various harmful bacteria without causing significant harm to the human body.Impact on Medicine and LegacyAlexander Fleming's discovery of penicillin was a game-changer in the medical field. Prior to this breakthrough, bacterial infections posed significant challenges and often led to fatalities. Penicillin transformed the treatment of infectious diseases, saving countless lives and providing an effective weapon against bacterial infections.Fleming's incredible contribution earned him recognition and accolades throughout his career. In 1945, he was awarded the Nobel Prize in Physiology or Medicine for his discovery of penicillin. His work spearheaded the development of antibiotics, revolutionizing modern medicine and opening doors to new research and advancements in healthcare.Beyond his scientific achievements, Alexander Fleming's humility and dedication to serving humanity left an indelible mark on the world. He strongly believed in the responsible use of antibiotics and warned against the risks of antibiotic resistance. Fleming's advocacy for responsible antibiotic usage continues to resonate with healthcare professionals worldwide.ConclusionAlexander Fleming's life and work continue to inspire generations of scientists and medical professionals. His accidental discovery of penicillin reshaped the field of medicine and saved countless lives. Fleming's relentless pursuit of scientific knowledge and his unwavering commitment to public health have left an enduring legacy. Today, his contributions serve as a constant reminder of the power of curiosity, research, and the potential for scientific breakthroughs to shape the future of healthcare.。

a r X i v :n l i n /0212002v 2 [n l i n .S I ] 21 M a r 2003Nonlinear superposition formula for N =1supersymmetric KdV Equation Q.P.Liu and Y.F.Xie Department of Mathematics,China University of Mining and Technology,Beijing 100083,China.Abstract In this paper,we derive a B¨a cklund transformation for the supersymmetric Korteweg-de Vries equation.We also construct a nonlinear superposition formula,which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea,Ramani and Grammaticos.The celebrated Korteweg-de Vries (KdV)equation was extended into super frame-work by Kupershmidt [3]in 1984.Shortly afterwards,Manin and Radul [7]proposed another super KdV system which is a particular reduction of their general supersymmet-ric Kadomstev-Petviashvili hierarchy.In [8],Mathieu pointed out that the super version of Manin and Radul for the KdV equation is indeed invariant under a space supersymmetric transformation,while Kupershmidt’s version does not.Thus,the Manin-Radul’s super KdV is referred to the supersymmetric KdV equation.We notice that the supersymmetric KdV equation has been studied extensively in lit-erature and a number of interesting properties has been established.We mention here the infinite conservation laws [8],bi-Hamiltonian structures [10],bilinear form [9][2],Darboux transformation [6].By the constructed Darboux transformation,Ma˜n as and one of us calculated the soli-ton solutions for the supersymmetric KdV system.This sort of solutions was also obtained by Carstea in the framework of bilinear formalism [1].However,these solutions are char-acterized by the presentation of some constraint on soliton parameters.Recently,using super-bilinear operators,Carstea,Ramani and Grammaticos [2]constructed explicitly new two-and three-solitons for the supersymmetric KdV equation.These soliton solutions are interesting since they are free of any constraint on soliton parameters.Furthermore,the fermionic part of these solutions is dressed through the interactions.In addition to the bilinear form approach,B¨a cklund transformation (BT)is also a powerful method to construct solutions.Therefore,it is interesting to see if the soliton solutions of Carstea-Ramani-Grammaticos can be constructed by BT approach.In this paper,we first construct a BT for the supersymmetric KdV equation.Then,we derive a nonlinear superposition formula.In this way,the soliton solutions can be producedsystematically.We explicitly show that the two-soliton solution of Carstea,Ramani and Grammaticos appears naturally in the framework of BT.To introduce the supersymmetric extension for the KdV equation,we recall some terminology and notations.The classical spacetime is(x,t)and we extend it to a super-spacetime(x,t,θ),whereθis a Grassmann odd variable.The dependent variable u(x,t) in the KdV equation is replaced by a fermionic variableΦ=Φ(x,t,θ).Now the super-symmetric KdV equation reads asΦt−3(ΦDΦ)x+Φxxx=0,(1)where D=∂∂x is the superderivative.Mathieu found the following supersymmetricversion of Gardner type mapΦ=χ+ǫχx+ǫ2χ(Dχ),(2) whereǫis an ordinary(bosonic)parameter.It is easy to show thatχsatisfies the following supersymmetric Gardner equationχt−3(χDχ)x−3ǫ2(Dχ)(χDχ)x+χxxx=0.(3) This map was used in[8]to prove that there exists an infinite number of conservation laws for the supersymmetric KdV equation(1).In the classical case,Gardner type of map was studied extensively by Kupershmidt[4].It is well known that such map may be used to construct interesting BT.We will show that it is also the case for the supersymmetric KdV equation.We notice that the supersymmetric Gardner equation(3)is invariant underǫ→−ǫ. The new solution of the supersymmetric KdV equation corresponding to with−ǫis denoted as˜Φ.Thus we have˜Φ=χ−ǫχx+ǫ2χ(Dχ).(4) From above relations(3-4),wefindΦ−˜Φ=2ǫχx,(5)Φ+˜Φ=2χ+2ǫχ(Dχ).(6) Let us introduce the potentials as followsΦ=Ψx,˜Φ=˜Ψx,thus,the equation(5)provides usχ=12(Ψ−˜Ψ)(DΨ−D˜Ψ),(8)whereλ=1/ǫis the B¨a cklund parameter.The transformation(8)is in fact the spatial part of BT.Its temporal counterpart can be easily worked out.We also remark here that the BT is reduced to the well known BT for the classical KdV equation ifη=0,as it should be.A BT can be used to generate special solutions.If we start with the trivial solution ˜Ψ=0,we obtain2a(ζ+θλ)eλx−λ3tΨ=−(Ψ1−Ψ0)(DΨ1−DΨ0),(10)2and1(Ψ0+Ψ2)x=λ2(Ψ2−Ψ0)+(Ψ3−Ψ1)(DΨ3−DΨ1),(12)2and1(Ψ2+Ψ3)x=λ1(Ψ3−Ψ2)+Subtraction (10)form (11),we have(Ψ2−Ψ1)x =λ2Ψ2−λ1Ψ1+(λ1−λ2)Ψ0+12Ψ2(D Ψ0)−12Ψ1(D Ψ1)+12Ψ0(D Ψ1),(14)similarly,from (12)and (13)we have(Ψ2−Ψ1)x = λ1−λ2+12(D Ψ2) Ψ3+12Ψ2(D Ψ2)−12(D Ψ1)−12(Ψ1−Ψ2)[(D Ψ3)+2λ1+2λ2−(D Ψ0)]=0.(16)The equation (16)is a differential equation for Ψ3.Solving it we obtainΨ3=Ψ0−(λ1+λ2)(Ψ1−Ψ2)λ1−λ2+v 1−v 2−(λ1+λ2)(η1−η2)(η1,x −η2,x )(λ1−λ2+v 1−v 2).It is clear that our nonlinear superposition formula reduces to the well-known superposition formula for the KdV as it should be.The advantage to have a superposition formula is that it is an algebraic one and can be used easily to find solutions.Using the solutions (9)as our seeds,we may construct a 2-soliton solution of (1)by means of our superposition formula.Indeed,letΨ0=0,Ψ1=−2a 1(ζ1+θλ1)e λ1x −λ31t1+a 2e λ2x −λ32tthen from our nonlinear superposition formula (17),we obtainΨ3=2 ζ1a 1e δ1−ζ2a 2e δ2+(ζ1−ζ2)a 1a 2e δ1+δ2+θ(λ1a 1e δ1−λ2a 2e δ2+(λ1−λ2)a 1a 2e δ1+δ2)whereδi=λi x−λ3i t+θζi(i=1,2).Now,takingλ1>λ2,a1=−λ1−λ2λ1+λ2we recover the2-soliton solution foundfirst by Carstea,Ramani and Grammaticos[2].As in the classical KdV case[11],we generate this2-soliton from a regular solution and a singular solution.We could continue this process to build the higher soliton solutions and the calculation will be tedious but straightforward.Acknowledgment The work is supported in part by National Natural Scientific Foun-dation of China(grant number10231050)and Ministry of Education of China. References[1]Carstea A S2000Extension of the bilinear formalism to supersymmetric KdV-typeequations Nonlinearity131645.[2]Carstea A S,Ramani A and Grammaticos B2001Constructing the soliton solutionsfor the N=1supersymmetric KdV hierarchy Nonlinearity141419.[3]Kupershmidt B A1984A super Korteweg-de Vries equation Phys.Lett.102A213.[4]Kupershmidt B A1981On the nature of Gardner transformation J.Math.Phys.22449;Kupershmidt B A1983Deformations of integrable systems Proc.R.Soc.Irish 83A45.[5]Liu Q P1995Darboux transformation for the supersymmetric KdV equations Lett.Math.Phys.35115.[6]Liu Q P and Ma˜n as M1997Darboux transformation for the Mann-Radul supersym-metric KdV equation Phys.Lett.B394337;Liu Q P and Ma˜n as M in Supersymmetry and Integrable Systems eds.H.Aratyn et al Lecture Notes in Physics502(Springer).[7]Manin Yu I and Radul A O1985A supersymmetric extension of the Kadomtzev-Petviashvili hierarchy Commun.Math.Phys.9865.[8]Mathieu P1988Supersymmetric extension of the Korteweg-de Vries equation J.Math.Phys.292499.[9]McArthur I N and Yung C M1993Hirota bilinearfform for the super KdV hierarchyMod.Phys.Lett.8A1739.[10]Oevel W and Popowicz Z1991The bi-Hamiltonian structure of fully supersymmetricKorteweg-de Vries systems Commun.Math.Phys.139441;Figueroa-O’Farril J,Mas J and Ramos E1991Integrability and biHamiltonian structure of the even order sKdV hierarchies Rev.Math.Phys.3479.[11]Wahlquist H D and Estabrook F B1974B¨a cklund transformation for solutions of theKorteweg-de Vries equation Phys.Rev.Lett.311386.。

