Some cubic modular identities of Ramanujan

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Multi-scale structural similarity for image quality assesment

MULTI-SCALE STRUCTURAL SIMILARITY FOR IMAGE QUALITY ASSESSMENT Zhou Wang1,Eero P.Simoncelli1and Alan C.Bovik2(Invited Paper)1Center for Neural Sci.and Courant Inst.of Math.Sci.,New York Univ.,New York,NY10003 2Dept.of Electrical and Computer Engineering,Univ.of Texas at Austin,Austin,TX78712 Email:zhouwang@,eero.simoncelli@,bovik@ABSTRACTThe structural similarity image quality paradigm is based on the assumption that the human visual system is highly adapted for extracting structural information from the scene,and therefore a measure of structural similarity can provide a good approxima-tion to perceived image quality.This paper proposes a multi-scale structural similarity method,which supplies moreﬂexibility than previous single-scale methods in incorporating the variations of viewing conditions.We develop an image synthesis method to calibrate the parameters that deﬁne the relative importance of dif-ferent scales.Experimental comparisons demonstrate the effec-tiveness of the proposed method.1.INTRODUCTIONObjective image quality assessment research aims to design qual-ity measures that can automatically predict perceived image qual-ity.These quality measures play important roles in a broad range of applications such as image acquisition,compression,commu-nication,restoration,enhancement,analysis,display,printing and watermarking.The most widely used full-reference image quality and distortion assessment algorithms are peak signal-to-noise ra-tio(PSNR)and mean squared error(MSE),which do not correlate well with perceived quality(e.g.,[1]–[6]).Traditional perceptual image quality assessment methods are based on a bottom-up approach which attempts to simulate the functionality of the relevant early human visual system(HVS) components.These methods usually involve1)a preprocessing process that may include image alignment,point-wise nonlinear transform,low-passﬁltering that simulates eye optics,and color space transformation,2)a channel decomposition process that trans-forms the image signals into different spatial frequency as well as orientation selective subbands,3)an error normalization process that weights the error signal in each subband by incorporating the variation of visual sensitivity in different subbands,and the vari-ation of visual error sensitivity caused by intra-or inter-channel neighboring transform coefﬁcients,and4)an error pooling pro-cess that combines the error signals in different subbands into a single quality/distortion value.While these bottom-up approaches can conveniently make use of many known psychophysical fea-tures of the HVS,it is important to recognize their limitations.In particular,the HVS is a complex and highly non-linear system and the complexity of natural images is also very signiﬁcant,but most models of early vision are based on linear or quasi-linear oper-ators that have been characterized using restricted and simplistic stimuli.Thus,these approaches must rely on a number of strong assumptions and generalizations[4],[5].Furthermore,as the num-ber of HVS features has increased,the resulting quality assessment systems have become too complicated to work with in real-world applications,especially for algorithm optimization purposes.Structural similarity provides an alternative and complemen-tary approach to the problem of image quality assessment[3]–[6].It is based on a top-down assumption that the HVS is highly adapted for extracting structural information from the scene,and therefore a measure of structural similarity should be a good ap-proximation of perceived image quality.It has been shown that a simple implementation of this methodology,namely the struc-tural similarity(SSIM)index[5],can outperform state-of-the-art perceptual image quality metrics.However,the SSIM index al-gorithm introduced in[5]is a single-scale approach.We consider this a drawback of the method because the right scale depends on viewing conditions(e.g.,display resolution and viewing distance). In this paper,we propose a multi-scale structural similarity method and introduce a novel image synthesis-based approach to calibrate the parameters that weight the relative importance between differ-ent scales.2.SINGLE-SCALE STRUCTURAL SIMILARITYLet x={x i|i=1,2,···,N}and y={y i|i=1,2,···,N}be two discrete non-negative signals that have been aligned with each other(e.g.,two image patches extracted from the same spatial lo-cation from two images being compared,respectively),and letµx,σ2x andσxy be the mean of x,the variance of x,and the covariance of x and y,respectively.Approximately,µx andσx can be viewed as estimates of the luminance and contrast of x,andσxy measures the the tendency of x and y to vary together,thus an indication of structural similarity.In[5],the luminance,contrast and structure comparison measures were given as follows:l(x,y)=2µxµy+C1µ2x+µ2y+C1,(1)c(x,y)=2σxσy+C2σ2x+σ2y+C2,(2)s(x,y)=σxy+C3σxσy+C3,(3) where C1,C2and C3are small constants given byC1=(K1L)2,C2=(K2L)2and C3=C2/2,(4)Fig.1.Multi-scale structural