BLUISH DISCOLORATION OF THEUMBILICUSIN THE DIAGNOSIS OF RUPTURED EXTRA\x=req-\UTERINE PREGNANCYEMIL NOVAK,M.D.BALTIMOREIn1919,Cullen1called attention to a"bluish dis-coloration of the umbilicus as a diagnostic sign where ruptured extra-uterine pregnancy exists."The patient on whom he had observed this sign was a woman of 38who had suffered with pain in the right lower abdomen for three weeks.The patient had not missed any menstrual period and there had been no uterine bleeding.In spite of the latter fact,Cullen states that "the bluish-black appearance of the navel unassociated with any history of injury,together with the mass to theright of the uterus,made thediagnosis of extra-uterinepregnancy relatively certain."This diagnosis was confirmedon opening the abdomen,which was found to be filledwith dark blood.Some re¬cent experiences with extra-uterine pregnancy have con¬vinced me of the value ofthis"blue belly-button"signof Cullen.REPORT OF CASESCase1.—History.—E.W.,aged19,nullipara,was seen in con¬sultation with Dr.W.G.Cop-page,Aug.21,1921.Menstru¬ation had been normal until June,the last period having occurred,June 6.There was no flow inJuly,but on August3uterinebleeding began and had beenpresent,though scantily and in¬termittently,up to the day thepatient was seen.There had beenpain in both sides of the lowerabdomen throughout the flow. Early on the day on which I saw her,this pain had become much more severe,and the patient soon complained of faint-ness and nausea.Weakness, pallor and shock became pro¬nounced,and the clinical picture,CuIIen's original picture,reprinted with permission from "Contributions to Medical and Biological Research,dedicated to Sir William Osier in honour of his Seventieth Birthday, July12,1921,by his pupils and co-workers."Bluish discol¬oration of the umbilicus associated with a ruptured extra-uterine pregnancy.This picture was obtained at operation three weeks after the first symptoms developed.The umbilicus itself has now turned a light green;above it the tissue has a faint bluish tinge;below the umbilicus the blue is marked.when I saw her,was the classical one of grave internal hemorrhage.Examination.—There was extreme pallor of the skin and mucous membranes,the lips being almost white.The pulse was140,the respiration shallow and feeble.The abdomen was full and rounded,with much tenderness over the entire lower zone.Percussion gave dulness over both flanks. Around the umbilicus was a well-defined areola of greenish-yellow discoloration,extending for about1cm(three-eighths inch)beyond the edges of the umbilical depression.Pelvic examination revealed the uterus to be slightly enlarged and pushed to the right by a large,very tender mass occupying the left iliac fossa.Examination of the blood disclosed only From the gynecologic department of Johns Hopkins Medical School.1.Cullen,T.S.:Bluish Discoloration of the Umbilicus as a Diag-nostic Sign Where Ruptured Extra-Uterine Pregnancy Exists,Contribu-tions to Medical and Biological Research,dedicated to Sir William Osler,1919.2,400,000red blood corpuscles to the cubic millimeter,with a hemoglobin of50per cent.The diagnosis of ruptured left tubai pregnancy was made,and the patient was at once sent into the South Baltimore General Hospital for operation. Operation.—Without going into details,suffice it to say that on opening the abdomen a large quantity of both liquid and clotted blood escaped.It had apparently been retained under considerable tension.The left tube was the seat of the gestation sac,and presented a ragged rupture about2cm. (three-quarters inch)in diameter.The pregnant tube was quickly removed in the usual manner.The patient made an uninterrupted recovery.Case2.—History.—L.H.,aged26,seen,Aug.26,1921, in consultation with Dr.L.J.Dobihal,had been married about seven years and had had two children,the younger five years previously.Since then there had been a number of abortions,all said to have been self-induced.The last reg¬ular menstrual period had occurred,July17.Uterine bleed¬ing reappeared.August5,and had been present constantly since then,although the amount was small.For two weeksthere had been frequent attacksof crampy pain in the lowerabdomen,especially in the rightside.These attacks had oftenbeen accompanied by nausea andfaintness.Examination.The generalcondition of the patient wasgood,the pulse being110,tem¬perature99.5 F.Abdominal ex¬amination revealed the abdomento be rather rounded,with aheavy layer of adipose tissue.There was some diffuse tender¬ness over the lower zone,espe¬cially on the right side.Percus¬sion disclosed movable dulness inboth flanks,and tympany else¬where.Surrounding the umbili¬cus there was a greenish-bluearea of pigmentation.Pelvicexamination revealed the uterusto be normal in size and posi¬tion.There was marked tender¬ness in the right side of thepelvis,where a small,tendermass could be indefinitely pal¬pated.The left side of thepelvis was negative.The bloodcount revealed11,000leukocytes.A diagnosis was made of righttubai pregnancy,and the patientwas sent into the hospital foroperation.Operation.—At the operation,August18,the right tube wasfound to be the seat of a spindle¬like enlargement,due to a righttubai pregnancy.The widest uiameter tne tuDe was only about3cm.(Wie inches), explaining how difficult its palpation would be ina stout person.The fimbriated extremity was stuffed with clots,the large amount of blood present in the pelvic and abdominal cavities being the result of tubai abortion.The right tube was excised and the appendix also removed.Recoveiry was uneventful.COMMENTThe explanation of this discoloration of the umbili¬cus in cases of extensive intraperitoneal hemorrhage is probably to be sought in the lymphatics of the umbilical region.The case of localized jaundice of the umbilicus reported in1905by Ransohoff,2to2.Ransohoff,Joseph:Gangrene of the Gallbladder:Rupture of the Common Bile Duct,with a New Sign,J. A.M. A.46:395(Feb.10)1906.which Cullen refers,is of great interest in this connec¬tion.It was one of rupture of the common bile duct, the abdominal cavity containing a large amount of free bile.The umbilicus in this patient was of a saffron-yellow hue,in sharp contrast to the skin over the rest of the abdomen.Ransohoff is inclined to believe that the phenomenon is the result of simple inhibition,but the more likely mechanism would seem to be that the bile pigments are deposited in the skin after absorption by the lymphatics.The abdominal wall is quite thin in the region of the umbilicus,and there is a rather rich anastomosis between the intraperitoneal and extra-peritoneal lymphatics at this point.This has been emphasized by Handley and others in connection with the extension of carcinoma of the liver and stomach to the periumbilical skin region.Reversely,it is not uncommon for carcinoma of the breast to make its way into the abdominal cavity by these channels,the cancer cells finding their way even to the ovary at times. The discoloration is not always bluish.In one of my cases it was a greenish-yellow,resembling a fading bruise.The different hues are unquestionably due to the differing degrees of oxidation of the deposited blood pigments,as in the case of the ordinary bruise. If this is true it would follow that a dark bluish dis¬coloration would indicate a recent hemorrhage,while a greenish-yellow or orange colored pigmentation would suggest that the intra-abdominal blood had been present for some time.It need scarcely be said that no discoloration at all can be expected when the hemor¬rhage is so cataclysmic that the patient comes under observation very soon after its occurrence,for the reason that there is insufficient time in such cases for the occurrence of absorption.Recently,Hellendall3has called attention to a similar discoloration of the umbilicus,and describes it as a new diagnostic sign.He is apparently ignorant of Cullen's article,published two years earlier.Furthermore,he evidently believes that it is observed only in cases of ruptured extra-uterine pregnancy associated with umbilical hernia,as in the case which he reports. There can be no doubt,therefore,that the credit for first describing this valuable sign belongs to American surgery.It is hardly necessary to say that Cullen's sign would hardly be positive in cases of extra-uterine pregnancy which are not associated with intraperitoneal hemor¬rhage of considerable degree.Nor need it be empha¬sized that severe intra-abdominal hemorrhage due to other causes than extra-uterine pregnancy might cause the umbilical discoloration as well.Cases of the latter type,however,are so rare that,to all intents and pur¬poses,the sign may be considered one especially appli¬cable to the diagnosis of ruptured extra-uterine gestation.While the recognition of severe intra-abdominal hemorrhage is often simple enough,there are not a few cases in which such a hemorrhage causes comparatively little general effect,and may be difficult of recognition.The demonstration of Cullen's sign in such cases will,I am sure,be of considerable value in diagnosis.26East Preston Street.3.Hellendall,H.:Ein neues Symptom der Extrauterinschwang-erschaft,Zentralbl. f.Gyn\l=a"\k.45:890,1921.Pellagra in Italy.—Lavoro publishes an account of the report by a committee appointed in1910,which found that pellagra declined during the war because of changes in the quality of the food for various reasons connected with the war,espe¬cially the scarcity of corn.THE ETIOLOGY OF ORTHOSTATICALBUMINURIAWILLY RIESER,MD.ANDSIDNEY L.RIESER,MD.NEW YORKThe finding of albumin in the urine is always of importance,and lays upon the medical attendant who makes the observation the obligation to determine,if possible,the cause of this sign and to appraise its sig-nificance.Few,if any,more interesting renal phe-nomena than orthostatic albuminuria come to our notice.It was observed as early as1887,and variously named cyclic,physiologic,intermittent,orthotic or orthostatic albuminuria.Orthostatic albuminuria is the most fitting appellation,as it designates an albu-minuria which occurs only in the upright standing posture.But none of these names adequately explain its pathogenesis.In our cases the albumin appears in urine voided from three to seven minutes after the erect posture has been assumed,and it disappears completely from the urine voided from three to seven minutes after the horizontal or lying posture is taken.Quantitatively, there is usually a heavy or very heavy trace.This condition is most commonly observed in chil-dren and adolescents,but is not uncommon in adults. One of our patients is24years old,the other,28.It occurs equally in the two sexes.These patients present a general relaxation and atonia of muscular and liga-mentous structures,both skeletal and visceral,which manifests itself in the long thorax,with dropped heart and low diaphragm,the scaphoid abdomen and the marked visceroptosis.The faulty ligamentous struc¬ture gives them an abnormal loose jointedness and hyperflexibility of the spine.Subjectively,they are of the asthénie type,easily fatigued,especially by effort in the standing posture, and most of all by the act of standing itself.They are subject to syncopal attacks,cardiac palpitation,head¬aches,and other evidences of vasomotor instability.Orthostatic albuminuria is differentiated from nephritis and the inflammatory albuminurias by its prompt and complete disappearance,on correction of the upright posture,or the assumption of the hori¬zontal.The urine secreted in the horizontal posture is normal in quantity and is free from any pathologic formed or chemical elements.During the albuminurie periods there is a marked oliguria,without increase of specific gravity.There is no impairment of salt and urea elimination.The pigments are increased,and phenolsulphonephthalein excretion is definitely dimin¬ished.Concomitant with the disappearance of the albumin,there is a return to normal elimination quan¬titatively and qualitatively.Hence it seems that the frequent and sudden albuminurie periods are attrib¬utable to renal dysfunction,not conditioned on intra-renal lesion,but on a factor which becomes operative through these postural changes,and which in turn suspends its influence in a similar manner.EXPLANATIONS OF THE PHENOMENONThe modus operandi of this postural change in the production of the albuminuria has been the subject of speculation and investigation for many years.The earliest observers held renal hyperpermeability,inflam-Read before the Society of Lebanon Hospital Alumni,Dec.13,1921.。

VITALI’S THEOREM AND WWKLDOUGLAS K.BROWNMARIAGNESE GIUSTOSTEPHEN G.SIMPSONAbstract.Continuing the investigations of X.Yu and others,westudy the role of set existence axioms in classical Lebesgue mea-sure theory.We show that pairwise disjoint countable additivityfor open sets of reals is provable in RCA0.We show that sev-eral well-known measure-theoretic propositions including the VitaliCovering Theorem are equivalent to WWKL over RCA0.1.IntroductionThe purpose of Reverse Mathematics is to study the role of set ex-istence axioms,with an eye to determining which axioms are needed in order to prove specific mathematical theorems.In many cases,it is shown that a specific mathematical theorem is equivalent to the set existence axiom which is needed to prove it.Such equivalences are often proved in the weak base theory RCA0.RCA0may be viewed as a kind of formalized constructive or recursive mathematics,with full clas-sical logic but severely restricted comprehension and induction.The program of Reverse Mathematics has been developed in many publica-tions;see for instance[5,10,11,12,20].In this paper we carry out a Reverse Mathematics study of some aspects of classical Lebesgue measure theory.Historically,the subject of measure theory developed hand in hand with the nonconstructive, set-theoretic approach to mathematics.Errett Bishop has remarked that the foundations of measure theory present a special challenge to the constructive mathematician.Although our program of Reverse Mathematics is quite different from Bishop-style constructivism,we feel that Bishop’s remark implicitly raises an interesting question:Which nonconstructive set existence axioms are needed for measure theory?VITALI’S THEOREM AND WWKL 2This paper,together with earlier papers of Yu and others [21,22,23,24,25,26],constitute an answer to that question.The results of this paper build upon and clarify some early results of Yu and Simpson.The reader of this paper will find that familiarity with Yu–Simpson [26]is desirable but not essential.We begin in section 2by exploring the extent to which measure theory can be developed in RCA 0.We show that pairwise disjoint countable additivity for open sets of reals is provable in RCA 0.This is in contrast to a result of Yu–Simpson [26]:countable additivity for open sets of reals is equivalent over RCA 0to a nonconstructive set existence axiom known as Weak Weak K¨o nig’s Lemma (WWKL).We show in sections 3and 4that several other basic propositions of measure theory are also equivalent to WWKL over RCA 0.Finally in section 5we show that the Vitali Covering Theorem is likewise equivalent to WWKL over RCA 0.2.Measure Theory in RCA 0Recall that RCA 0is the subsystem of second order arithmetic with∆01comprehension and Σ01induction.The purpose of this section is toshow that some measure-theoretic results can be proved in RCA 0.Within RCA 0,let X be a compact separable metric space.We define C (X )= A,the completion of A ,where A is the vector space of rational “polynomials”over X under the sup-norm, f =sup x ∈X |f (x )|.For the precise definitions within RCA 0,see [26]and section III.E of Brown’s thesis [4].The construction of C (X )within RCA 0is inspired by the constructive Stone–Weierstrass theorem in section 4.5of Bishop and Bridges [2].It is provable in RCA 0that there is a natural one-to-one correspondence between points of C (X )and continuous functions f :X →R which are equipped with a modulus of uniform continuity ,that is to say,a function h :N →N such that for all n ∈N and x ,y ∈Xd (x,y )<12n .Within RCA 0we define a measure (more accurately,a nonnegative Borel probability measure)on X to be a nonnegative bounded linear functional µ:C (X )→R such that µ(1)=1.(Here µ(1)denotes µ(f ),f ∈C (X ),f (x )=1for all x ∈X .)For example,if X =[0,1],the unit interval,then there is an obvious measure µL :C ([0,1])→R given by µL (f )= 10f (x )dx ,the Riemann integral of f from 0to 1.We refer to µL as Lebesgue measure on [0,1].There is also the obvious generalization to Lebesgue measure µL on X =[0,1]n ,the n -cube.VITALI’S THEOREM AND WWKL 3Definition 2.1(measure of an open set).This definition is made in RCA 0.Let X be any compact separable metric space,and let µbe any measure on X .If U is an open set in X ,we defineµ(U )=sup {µ(f )|f ∈C (X ),0≤f ≤1,f =0on X \U }.Within RCA 0this supremum need not exist as a real number.(Indeed,the existence of µ(U )for all open sets U is equivalent to ACA 0over RCA 0.)Therefore,when working within RCA 0,we interpret assertions about µ(U )in a “virtual”or comparative sense.For example,µ(U )≤µ(V )is taken to mean that for all >0and all f ∈C (X )with 0≤f ≤1and f =0on X \U ,there exists g ∈C (X )with 0≤g ≤1and g =0on X \V such that µ(f )≤µ(g )+ .See also [26].Some basic properties of Lebesgue measure are easily proved in RCA 0.For instance,it is straightforward to show that the Lebesgue measure of the union of a finite set of pairwise disjoint open intervals is equal to the sum of the lengths of the intervals.We define L 1(X,µ)to be the completion of C (X )under the L 1-norm given by f 1=µ(|f |).(For the precise definitions,see [5]and[26].)In RCA 0we see that L 1(X,µ)is a separable Banach space,but to assert within RCA 0that points of the Banach space L 1(X,µ)represent measurable functions f :X →R is problematic.We shall comment further on this question in section 4below.Lemma 2.2.The following is provable in RCA 0.If U n ,n ∈N ,is a sequence of open sets,then µ∞ n =0U n ≥lim k →∞µ k n =0U n .Proof.Trivial.Lemma 2.3.The following is provable in RCA 0.If U 0,U 1,...,U k is a finite,pairwise disjoint sequence of open sets,then µ k n =0U n ≥k n =0µ(U n ).Proof.Trivial.An open set is said to be connected if it is not the union of two disjoint nonempty open sets.Let us say that a compact separable metric space X is nice if for all sufficiently small δ>0and all x ∈X ,the open ballB (x,δ)={y ∈X |d (x,y )<δ}VITALI’S THEOREM AND WWKL4 is connected.Such aδis called a modulus of niceness for X.For example,the unit interval[0,1]and the n-cube[0,1]n are nice, but the Cantor space2N is not nice.Theorem2.4(disjoint countable additivity).The following is prov-able in RCA0.Assume that X is nice.If U n,n∈N,is a pairwise disjoint sequence of open sets in X,thenµ∞n=0U n=∞n=0µ(U n).Proof.Put U= ∞n=0U n.Note that U is an open set.By Lemmas2.2and2.3,we have in RCA0thatµ(U)≥ ∞n=0µ(U n).It remainsto prove in RCA0thatµ(U)≤ ∞n=0µ(U n).Let f∈C(X)be suchthat0≤f≤1and f=0on X\U.It suffices to prove thatµ(f)≤∞n=0µ(U n).Claim1:There is a sequence of continuous functions f n:X→R, n∈N,defined by f n(x)=f(x)for all x∈U n,f n(x)=0for all x∈X\U n.To prove this in RCA0,recall from[6]or[20]that a code for a continuous function g from X to Y is a collection G of quadruples (a,r,b,s)with certain properties,the idea being that d(a,x)<r im-plies d(b,g(x))≤s.Also,a code for an open set U is a collection of pairs(a,r)with certain properties,the idea being that d(a,x)<r im-plies x∈U.In this case we write(a,r)<U to mean that d(a,b)+r<s for some(b,s)belonging to the code of U.Now let F be a code for f:X→R.Define a sequence of codes F n,n∈N,by putting(a,r,b,s) into F n if and only if1.(a,r,b,s)belongs to F and(a,r)<U n,or2.(a,r,b,s)belongs to F and b−s≤0≤b+s,or3.b−s≤0≤b+s and(a,r)<U m for some m=n.It is straightforward to verify that F n is a code for f n as required by claim1.Claim2:The sequence f n,n∈N,is a sequence of elements of C(X). To prove this in RCA0,we must show that the sequence of f n’s has a sequence of moduli of uniform continuity.Let h:N→N be a modulus of uniform continuity for f,and let k be so large that1/2k is a modulus of niceness for X.We shall show that h :N→N defined by h (m)=max(h(m),k)is a modulus of uniform continuity for all of the f n’s.Let x,y∈X and m∈N be such that d(x,y)<1/2h (m). To show that|f n(x)−f n(y)|<1/2m,we consider three cases.Case1:VITALI’S THEOREM AND WWKL5 x,y∈U n.In this case we have|f n(x)−f n(y)|=|f(x)−f(y)|<1VITALI’S THEOREM AND WWKL 6From (1)we see that for each >0there exists k such that µ(f )− ≤ kn =0µ(f n ).Thus we haveµ(f )− ≤kn =0µ(f n )≤k n =0µ(U n )≤∞ n =0µ(U n ).Since this holds for all >0,it follows that µ(f )≤ ∞n =0µ(U n ).Thus µ(U )≤ ∞n =0µ(U n )and the proof of Theorem 2.4is complete.Corollary 2.5.The following is provable in RCA 0.If (a n ,b n ),n ∈N is a sequence of pairwise disjoint open intervals,then µL ∞ n =0(a n ,b n ) =∞ n =0|a n −b n |.Proof.This is a special case of Theorem 2.4.Remark 2.6.Theorem 2.4fails if we drop the assumption that X is nice.Indeed,let µC be the familiar “fair coin”measure on the Cantor space X =2N ,given by µC ({x |x (n )=i })=1/2for all n ∈N and i ∈{0,1}.It can be shown that disjoint finite additivity for µC is equivalent to WWKL over RCA 0.(WWKL is defined and discussed in the next section.)In particular,disjoint finite additivity for µC is not provable in RCA 0.3.Measure Theory in WWKL 0Yu and Simpson [26]introduced a subsystem of second order arith-metic known as WWKL 0,consisting of RCA 0plus the following axiom:if T is a subtree of 2<N with no infinite path,thenlim n →∞|{σ∈T |length(σ)=n }|VITALI’S THEOREM AND WWKL 7see also Sieg [18].In this sense,every mathematical theorem provable in WKL 0or WWKL 0is finitistically reducible in the sense of Hilbert’s Program;see [19,6,20].Remark 3.2.The study of ω-models of WWKL 0is closely related to the theory of 1-random sequences,as initiated by Martin-L¨o f [16]and continued by Kuˇc era [7,13,14,15].At the time of writing of [26],Yu and Simpson were unaware of this work of Martin-L¨o f and Kuˇc era.The purpose of this section and the next is to review and extend the results of [26]and [21]concerning measure theory in WWKL 0.A measure µ:C (X )→R on a compact separable metric space X is said to be countably additive if µ∞ n =0U n =lim k →∞µ k n =0U n for any sequence of open sets U n ,n ∈N ,in X .The following theorem is implicit in [26]and [21].Theorem 3.3.The following assertions are pairwise equivalent over RCA 0.1.WWKL.2.(countable additivity)For any compact separable metric space Xand any measure µon X ,µis countably additive.3.For any covering of the closed unit interval [0,1]by a sequence of open intervals (a n ,b n ),n ∈N ,we have ∞n =0|a n −b n |≥1.Proof.That WWKL implies statement 2is proved in Theorem 1of [26].The implication 2→3is trivial.It remains to prove that statement 3implies WWKL.Reasoning in RCA 0,let T be a subtree of 2<N with no infinite path.PutT ={σ i |σ∈T,σ i /∈T,i <2}.For σ∈2<N put lh(σ)=length of σanda σ=lh(σ)−1n =0σ(n )2lh(σ).Note that |a σ−b σ|=1/2lh(σ).Note also that σ,τ∈2<N are incompa-rable if and only if (a σ,b σ)∩(a τ,b τ)=∅.In particular,the intervals (a τ,b τ),τ∈ T,are pairwise disjoint and cover [0,1)except for some of the points a σ,σ∈2<N .Fix >0and put c σ=a σ− /4lh(σ),d σ=a σ+ /4lh(σ).Then the open intervals (a τ,b τ),τ∈ T,(c σ,d σ),VITALI’S THEOREM AND WWKL 8σ∈2<N and (1− ,1+ )form a covering of [0,1].Applying statement 3,we see that the sum of the lengths of these intervals is ≥1,i.e. τ∈ T12lh(τ)=1.From this,equation (2)follows easily.Thus we have proved that state-ment 3implies WWKL.This completes the proof of the theorem.It is possible to take a somewhat different approach to measure the-ory in RCA 0.Note that the definition of µ(U )that we have given (Definition 2.1)is extensional in RCA 0.This means that if U and V contain the same points then µ(U )=µ(V ),provably in RCA 0.An alternative approach is the intensional one,embodied in Definition 3.4below.Recall that an open set U is given in RCA 0as a sequence of basic open sets.In the case of the real line,basic open sets are just intervals with rational endpoints.Definition 3.4(intensional Lebesgue measure).We make this defini-tion in RCA 0.Let U = (a n ,b n ) n ∈N be an open set in the real line.The intensional Lebesgue measure of U is defined by µI (U )=lim k →∞µL k n =0(a n ,b n ) .Theorem 3.5.It is provable in RCA 0that intensional Lebesgue mea-sure µI is countably additive on open sets.In other words,if U n ,n ∈N ,is a sequence of open sets,then µI∞ n =0U n =lim k →∞µI k n =0U n .Proof.This is immediate from the definitions,since ∞n =0U n is defined as the union of the sequences of basic open intervals in U n ,n ∈N .Returning now to WWKL 0,we can prove that intensional Lebesgue measure concides with extensional Lebesgue measure.In fact,we have the following easy result.Theorem 3.6.The following assertions are pairwise equivalent over RCA 0.VITALI’S THEOREM AND WWKL91.WWKL.2.µI(U)=µL(U)for all open sets U⊆[0,1].3.µI is extensional on open sets.In other words,for all open setsU,V⊆[0,1],if∀x(x∈U↔x∈V)thenµI(U)=µI(V).4.For all open sets U⊇[0,1],we haveµI(U)≥1.Proof.This is immediate from Theorems3.3and3.5.4.More Measure Theory in WWKL0In this section we show that a good theory of measurable functions and measurable sets can be developed within WWKL0.Wefirst consider pointwise values of measurable functions.Our ap-proach is due to Yu[21,24].Let X be a compact separable metric space and letµ:C(X)→R be a positive Borel probability measure on X.Recall that L1(X,µ)is defined within RCA0as the completion of C(X)under the L1-norm.In what sense or to what extent can we prove that a point of the Banach space L1(X,µ)gives rise to a function f:X→R?In order to answer this question,recall that f∈L1(X,µ)is given by a sequence f n∈C(X),n∈N,which converges to f in the L1-norm; more preciselyf n−f n+1 1≤12nfor all n,and|f m(x)−f m (x)|≤12k.VITALI’S THEOREM AND WWKL10 Then for x∈C fnand m ≥m≥n+2k+2we have|f m(x)−f m (x)|≤m −1i=m|f i(x)−f i+1(x)|≤∞i=n+2k+2|f i(x)−f i+1(x)|≤12k.We need a lemma:Lemma4.2.The following is provable in RCA0.For f∈C(X)and >0,we haveµ({x|f(x)> })≤ f 1/ .Proof.Put U={x|f(x)> }.Note that U is an open set.If g∈C(X),0≤g≤1,g=0on X\U,then we have g≤|f|, hence µ(g)=µ( g)≤µ(|f|)= f 1,henceµ(g)≤ f 1/ .Thus µ(U)≤ f 1/ and the lemma is proved.Using this lemma we haveµ(X\C fnk )=µx∞i=n+2k+2|f i(x)−f i+1(x)|>12i=1VITALI’S THEOREM AND WWKL 11hence by countable additivityµ(X \C f n )≤∞ k =0µ(X \C f nk )≤∞k =012n .This completes the proof of Proposition 4.1.Remark 4.3(Yu [21]).In accordance with Proposition 4.1,forf = f n n ∈N ∈L 1(X,µ)and x ∈ ∞n =0C f n ,we define f (x )=lim n →∞f n (x ).Thus we see thatf (x )is defined on an F σset of measure 1.Moreover,if f =g in L 1(X,µ),i.e.if f −g 1=0,then f (x )=g (x )for all x in an F σset of measure 1.These facts are provable in WWKL 0.We now turn to a discussion of measurable sets within WWKL 0.We sketch two approaches to this topic.Our first approach is to identify measurable sets with their characteristic functions in L 1(X,µ),accord-ing to the following definition.Definition 4.4.This definition is made within WWKL 0.We say that f ∈L 1(X,µ)is a measurable characteristic function if there exists a sequence of closed setsC 0⊆C 1⊆···⊆C n ⊆...,n ∈N ,such that µ(X \C n )≤1/2n for all n ,and f (x )∈{0,1}for all x ∈ ∞n =0C n .Here f (x )is as defined in Remark 4.3.Our second approach is more direct,but in its present form it applies only to certain specific situations.For concreteness we consider only Lebesgue measure µL on the unit interval [0,1].Our discussion can easily be extended to Lebesgue measure on the n -cube [0,1]n ,the “fair coin”measure on the Cantor space 2N ,etc .Definition 4.5.The following definition is made within RCA 0.Let S be the Boolean algebra of finite unions of intervals in [0,1]with rational endpoints.For E 1,E 2∈S we define the distanced (E 1,E 2)=µL ((E 1\E 2)∪(E 2\E 1)),the Lebesgue measure of the symmetric difference of E 1and E 2.Thus d is a pseudometric on S ,and we define S to be the compact separable metric space which is the completion of S under d .A point E ∈ S is called a Lebesgue measurable set in [0,1].VITALI’S THEOREM AND WWKL 12We shall show that these two approaches to measurable sets (Defi-nitions 4.4and 4.5)are equivalent in WWKL 0.Begin by defining an isometry χ:S →L 1([0,1],µL )as follows.For 0≤a <b ≤1defineχ([a,b ])= f n n ∈N ∈L 1([0,1],µL )where f n (0)=f n (a )=f n (b )=f n (1)=0and f n a +b −a 2n +1=1and f n ∈C ([0,1])is piecewise linear otherwise.Thus χ([a,b ])is a measurable characteristic function corresponding to the interval [a,b ].For 0≤a 1<b 1<···<a k <b k ≤1defineχ([a 1,b 1]∪···∪[a k ,b k ])=χ([a 1,b 1])+···+χ([a k ,b k ]).It is straightforward to prove in RCA 0that χextends to an isometryχ: S→L 1([0,1],µL ).Proposition 4.6.The following is provable in WWKL 0.If E ∈ Sis a Lebesgue measurable set,then χ(E )is a measurable characteristic function in L 1([0,1],µL ).Conversely,given a measurable characteristic function f ∈L 1([0,1],µL ),we can find E ∈ Ssuch that χ(E )=f in L 1([0,1],µL ).Proof.It is straightforward to prove in RCA 0that for all E ∈ S , χ(E )is a measurable characteristic function.For the converse,let f be a measurable characteristic function.By Definition 4.4we have that f (x )∈{0,1}for all x ∈ ∞n =0C n .ByProposition 4.1we have |f (x )−f 3n +3(x )|<1/2n for all x ∈C f n .Put U n ={x ||f 3n +3(x )−1|<1/2n }and V n ={x ||f 3n +3(x )|<1/2n }.Then for n ≥1,U n and V n are disjoint open sets.Moreover C n ∩C f n ⊆U n ∪V n ,hence µL (U n ∪V n )≥1−1/2n −1.By countable additivity(Theorem 3.3)we can effectively find E n ,F n ∈S such that E n ⊆U n and F n ⊆V n and µL (E n ∪F n )≥1−1/2n −2.Put E = E n +5 n ∈N .It is straightforward to show that E belongs to S and that χ(E )=f in L 1([0,1],µL ).This completes the proof.Remark 4.7.We have presented two notions of Lebesgue measurable set and shown that they are equivalent in WWKL 0.Our first notion (Definition 4.4)has the advantage of generality in that it applies to any measure on a compact separable metric space.Our second no-tion (Definition 4.5)is advantageous in other ways,namely it is more straightforward and works well in RCA 0.It would be desirable to find a single definition which combines all of these advantages.VITALI’S THEOREM AND WWKL 135.Vitali’s TheoremLet S be a collection of sets.A point x is said to be Vitali covered by S if for all >0there exists S ∈S such that x ∈S and the diameter of S is less than .The Vitali Covering Theorem in its simplest form says the following:if I is a sequence of intervals which Vitali covers an interval E in the real line,then I contains a countable,pairwise disjoint set of intervals I n ,n ∈N ,such that ∞n =0I n covers E except for a set of Lebesgue measure 0.The purpose of this section is to show that various forms of the Vitali Covering Theorem are provable in WWKL 0and in fact equivalent to WWKL over RCA 0.Throughout this section,we use µto denote Lebesgue measure.Lemma 5.1(Baby Vitali Lemma).The following is provable in RCA 0.Let I 0,...,I n be a finite sequence of intervals.Then we can find a pair-wise disjoint subsequence I k 0,...,I k m such thatµ(I k 0∪···∪I k m )≥1VITALI’S THEOREM AND WWKL 14I =[2a −b,2b −a ].)Thusµ(I 0∪···∪I n )≤µ(I k 0∪···∪I k m )≤µ(I k 0)+···+µ(I k m )=3µ(I k 0)+···+3µ(I k m )=3µ(I k 0∪···∪I k m )and the lemma is proved.Lemma 5.2.The following is provable in WWKL 0.Let E be an in-terval,and let I n ,n ∈N ,be a sequence of intervals.If E ⊆ ∞n =0I n ,then µ(E )≤lim k →∞µ k n =0I n .Proof.If the intervals I n are open,then the desired conclusion follows immediately from countable additivity (Theorem 3.3).Otherwise,fix >0and let I n be an open interval with the same midpoint as I n andµ(I n )=µ(I n )+µ(E \A ).(3)VITALI’S THEOREM AND WWKL 15To prove the claim,use Lemma 5.2and the Vitali property to find a finite set of intervals J 1,...,J l ∈I such that J 1,...,J l ⊆E \A andµ(E \(A ∪J 1∪···∪J l ))<13µ(J 1∪···∪J l ).We then have µ(E \(A ∪I 1∪···∪I k ))<212µ(E \A )≤212µ(E \A )=34nµ(E ).Then by countable additivity we have µ E \∞ n =1A n =0and the lemma is proved.Remark 5.4.It is straightforward to generalize the previous lemma to the case of a Vitali covering of the n -cube [0,1]n by closed balls or n -dimensional cubes.In the case of closed balls,the constant 3in the Baby Vitali Lemma 5.1is replaced by 3n .Theorem 5.5.The Vitali theorem for the interval [0,1](as stated in Lemma 5.3)is equivalent to WWKL over RCA 0.Proof.Lemma 5.3shows that,in RCA 0,WWKL implies the Vitali theorem for intervals.It remains to prove within RCA 0that the Vitali theorem for [0,1]implies WWKL.Instead of proving WWKL,we shall prove the equivalent statement 3.3.3.Reasoning in RCA 0,suppose thatVITALI’S THEOREM AND WWKL 16(a n ,b n ),n ∈N ,is a sequence of open intervals which covers [0,1].Let I be the countable set of intervals (a nki ,b nki )= a n +i k(b n −a n ) where i,k,n ∈N and 0≤i <k .Then I is a Vitali covering of [0,1].By the Vitali theorem for intervals,I contains a sequence of pairwise disjoint intervals I m ,m ∈N ,such that µ ∞ m =0I m ≥1.By disjoint countable additivity (Corollary 2.5),we have∞m =0µ(I m )≥1.From this it follows easily that∞n =0|a n −b n |≥1.Thus we have 3.3.3and our theorem is proved.We now turn to Vitali’s theorem for measurable sets.Recall our discussion of measurable sets in section 4.A sequence of intervals I is said to almost Vitali cover a Lebesgue measurable set E ⊆[0,1]if for all >0we have µL (E \O )=0,where O = {I |I ∈I ,diam(I )< }.Theorem 5.6.The following is provable in WWKL 0.Let E ⊆[0,1]be a Lebesgue measurable set with µ(E )>0.Let I be a sequence of intervals which almost Vitali covers E .Then I contains a pairwise disjoint sequence of intervals I n ,n ∈N ,such that µ E \∞ n =0I n =0.Proof.The proof of this theorem is similar to that of Lemma 5.3.The only modification needed is in the proof of the claim.Recall from Definition 4.5that E =lim n →∞E n where each E n is a finite union of intervals in [0,1].Fix m so large thatµ((E \E m )∪(E m \E ))<1VITALI’S THEOREM AND WWKL 17andµ(E m \(A ∪J 1∪···∪J l ))<136µ(E \A )<236µ(E \A )≤236µ(E \A )<236µ(E \A )=3,The Baire category theorem in weak subsystems of second order arith-metic ,Journal of Symbolic Logic 58(1993),557–578.7.O.Demuth and A.Kuˇc era,Remarks on constructive mathematical analysis ,[3],1979,pp.81–129.8.H.-D.Ebbinghaus,G.H.M¨u ller,and G.E.Sacks (eds.),Recursion Theory Week ,Lecture Notes in Mathematics,no.1141,Springer-Verlag,1985,IX +418pages.VITALI’S THEOREM AND WWKL189.Harvey Friedman,unpublished communication to Leo Harrington,1977.10.Harvey Friedman,Stephen G.Simpson,and Rick L.Smith,Countable algebraand set existence axioms,Annals of Pure and Applied Logic25(1983),141–181.11.,Randomness and generalizations offixed point free functions,[1],1990, pp.245–254.15.,Subsystems of Second Order Arithmetic,Perspectives in Mathematical Logic,Springer-Verlag,1998,XIV+445pages.21.Xiaokang Yu,Measure Theory in Weak Subsystems of Second Order Arithmetic,Ph.D.thesis,Pennsylvania State University,1987,vii+73pages.22.,Riesz representation theorem,Borel measures,and subsystems of sec-ond order arithmetic,Annals of Pure and Applied Logic59(1993),65–78. 24.,A study of singular points and supports of measures in reverse mathe-matics,Annals of Pure and Applied Logic79(1996),211–219.26.Xiaokang Yu and Stephen G.Simpson,Measure theory and weak K¨o nig’slemma,Archive for Mathematical Logic30(1990),171–180.E-mail address:dkb5@,giusto@dm.unito.it,simpson@ The Pennsylvania State University。

抑郁症诊断治疗中超声技术的运用及发展-医学技术论文-基础医学论文-医学论文——文章均为WORD文档，下载后可直接编辑使用亦可打印——摘要：超声波作为一种机械振动波，兼具波动效应、力学效应和热效应，这3种效应在临床中均有较大应用价值，可用于疾病的成像诊断、辅助给药、调控以及热消融治疗等. 超声技术所具有的非侵入性、穿透力强、空间分辨率高等特性，使其在神经系统疾病的诊断和治疗中具有广泛的应用前景. 而抑郁症作为一种常见的精神疾病，其诊断和治疗都面临很大的困难. 因此，大量学者将超声技术应用于抑郁症诊疗. 本文主要从超声成像、超声定点给药、超声调控、超声导抑郁几个方面总结近十年来超声技术在抑郁症中的应用，以期为研究抑郁症发病机制及诊疗提供一定的参考和帮助.关键词：超声成像; 超声神经调控; 超声定点给药; 抑郁症;Abstract：As a kind of mechanical vibration wave, ultrasonic has ripple effect, mechanical effect and thermal effect，all of which have significant application value in clinical. Thus it can be used for diagnosisof disease, auxiliary dosage, regulation and thermal ablation etc. Ultrasonic techniques of non-invasive, strong penetrating power and high spatial resolution characteristics, also make it widely application in the diagnosis and treatment of disease of the nervous system. Depression, a common psychiatric disorder, is facing great challenges in diagnosis and treatment. A large number of researchers have applied ultrasonic technology in depression. This review mainly summarized the application of ultrasonic technology in depression in recent ten years from the aspects of ultrasonic imaging, ultrasonic fixed-point drug delivery, ultrasonic regulation and ultrasound-induced depression, hoping to provide certain reference and help for the study of the pathogenesis, diagnosis and treatment of depression.Keyword：ultrasound imaging; ultrasound neuromodulation; ultrasound site-specific administration; depression;人耳能听到的声波频率为20~20000 Hz，超声是频率高于人耳听阈上限的声波. 超声兼具波动效应、力学效应和热效应. 如图1所示，波动效应可用于成像-诊断，如B超、彩超、造影等，这部分超声波频率为1~5MHz. 医学上最早利用超声波是在1942年，奥地利医生杜西克首次用超声技术扫描脑部结构；到了60年代医生们开始将超声波应用于腹部器官的探测. 超声的力学效应可用于操控给药以及调控，同时，所用的声强以及激励时间要比成像所用的要高；而超声的热效应可用于病变组织的消融，所需声强及激励时间更高，比如高强度聚焦超声[1]，主要应用在前列腺癌、胰腺癌、肝癌、子宫肌瘤、部分骨肿瘤等疾病的治疗.鉴于这些特殊的效应，超声技术在神经系统疾病中的应用也越来越广泛.图1 超声的声学效应及生物医学应用[1]Fig.1 Acoustic effects of ultrasound and biomedicalapplications[1]抑郁症作为一种常见的精神疾病，其诊断和治疗都存在较大的困难. 首先，关于抑郁症的发病机制尚不明确，而临床症状又具有较大的个体差异性，目前临床诊断主要通过患者自身或者家属的主诉测评以及各种抑郁量表测评，同时抑郁症又包含多个亚型，缺乏较为客观的指标，容易与精神症阴性症状、躁狂症混淆而发生误诊，特别是一些缺乏临床经验的医师[2]. 其次，目前抑郁症的首要治疗方式仍然是药物治疗，但药物起效较慢，且有相当一部分抑郁症患者对于抗抑郁药物抵抗[3]. 而随着社会发展，人们生活压力增加，抑郁症的发病率越来越高. 世界卫生组织在2017年最新发布的报告中指出，全球抑郁症患者人数约3.22亿，患病率4.4%，我国抑郁症患病率约为4.2%[4]. 抑郁症是世界第四大疾病，预计到2020年将成为世界第二大疾病，但我国对抑郁症的医疗防治还处在识别率低的局面，地级市以上的医院对其识别率不足20%，只有不到10%的患者接受了相关的药物治疗. 而且，抑郁症的发病(和自杀)已开始出现低龄(大学，乃至中小学生群体)化趋势. 这是由于在青少年时期是大脑生长发育的关键时期，脑网络尚处在未成熟的阶段，脑网络之间的连接、整合正处在不断细化中，还缺乏较完善的调控机制[5]，个体对父母关系、外部生活等不能很好地应对，容易导致抑郁症的发生.超声技术因其非侵入性、强穿透力和高空间分辨率等优点，既能够对抑郁患者深部脑区成像进行辅助治疗又可以克服血脑屏障对抗抑郁药物的,其自身的调控作用还可以对抑郁达到一定的治疗效果. 因此，超声技术的运用有望在抑郁症的诊疗及评价中发挥重要作用. 本文主要总结了超声在抑郁症诊疗中的作用及应用进展，主要包括超声分子成像，超声定点给药,超声神经调控等. 以期为诊疗抑郁症、研究抑郁症发病机制以及研发抗抑郁药物提供一定的参考和帮助.1、超声成像与抑郁症经颅超声成像(transcranial sonography，TCS)已成为中枢神经系统退行性疾病有效、可靠的辅助诊断工具. 与磁共振成像(magnetic resonance imaging，MRI)或电子计算机断层扫描(computed tomography，CT)等其他神经成像方式相比，TCS具有伪影敏感性低、成本低和重复性好[6]、高抵抗性等优点，可以在便携式机器上进行. 在现代高分辨率成像技术的帮助下，TCS可以获得较好的脑深部结构成像分辨率，在临可与MRI相媲美[7]. 比如，目前高端临床TCS系统显示脑深部回声结构，图像分辨率高达0.7 mm1 mm，甚至高于临床条件下的MRI[8]. 更重要的是，TCS能够检测到其他成像模式下看不到的深部脑区的异常，比如中脑结构和基底神经节的病变. 除了对脑组织成像，超声技术在脑血管成像中也有较好的效果. 例如经颅多普勒超声成像(transcranial Doppler，TCD)检测颅内脑底动脉环上的各个主要动脉血流动力学及各血流生理参数，可床旁操作、方便、灵活、可重复操作. 其费用较磁共振血管成像MRA(magnetic resonance angiography)、数字减影血管造影(digital subtraction angiography，DSA)要低，更重要的是TCD的优势是对弯曲部分血管的成像效果最佳，能更好地判断该区域血管是否有狭窄. 近年来，一种新的超声神经成像技术，组织搏动性成像(tissue pulsatility imaging，TPI)，对检测脑部小体积搏动性的变化具有良好的敏感性，可用于识别中年抑郁症早期和微妙的脑血管功能变化. 利用TPI发现(1.82 MHz)，在中年抑郁症患者中脑组织搏动增加，提示抑郁症候群存在早期微小的血管损伤[9].1.1、利用经颅超声成像研究抑郁症发病相关脑区有研究显示，抑郁症的发生与脑部深度核团的功能失调有关，病变脑区大多集中在大脑中线位置，抑郁症患者脑区的结构、功能以及脑区之间的功能连接都存在异常[10]，这些脑区包含前额叶皮层(prefrontal cortex, PFC)、前扣带回(anterior cingulate cortex, ACC)、后扣带回(posterior cingutate cortex, PCC)，还有比较深层的纹状体(striatum)、杏仁核(amygdata)、海马(hippocampal formation，HF)和丘脑(thalamus)等脑区. 这些中线脑区横跨多个脑网络，这些网络在情绪调节、记忆、内部心理活动以及认知过程中注意资源的分配等方面具有重要的作用[10]. 黑质存在于大脑脚底和中脑被盖之间，见于中脑全长. 黑质细胞富含黑色素，是脑内合成多巴胺的主要核团，而多巴胺神经元的病变与抑郁症有关. 因此，目前关于超声成像诊断抑郁症针对的深部脑区是脑干中线和黑质，主要表现为脑干中线的低回声和黑质的高回声[11].早在1995年，Becker等(2.25 MHz)采用经颅彩色多普勒超声(transcranial color-coded real-time sonography，TCCS)对重度抑郁症、双相情感障碍和精神症患者的脑干中线（BR）回声性进行了评估，并与健康成年人进行了比较，结果发现仅在重度抑郁症患者中检测到BR回声显着降低，说明BR结构的异常与单相抑郁症的发生相关. 自此，这一解剖学区域成为研究抑郁症发病机制的焦点[12]. 而后，还有学者将BR回声作为主要指标对其他神经系统疾病伴发的抑郁进行了深入研究，例如帕金森伴发抑郁，氏舞蹈症（HD）伴抑郁等[13,14]，这些研究都发现抑郁后的BR回声显着降低(2.25 MHz)，并且在运动症状出现之前就检测到BR的回声减弱，这为伴发抑郁的早期诊断提供了客观有效的方法.除了BR回声减弱之外，黑质回声强度也存在异常增加的现象. 例如，在重度抑郁症患者中(2.5 MHz)黑质(SN)高回声频率有所增加[15]，这可能与SN、邻近腹侧被盖区、黑纹状体出现的多巴胺能系统的改变和功能障碍有关[16]. 在一项为期10年的随访研究中，研究人员利用SN高回声性、轻度不对称运动减慢和嗅觉减退联合预测PD 的后续发展，结果发现这种方法的敏感性为100%，特异性为98%，阳性预测率为75% [17].RN是位于BR附近狭窄区域内数个核团的总称，是脑内含有五羟色胺能神经元的主要部位，因此TCS在RN成像对于早期抑郁的诊断有一定的价值[18]. 鉴于之前的研究方法在一定程度上缺乏定量和客观的指标，Liu等采用血小板五羟色胺(5-HT)水平作为抑郁的客观指标，结果表明，在5-HT水平无差异的条件下，PD伴抑郁症患者的中缝核(RN)回声异常降低显着高于PD非抑郁症患者和健康对照组，且RN异常与抑郁程度无相关性.在一项应用三维定量SN的研究中(2.0~3.5 MHz)，对PD诊断的灵敏度和特异性分别高达91%和73%[19].但应用TCS脑深部成像对抑郁症进行诊断还需考虑许多问题. 脑深部结构如SN的TCS成像可靠性主要取决于两个因素：一是操作者的技术，包括临床经验、解剖熟悉程度、伪像的识别能力、技术熟练程度等，针对该问题可以通过应用优化技术自动检测和数字化图像分析来解决；二是颞听骨窗的质量[19,20]，在骨窗质量较差的情况下，可以通过降低超声的频率来进行改善.1.2、超声成像评估抑郁程度目前对于抑郁程度的分类标准还不完善，对抑郁症的诊断主要是通过汉密尔顿抑郁量表、贝克抑郁量表等量表的形式来进行，但这种量表的方法与抑郁症患者的主观意愿，诚信度以及病耻感都有很大的关系，因此对于抑郁程度的判定很难把握. 最近的一项研究利用TCS成像(2.5 MHz)，以红核为内标对BR进行1~4级半定量分级，1~3级可判定为异常，PD伴抑郁患者和单纯抑郁患者的BR异常率均显着升高，大部分轻度抑郁患者的TCS评分为3级，中度抑郁患者的TCS 评分为2~3级，重度抑郁患者的TCS评分为1级，不同的BR回声度反映了患者中线结构损伤程度的不同[21]，因此利用TCS成像可以用于评估患者的抑郁程度. 经颅多普勒超声是早期发现、评估和管理有痴呆风险的血管抑郁症患者的有效的工具[22].2、超声技术在抑郁症治疗中的应用2.1、聚焦超声开放血脑屏障辅助药物治疗抑郁症目前抑郁症最常用的方法仍然是药物治疗. 但血脑屏障(blood-brain barrier, BBB)的存在使到达目标靶区的抗抑郁药物浓度降低，降低了抗抑郁药物的疗效. BBB屏障是一种特化的非渗透性屏障，由紧密连接的内皮细胞、厚实的基底膜和星形胶质细胞组成. 内皮细胞之间的紧密连接以及多重耐药通路(multidrug resistance, MDR)中的酶、受体、转运蛋白、外排泵等限制了血管腔隙分子通过细胞旁路或转细胞运输途径进入大脑[23]. 虽然BBB的存在能够保护大脑不受细菌和其他有害物质的侵害，但也使得98%的小分子药物，甚至几乎100%的大分子药物都被排除在脑实质之外[24].近期动物研究表明，经颅聚焦超声(transcranial focused ultrasound, tFUS)可以持续短暂打开BBB(Ispta=0.2~11.5 W/cm2，频率=1.63 MHz)而不会造成神经组织的损伤[25],且BBB的打开是可逆的. 将超声探头的压电材料制成圆弧状，或利用电相位调制聚焦传输的超声能量，可以实现聚焦超声(FUS)，FUS可以在体内某一焦点内无创积累声能，对周围组织和近场的生物效应可以忽略不计[26]. 将tFUS(频率=1 MHz，Ispta=2.0 W/cm2)与微泡相结合(MB-FUS)，可以降低BBB 开放所需要的超声能量，进一步降低脑部热损伤发生的概率，与传统的脑部药物递送方法如高渗亲脂化学药物的输注相比，MB-FUS是一种完全无创的局部过程，可最大限度地减少非预期的靶外效应. 此外，这种可恢复的MB-FUS技术可以提供一个长达数小时的时间窗，这不仅有利于药物进入中枢神经系统，还可增强药物的渗透性和保留率[25]. Xie等[27]首先将这种技术应用于猪模型中(频率=1 MHz)，证实了无论是蛋白质包裹的全氟碳微泡还是脂包裹的全氟碳微泡，都可以显着提高BBB的渗透率. Liu等开展的另一项研究表明，使用更低频率的超声(28 kHz)可以在猪体内实现BBB的开放. Treat等[28]利用MB-FUS 技术(频率=1.5或1.7 MHz，0.06~3.0 W/cm2)成功将阿霉素递送至正常大鼠大脑. 陈芸等利用MRI引导下的低频聚焦超声联合载GDNF微泡靶向开放BBB，增加了中枢神经系统中胶质细胞源性神经营养因子GDNF的含量，且通过这种方法逆转大鼠的抑郁样行为，达到与脑内微注射GDNF相同的效果，避免了脑内微注射对脑组织的损伤，进一步增加了神经营养因子在治疗脑疾病方面的优势[29]，Fan等将这种方法用于灵长类动物中，利用MRI引导的聚焦超声系统(magnetic resonance guided focused ultrasound system，MRgFUS，220 kHz)在恒河猴身上进行了实验，将海马、外侧膝状体核、初级视觉皮层作为目标靶区. MRI显示在灰质结构中局部BBB被破坏，而在其他结构未见损伤(超声波持续时间150s，脉冲时间10ms，脉冲重复频率10 Hz，峰值负压在130~300 kPa). 动物恢复后，行为和视觉均未见异常,说明超声处理过程未造成功能损伤[26].2.2、超声热消融治疗抑郁症最近的一项研究表明，MRgFUS(频率=650 kHz)作为一种微创热损伤技术，将其应用于人类内囊前肢(ALIC)治疗重度抑郁症(右侧ALIC最高温度为53 ℃,左侧ALIC最高温度为54 ℃)，取得了一定疗效，且在治疗一年后仍有效果[30]. 目前这项技术应用于人类所面临的主要障碍是颅骨，由于人的颅骨的厚度和形状不规则，FUS在通过颅骨不同部位时会发生衰减和偏转[31]. 此外，FUS的高衰减会导致颅骨温度升高. Clement等[32]提出的半球形相控阵可以解决这些问题，该阵列的驱动频率为665 kHz，降低了颅骨对超声波能量的吸收，阵列由个元素组成，可以单独驱动这些元素来校正光束的像差，此外，还利用主动冷却系统将颅骨外表面和头皮的温度控制在安全范围内.综上，目前利用FUS技术治疗脑部疾病的研究已经有很多，包括脑瘤、PD、氏病、阿尔茨海默症等，都取得了一定的疗效，这为抑郁症的治疗提供了新的思路.2.3、低强度聚焦超声调控抑郁症的研究之前我们已经发文综述了低强度聚焦超声(LIFU)对中枢神经系统的调控作用[33]. LIFU不仅能够对大脑皮层脑区进行神经调控，还能非侵入性地刺激深部脑区，如海马、丘脑等，实现对大脑深部组织的功能调节，对于治疗神经系统疾病具有重要的应用价值. 不仅如此，LIFU的时间分辨率和空间分辨率都很高，比如，有研究表明利用LIFU 刺激小鼠的运动皮层，尾巴运动的潜伏期可小于50 ms，而LIFU的空间分辨率能够达到mm量级. 这种高效的分辨率有助于实现实时精确的神经调控. 而其神经机制在于LIFU可以通过机械振动激活(Isppa=3 W/cm2；频率=0.35 MHz)或抑制(Isppa=5 W/cm2；频率=0.35 MHz)神经元活动，从而改变行为学和电生理过程[34,35].在利用电生理技术来研究超声的作用机制时，存在一个令人头痛的问题，那就是商用超声换能器的体积与经典的电生理技术并不兼容，这就导致利用膜片钳在单细胞水平上研究超声的生物物理转导机制是比较困难的，针对这一问题，2018年，Lin等[36]发明了一种新型的超声调节芯片，利用该芯片来刺激海马切片，并用全细胞膜片钳记录研究了超声对锥体神经元离子通道水平的影响. 这种新型神经调节芯片的产生为研究超声的神经调节机制提供了一种简单而有力的工具.之前也有研究证明LIFU(Ispta=86 mW/cm2，频率=0.5 MHz; Ispta=400 mW/cm2，频率=500 kHz)在缺血性脑损伤[37,38]、癫痫(中心频率=30MHz)[39]和阿尔茨海默症(平均峰值压强=0.7 MPa, 频率=1 MHz)[40,41]中具有明显的治疗作用. 那么在抑郁症这一疾病中，也有研究者发现了超声的治疗作用. 我们知道，抑郁症的发病机制与海马区神经再生的减少和大脑内源性神经营养因子(brain derived neurotrophic factor, BDNF)的下降有关，而抗抑郁药物往往是使两者的发生上调[42,43]. 根据这一现象，就有研究者猜测，LIFU或许是通过增加BDNF含量来达到治疗抑郁症的目的[44,45]. 为了验证这一猜想，有研究者进行了一系列的动物研究，结果表明，LIFU(频率=0.25~0.50 MHz; Isppa=0.075~0.229 W/cm2; Ispta=21~163 mW/cm2)确实能够提高海马区BDNF的表达[46]，并显着促进了背侧海马齿状回区的神经增殖(频率=1.68MHz；平均峰值压强=0.960.3MPa)[47].在靶区的选择方面，有研究表明，前额叶皮层(prefrontal cortex, PFC)是LIFU最容易靶向的区域，而其他脑深部结构则被致密的白质束覆盖，这些白质束可以对LIFU吸收或散射[46]. 此外，越来越多的临床证据也表明了TMS用于抑郁患者左侧PFC的有效性和安全性[48]. 最近的一项研究将LIFU(频率=0.5 MHz；Isppa=7.59 W/cm2；Ispta=4.55 W/cm2)应用于大鼠的前边缘皮层，有研究表明，大鼠的前边缘皮层与人类大脑PFC同源[49]，研究LIFU对大鼠抑郁的治疗效果. 这项研究表明，LIFU能够改善抑郁模型大鼠的抑郁样行为，增加BDNF的表达量，且未对脑组织造成损伤,该研究是LIFU首次运用到大鼠抑郁模型中，为LIFU的抗抑郁作用提供了直接的证据[50].目前关于超声对抑郁症患者的调控作用研究比较少，但超声对于正常人体的神经调控研究较多. Fomenko等[51]通过电子数据库检索总结了有关人体超声神经调节的文献，结果发现，LIFU可以通过抑制皮层发电位，影响感觉器官改变感觉/运动结果,来影响人类的大脑活动.3、超声导抑郁症前面提到超声可以对抑郁症进行辅助诊断和治疗，但是有一些观点认为超声也能够导抑郁症的发生. 目前最常用的建立抑郁症模型的方法是使用物理应激源，如约束( )、足部电击休克、高温、剥夺食物和水、寒冷等[52,53,54,55]. 但这种方法的稳定性不高，模型复制困难，且长时间使用可能会使机体产生免疫，从而不再受物理应激的影响. Beckett首先观察到超声发抑郁症的现象，该研究应用22 kHz的超声频率和至少65 dB的超声强度，发现在该超声参数下能够引起鼠的警戒反应，并导鼠的逃逸和僵直反应. 在这项研究中，1 min 的超声波辐照改变了大鼠的运动行为，而安定则消除了这种行为. Oliviera等[56]的研究表明，在频率为22 kHz的超声波辐照下一小时,会影响中枢血清素能的传递以及大鼠的抑制性回避行为. Anna Morozova等对大鼠和小鼠施加不可预见的交替频率为20~25 kHz(与负性情绪有关)和25~45 kHz(与中性情绪有关)的超声波来建立抑郁模型[57]. 结果发现，使用上述参数，产生的抑郁行为学较稳定，便于观察. 在该研究中，对Wistar大鼠和Balb/c小鼠施加了3周的超声，结果表明，超声减少了大鼠和小鼠对蔗糖水的摄取量，游泳测试中漂浮行为(绝望行为)增加，社会互动能力和运动能力下降，对大鼠的包括海马在内的多个脑区的mRNA水平分析显示，五羟色胺转运体(serotonin transporter, SERT)、5-HT1A和5-HT2A受体表达增加，BDNF 的表达减少，血管内皮生长因子含量也下降，上述参数引起的行为学和生理变化，大部分可以通过服用氟西汀来缓解，这表明该频率范围内的超声确实有可能引发抑郁症.那么为什么超声波能够发抑郁症呢？尽管啮齿类动物在超声波范围内传递的特定物种信息的性质尚不完全清楚，但研究发现，小鼠和大鼠对于特定频率范围内的声音所表现出的情感敏感性在很大程度上是重叠的. 例如，在诸如社交失败、疼痛、母性分离等情况下，小鼠和大鼠都能发出20~25 kHz的声波[58,59,60,61]. 50 kHz及以上的声波是小鼠在积极的经历中产生的，被认为是积极情绪的表现，特别是在母狗与幼犬的互动、交配以及其他的积极社会交互活动中,动物会发出这个频率范围的声波[62,63, ,65,66].综上所述，既然超声可以导抑郁症的发生，那么利用超声建立抑郁症模型或许有助于提高临床抑郁模型的有效性，从而推进抑郁症的转化研究和抗抑郁药物的研发.4 、总结与展望通过以上文献调研，我们发现，不同声强和频率的超声波具有不同的效应，因而可应用的领域比较广泛,具有很大的应用价值.近50年来，超声检查作为一种影像学诊断方法以其用途广、价格低、携带方便和高效可靠的性能成为医学中不可缺少的检查手段，随着成像技术和多普勒技术的发展和改进以及超声对比剂的出现，超声的应用价值进一步提高，成为诊断抑郁症的有效的辅助诊断工具.一直以来，药物治疗抑郁症面临一大障碍BBB，BBB的存在使抗抑郁药物到达目标靶区的浓度降低，抗抑郁效果也不尽人意，而超声的出现打破了BBB对抗抑郁药物输送的障碍，极大地提高了其抗抑郁效果.近几年，研究学者们又发现超声刺激的神经调控作用，这种非侵入式的方法引起了研究学者们的广泛关注，通过研究发现低强度超声刺激可以提高BDNF水平，促进神经发生，而高强度聚焦超声的热消融效应可以消融抑郁患者的病变脑区，这些方法都对抑郁有一定的治疗效果.此外，超声还可以发抑郁，这对于我们建立更加有效的抑郁模型，对抑郁症的深入研究，抗抑郁药物的开发以及抗抑郁药物的效果评估都有重要意义.超声在神经系统疾病中的应用潜力是巨大的，尤其是利用低强度聚焦超声刺激进行神经调控的领域. 目前神经调控技术已有电刺激、磁刺激、光遗传等，但非侵入式电、磁的聚焦性差、空间分辨率不高等局限性限制了其进一步的应用，光遗传学是一种高精度的可操控单个神经元活性的高空间分辨率和细胞特异性的技术，但同时也需要进行病毒转录以及高精度手术，目前尚未批准应用于临床. 低强度聚焦超声刺激作为一种新型的脑刺激技术，具有无创、靶向性好、聚焦效果好、空间分辨率高的优势，可以定点将声能传送到我们想要的脑区，与MRI技术结合后更是相得益彰，针对神经系统疾病如抑郁症、慢性疼痛、帕金森病等具有广泛的应用价值. 但超声不同的强度、频率、调制范式的不同组合以及动物麻醉水平和超声换能器的固有属性都会对其所产生的神经调控效应有一定的影响，而目前关于超声参数对其神经调控效应的影响还未形成标准. 未来可注重定量研究不同的超声参数以及调制范式所产生的神经调控效应. 在超声神经调控机制方面，特别是活体动物的作用机制目前存在争议，但其机械效应的作用不可否认，未来在这一领域尚需要广大研究人员进行深入探究.。

漫谈扭结看到自然科学版有一个关于电磁场的讨论, 激发了我写点东西的冲动.一直以来, 对物理的兴趣都不比数学少. 所以偷偷摸摸看了一些物理的东西. 多半都是半懂不懂了, 但也有一点小体会.历史上数学和物理有几次神秘的相互作用, 第一次是牛顿力学体系的建立, 物理学的需要直接导致微积分这个强大的数学工具的诞生; 第二次是爱因斯坦的广义相对论, 让黎曼几何这个当时非主流的数学理论成为理论物理学家的必备知识; 第三次是冯.诺依曼为量子力学建立数学基础的尝试, 极大地推动了泛函分析的诞生和发展; 第四次就到了前几天热烈讨论的杨振宁, 他的规范场论就是数学上正在发展的纤维丛理论.从这几次联系看来, 物理和数学就象陈省身在他的微分几何讲义后记中所画的那个图一样, 是两条时分时合的曲线. 在广义相对论之后, 包括爱因斯坦在内的很多物理学家都尝试用数学来解释一切, 但是他们不仅失败了, 还遭到了新兴的量子力学的冷落. 量子力学使用了一些简单的数学工具, 但是牺牲了严格的数学推理. 这时候冯.诺依曼出来了, 他成功地用无穷维空间的算子理论阐述了量子力学, 并极大地发展了泛函分析这一数学分支. 然而量子力学受到自身的推动以及来自相对论的改造迅速地进化到量子场论这个至今无法从数学上理解的诡异理论. 数学再一次失去了在物理中的重要地位. 物理学家们按照他们自己的逻辑将量子场论发展得离数学很远很远, 他们普遍认为数学的能力已经到达极限. 广义相对论这个美妙的几何理论被称为经典理论. 经典这个词, 往往意味着过时. 几何已经被物理学家抛弃. 这时, 杨振宁-米尔斯找到了一个理论,可以用来解释强相互作用. 这个理论被叫做规范理论. 这个名字可能来源于电磁场的各种规范(库仑规范, 洛伦兹规范,...), 本意应该是让电磁场的矢量势和标量势固定的一个机制. 可能当时的物理学家愿意学习新数学的人很少, 过了好几年以后杨振宁才知道, 这个规范理论在数学里已经被研究过, 有一整套的概念和方法, 这就是纤维丛上的联络理论. 于是几何以为自己重新夺回了物理理论的解释权. 没想到这个规范场论仍然需要被量子场论改造以后才能用, 这样量子场论这个魔鬼又一次给了几何沉重一击, 因为改造后的量子规范场论成了一个更邪恶的魔鬼, 完全失去了几何意义. 数学和物理又一次分道扬镳. 这一分就是二十年.但是严重的分裂之后总是大统一, 中国历史的规律同样适用于数学物理.在80年代一代大牛牛的工作以后, 数学和理论物理终于有了现在的全面融合, 形成了一个数理共荣圈.80年代, 一方面理论物理有了大发展. 超弦理论兴起, 作为唯一的大统一理论候选者, 带动了数学很多分支的进步. 把拓扑, 代数几何, 数论的相关理论融为一炉. 这个方面我是门外汉, 基本一无所知, 希望有同学介绍一下. 另一方面数学里的低维拓扑方向有了突破, 使得扭结这个古老的对象被推至一个中心地位.我知道扭结这个东西, 是听了北大的王诗窚老师关于扭结的一个报告.当时真的是非常地孤陋寡闻, 觉得一切对我来说新奇的理论都是新理论.所以觉得扭结是个新方向, 嗯, 很好, 以后就搞它了. 到了美国才知道,我老板在我出生的时候就写了一篇关于扭结的论文, 而扭结方面的新进展在我小学没毕业的时候就发生了. 想起来就觉得悲哀, 中国的教育真的很毁人. 整个中学时代就是在浪费时间, 6年时间, 可以接受的知识绝对比我们实际上接受的多得多. 王老师的报告还是很精彩的, 虽然最后我不免还是睡着了, 因为听不懂DNA这些所谓的扭结理论的应用. 下面我就转贴一篇王老师的讲稿.王老师的这个讲稿应该是配合道具的, 所以看起来有些费劲. 而且他谈的是在生物, 化学中的应用, 这些并非是扭结的真正意义. 记得当时我出国, 好多亲友问我学的东西有什么用, 我总是说, 可以用来设计立体交通. 现在看来当然是扯淡.三维空间中的一个闭合的圈, 可能根本没打结, 但是仍然可以看上去很复杂, 比如把它揉成一团. 一个自然而根本的问题是, 如果不动手去解,单凭观察, 怎么能判断它到底有没有打结? 这个问题到现在还没有解决.这个问题在数学上就是不变量的问题. 我们想找一个量, 数量或者更广泛的量, 这个量在"解结" 这个过程中是不变的. "解结" 是个怎么样的过程呢? 就是一种变形, 而在变形过程中保持某种"连续性", 简单来说, 就是不能剪断绳圈. 圈在空间的形态在拓扑上叫做从圆到三维空间的一个"嵌入" (imbedding). 如果一个形态可以通过连续变形成为另一个形态, 我们就说这两个"嵌入"是"同痕的"(isotopic). 这个连续变形就叫一个"同痕"(isotopy). 直观上, 两个同痕的嵌入当然是同一个扭结, 因为跟打结有关的性质是不会在连续变形的过程中发生变化的.显然同痕是一个等价关系, 所有的嵌入在这个等价关系下可以分成等价类.每一个等价类对应一个扭结. 有了等价类, 自然就有不变量问题. 就是说,一个等价类里的不同元素有哪些共同的数字特征? 这些数字特征将有可能区分不同的等价类. 所以打没打结的问题就是: 找一个平凡扭结的完全不变量. 平凡扭结就是本质上没打结的圈, 完全不变量就是说, 所有没打结的圈的形态都有一个相同的数字特征, 而所有打结的圈的形态的这个数字特征将与没打结的那些不同. 熟悉线性代数的同学可能想到这个例子:线性变换. 一个线性变换可以有不同的矩阵表示, 这些矩阵都是相似的.所有矩阵在相似关系下分成等价类. 每一个等价类对应一个变换. 如果我们想知道一个矩阵是不是代表恒等变换, 我们可以看它所有的特征值以及所有循环子空间的维数. 如果都是1, 它就代表恒等变换, 如果有一个不是1, 它就不代表恒等变换. 所以数字集合{特征值, 循环子空间维数} 是一个完全不变量. 这个例子其实不太恰当, 因为恒等变换的矩阵等价类里只有一个元素, 就是单位矩阵, 所以不变量可以取作单位矩阵自己. 而在扭结的情况, 平凡扭结的形态有无穷多.至今, 扭结不变量有很多, 但完全的不变量, 一个都没有. 也就是说, 至今还没有找到一个不变量可以区分平凡扭结和非平凡扭结.80年代以前的几十年, Alexander Polynomials 一直是唯一的数值扭结不变量. 它的构造基于空间挖去扭结以后的拓扑结构. 到了1984年, Jones在研究冯.诺依曼代数的时候偶然发现了一个新的扭结不变量, 现在称为Jones Polynomials. 这个不变量的最初构造非常精巧, 涉及很多高深的代数知识. 但是经过几个大牛牛的研究, 这个不变量有了很多种解释. 看待它的方式多了, 对它就了解得更清楚了.这个Jones Polynomial理论被证实与其他分支有着广泛而微妙的联系. Jones自己走的路子是通过算子代数; 后来他自己同L.Kauffman,V.Turaev 发现了从统计力学模型出发的构造方法. 这个方法应该是最初等的, 最容易被接受的. 基本想法就是把扭结在每个重叠点处"解开"成为一些不相交的平凡投影(平面圆圈). 每个重叠点有两种解法,如果扭结的一个投影有三个重叠点, 这个投影就有8种解法. 每个解法叫做一个"态", 每个态联系一个单项式, 我们把所有态的单项式加起来,就得到一个多项式, 再用一个其他的数字(自绕数)修正一下, 就得到这个扭结的Jones Polynomial. 这种构造方法在统计力学里称为"配分函数" 或"状态和"; 同时V.Drinfeld在研究Hopf代数的时候发现了另一种构造方法, 跟Hopf代数的交换性质有关系, 叫做"R矩阵". 这种方法成为现在广泛使用的扭结不变量构造方法;这些方法有个共同的不足之处: 都依赖于扭结的二维投影. 算子代数和Hopf代数的构造都要先用一个"辫子"来表示扭结, 而统计力学的构造显然需要一个投影. 一些数学家不太满意这种情况, 因为早在一百多年前高斯就"内在"地构造了一个整数值的不变量, 用来研究两个扭结是怎么"链接" 起来的. 这个整数实际上是其中一个扭结对另一个扭结的"环绕数". 但是高斯用一个二重三维曲线积分算出了这个整数. 他的想法可能来自于当时的电磁学, 把两个扭结看成空间的两个环形电流, 然后计算它们的相互作用. 高斯这个"内在"的三维构造巧夺天工, 成为后来的数学家极欲模仿的典范. 所以在1988年一个纪念Hermann Weyl的讲座上, M.Atiyah提出了这个问题: 寻求Jones Polynomial的一个三维的内在构造. E.Witten立即投入到这个问题中, 在1989年发表了至今在拓扑学领域引用次数最高的"Qantum Field Theory and the Jones Polynomial", 给了Jones的理论一个基于量子场论的解释. 这种用量子场论观点研究拓扑学的方式叫做"拓扑量子场论"(Topological Qantum Field Theory). 几何与物理又一次走到了一起.Witten的理论是一个量子规范场论. 我正式学习规范场论是在这边的微分几何课上. 老师是日本人, 年纪轻轻, 在他的领域已经举足轻重.曾经问过他每天花多少时间来思考数学, 回答是每时每刻. 总觉得很多日本人有一股劲儿, 好像小平邦彦. 现在中国数学落后日本这么多,也无话可说, 人家就是勤奋. 当时在微分几何课程的广告上写的授课内容是: Gauge Theory; Hodge Theory; Morse Theory. 很酷. 在我们这样的学校, 有这么一门课真的是很不容易.所以把这三个理论放在一门课里讲, 因为Hodge理论的对象--Laplace方程, 如果未知函数是二次形式, 就是规范群为U(1)的杨振宁-米尔斯方程. 即, Maxwell方程组. 而Witten的论文"Supersymmetry and Morse Theory" 将微分拓扑中的Morse理论解释为一个超对称模型: 黎曼流形上的偶数次形式是玻色态, 奇数次形式是费米态, Q1=d+d* 和Q2=i(d-d*) 是两个超对称算子, 它们把费米态映到玻色态, 把玻色态映到费米态, 而且反交换.系统的哈密顿量H=Q1Q1+Q2Q2=dd*+d*d 就是流形上的Laplace算子(动能).所以寻找超对称的真空态的问题, 即求解Q1|0>=0, Q2|0>=0, 等价于求解黎曼流形上的Laplace方程. 如果引进相互作用(流形上的一个Morse函数), 那么这个超对称的量子力学模型在经典近似下给出Morse不等式.在经典的层面上, 规范理论是很"整齐"的理论. 比如经典电磁学就是U(1)主丛上的规范理论; 磁单极子是二维球面上一个非平凡U(1)丛的一个联络,杨振宁-米尔斯瞬子是四维球面上一个SO(3)主丛的一个联络; 等等非常漂亮的结论. 但是任何理论都要量子化, 规范理论也不例外. 与扭结相关的规范理论采用路径积分量子化. 路径积分最初由Dirac想到, 在他的"量子力学原理" 中提到过, 并注明说"不关心高等动力学的同志可以略去这一节", 可见是很费解的东西. 主要想法是在量子力学中重建最小作用量原理. 量子力学的最初形式都是哈密顿模式: 矩阵力学模仿正则方程, 波动力学模仿Hamilton- Jacobi方程, Dirac的变换理论又是模仿正则变换. 而用变分法从最小作用量原理导出Lagrange方程也是经典力学里很漂亮的办法, 而且将时间空间同等看待, 最容易与相对论结合. 后来Feynmann得到了一个理想的表达, 称为路径积分, 实际上是构造Schr?dinger方程的格林函数的方法. 经过搞数学的Kac严格化, 成为对一类抛物型微分方程构造格林函数的一般方法, 是概率论与随机过程应用在数学物理上的典范. 对热传导方程来说, 粒子的动能是通过混乱的布朗运动传递的, 传递的路线是不可预知的, 于是可以赋予每条可能的路线一个概率, 格林函数(传播子)就是这些路线效果的期望值. 但是Schr?din ger方程是一个很奇怪的方程, 形式上是抛物型, 所以可以用同样的办法构造传播子, 然而赋予每条路线的那个权重没有概率的解释, 因为在时间导数的前头有个虚数单位i, 这个i使得本该是概率的那个权重变成了一个模一的复数. 而传播过程不再是超距的, 而是有限速度的. 换言之, 它实质上描述波动.所以这个传播子是很难从数学上理解的东西, 无穷维空间测度论的解释只适合热传导的情况. 不知道有没有同学清楚这个传播子的数学解释, 希望可以讨论一下. 量子力学的情况已经这么复杂, 推广到场论上去的路径积分简直就是一个灾难.经典力学里粒子的基本力学变量是坐标和与之共轭的动量, 其他力学变量是它们的函数. 而粒子的"运动"是相空间的一条曲线. 所谓作用量是所有"运动"的空间上的泛函. 这里我用"函数"来代表复合关系, 只跟变量的取值有关; 泛函代表映射关系, 跟变量的形式(整个运动过程)有关.比如能量就是动量的一个函数, 每个时刻都有一个值, 这个值只与那个时刻的坐标,动量的值有关; 而作用量是Lagrange函数对时间的积分,只对时间段有意义, 与坐标, 动量随时间的变换有关, 与某时刻的值无关, 是"运动"的泛函. 现在运用场论的观点, 把"运动"看作一维时间上的一个"场", 就是说, 三个坐标和三个动量的值在时间上的分布. 那么能量就是场的函数, 而作用量是场的泛函. 记场为C: t--> R^6,定义泛函x_t, p_t 为x_t(C)=x(C(t)), p_t(C)=p(C(t)). 如果有经典力学变量f(x,p), 那么量子化以后, 这个力学变量在t时刻的期望将是:E[f(x_t,p_t)], 这里的测度空间是{所有可能的场C}, 概率密度是:pdf(C)= exp{iS(C)}, S(C)=\int L(x_t(C),p_t(C))dt 是作用量.写开那个期望就是\int f(x_t(C),p_t(C))exp{iS(C)}dC.相对论的情形基本上是上面的推广, 有一点点区别. 基本力学变量是在时空分布的场, 作用量是场的泛函, 其他力学变量, 与单粒子的情况不同, 一般是场的泛函而不是场的函数, 这是因为在一个时空点的场的值不能提供关于能量等我们关心的力学变量的信息, 而是要计及整个场的分布. 如果用A:R^4 --> V 来表示时空中取值在V中的场, 那么量子化后一个力学变量f(A) 的期望是\int f(A)exp{iS(A)}dA, 积分的空间是{所有可能的场A}.回到扭结问题. 现在来看三维流形上的规范场, 就是三维流形上某个主丛的联络. {丛上所有联络} 就是我们量子化的时候要在上面积分的空间. (这个空间上到底有没有一个测度使得积分有意义还是一个根本的未解决问题, 所以在这里我们已经失去了数学上的严格) 我们需要某个力学变量的期望值, 这个力学变量就是扭结与联络的一个"配对", 计为<K,A>, 从数学上来说就是联络A沿扭结K的"和乐"(holonomy)的迹(trace), 取数量值. 所以这个由固定的扭结决定的力学变量是{丛上所有联络}这个空间上的泛函. 这个泛函(力学变量)的期望值就是扭结K的一个拓扑不变量: Z(K)=\int <K,A> exp{i*CS(A)} dA.这里的作用量是一个特殊的作用量: Chern-Simons invariant (Chern-Sim onsnumber) CS(A). 这是一个共形不变量, 也是一个局部规范不变量, 这个不变量也是90年代低维几何拓扑的中心议题之一.这个不变量的定义完全是形式的, 其中含有很可能没有意义的路径积分. 从这个形式的定义中解读不变量的信息有两个办法: 一个是Witten的办法, 观察和玩弄这个形式的表达式, 把流形分割成几个与黎曼曲面同伦的部分,再结合一些正则量子化方法和moduli space的理论, 证明这个不变量的一些性质. 这篇论文是拓扑量子场论的经典之作, 体现了Witten这个牛牛深不见底的学识和海阔天空的想象力. 估计够我学十好几年的. 另一路也是几个牛牛在搞, 顺便说一句, 这些牛牛多半都是犹太的. Dror Bar-Natan的博士论文就是关于这个不变量的, 名叫"Perturbative Aspects of the Chern-Simons Topol ogicalQuantum Field Theory", 用微扰展开, Feynmann图等技巧避开了形式定义不严格的问题, 证明了很多结果, 并通过Feynmann图与另一族重要的扭结不变量----Vassiliev不变量联系了起来. 这一联系可不得了, 几个牛牛过来一插手, 把这个理论整得有如天书一般, 完全看不懂了. 其中包括Kontsevich, W.Thurs ton的儿子D.Thurston, 还有什么Rozansky, 以及Witten自己. 这些人里, Witten和Kontsevich是泰山北斗, 个人认为比牛顿牛多了; Dror Bar-Nata n的一篇关于Vassiliev不变量的论文引用次数排名居高不下, 可与Witten的那个相媲美, 博士论文又那么牛逼; D.Thurston本科的论文我就看不懂, 博士论文更是具有独创性, 概念符号都是自己发明的, 开创了一个新的课题. 虽然我还没来得及参详, 我一个同学已经跟我吹了好多次了, 搞得我现在也对这个Thurston崇拜得不得了. 他现在也是Fields奖热门人选.。

达尔克罗兹方法体态律动英文书名The Dalcroze Method, known for its unique approach to music education, emphasizes the integration of body, ear, and mind through movement and rhythm.This pedagogical technique, developed by Emile Jaques-Dalcroze, allows students to experience music in akinesthetic way, fostering a deeper understanding of musical concepts such as tempo, dynamics, and phrasing.In classrooms where the Dalcroze Method is applied,you'll find students engaging in eurhythmics, a form of movement that translates the flow and structure of music into physical gestures.The method's effectiveness transcends the boundaries of music education, as it also enhances cognitive and motor skills, making it a holistic learning experience for all ages.By incorporating the Dalcroze Method into their teaching, educators can unlock a new dimension of musical appreciation and expression, enriching the learning journey for their students.This approach to music education is not just aboutlearning to play an instrument or sing; it's about feelingthe music and moving with it, creating a profound connection between the art and the individual.The Dalcroze Method's influence can be seen in the way it has inspired other disciplines to adopt a more embodied approach to learning, recognizing the power of movement in enhancing cognitive and creative processes.For young learners, the Dalcroze Method can be a playful introduction to the world of music, where they can explore the joy of rhythm and melody through dance and movement.As students progress, the method offers a sophisticated toolkit for musicians, helping them to develop a nuanced and expressive performance style that is grounded in a deep understanding of music's physicality.In essence, the Dalcroze Method is a testament to the belief that music is not just an auditory experience but a multisensory journey that can be fully embraced through the body's natural language of movement.。