similarity measurement system.L:low-passﬁltering;2↓:downsampling by2. respectively.L is the dynamic range of the pixel values(L=255for8bits/pixel gray scale images),and K1 1and K2 1aretwo scalar constants.The general form of the Structural SIMilarity(SSIM)index between signal x and y is deﬁned as:SSIM(x,y)=[l(x,y)]α·[c(x,y)]β·[s(x,y)]γ,(5)whereα,βandγare parameters to deﬁne the relative importanceof the three components.Speciﬁcally,we setα=β=γ=1,andthe resulting SSIM index is given bySSIM(x,y)=(2µxµy+C1)(2σxy+C2)(µ2x+µ2y+C1)(σ2x+σ2y+C2),(6)which satisﬁes the following conditions:1.symmetry:SSIM(x,y)=SSIM(y,x);2.boundedness:SSIM(x,y)≤1;3.unique maximum:SSIM(x,y)=1if and only if x=y.The universal image quality index proposed in[3]corresponds to the case of C1=C2=0,therefore is a special case of(6).The drawback of such a parameter setting is that when the denominator of Eq.(6)is close to0,the resulting measurement becomes unsta-ble.This problem has been solved successfully in[5]by adding the two small constants C1and C2(calculated by setting K1=0.01 and K2=0.03,respectively,in Eq.(4)).We apply the SSIM indexing algorithm for image quality as-sessment using a sliding window approach.The window moves pixel-by-pixel across the whole image space.At each step,the SSIM index is calculated within the local window.If one of the image being compared is considered to have perfect quality,then the resulting SSIM index map can be viewed as the quality map of the other(distorted)image.Instead of using an8×8square window as in[3],a smooth windowing approach is used for local statistics to avoid“blocking artifacts”in the quality map[5].Fi-nally,a mean SSIM index of the quality map is used to evaluate the overall image quality.3.MULTI-SCALE STRUCTURAL SIMILARITY3.1.Multi-scale SSIM indexThe perceivability of image details depends the sampling density of the image signal,the distance from the image plane to the ob-server,and the perceptual capability of the observer’s visual sys-tem.In practice,the subjective evaluation of a given image varies when these factors vary.A single-scale method as described in the previous section may be appropriate only for speciﬁc settings.Multi-scale method is a convenient way to incorporate image de-tails at different resolutions.We propose a multi-scale SSIM method for image quality as-sessment whose system diagram is illustrated in Fig. 1.Taking the reference and distorted image signals as the input,the system iteratively applies a low-passﬁlter and downsamples theﬁltered image by a factor of2.We index the original image as Scale1, and the highest scale as Scale M,which is obtained after M−1 iterations.At the j-th scale,the contrast comparison(2)and the structure comparison(3)are calculated and denoted as c j(x,y) and s j(x,y),respectively.The luminance comparison(1)is com-puted only at Scale M and is denoted as l M(x,y).The overall SSIM evaluation is obtained by combining the measurement at dif-ferent scales usingSSIM(x,y)=[l M(x,y)]αM·Mj=1[c j(x,y)]βj[s j(x,y)]γj.(7)Similar to(5),the exponentsαM,βj andγj are used to ad-just the relative importance of different components.This multi-scale SSIM index deﬁnition satisﬁes the three conditions given in the last section.It also includes the single-scale method as a spe-cial case.In particular,a single-scale implementation for Scale M applies the iterativeﬁltering and downsampling procedure up to Scale M and only the exponentsαM,βM andγM are given non-zero values.To simplify parameter selection,we letαj=βj=γj forall j’s.In addition,we normalize the cross-scale settings such thatMj=1γj=1.This makes different parameter settings(including all single-scale and multi-scale settings)comparable.The remain-ing job is to determine the relative values across different scales. Conceptually,this should be related to the contrast sensitivity func-tion(CSF)of the HVS[7],which states that the human visual sen-sitivity peaks at middle frequencies(around4cycles per degree of visual angle)and decreases along both high-and low-frequency directions.However,CSF cannot be directly used to derive the parameters in our system because it is typically measured at the visibility threshold level using simpliﬁed stimuli(sinusoids),but our purpose is to compare the quality of complex structured im-ages at visible distortion levels.3.2.Cross-scale calibrationWe use an image synthesis approach to calibrate the relative impor-tance of different scales.In previous work,the idea of synthesizing images for subjective testing has been employed by the“synthesis-by-analysis”methods of assessing statistical texture models,inwhich the model is used to generate a texture with statistics match-ing an original texture,and a human subject then judges the sim-ilarity of the two textures [8]–[11].A similar approach has also been qualitatively used in demonstrating quality metrics in [5],[12],though quantitative subjective tests were not conducted.These synthesis methods provide a powerful and efﬁcient means of test-ing a model,and have the added beneﬁt that the resulting images suggest improvements that might be made to the model[11].M )distortion level (MSE)12345Fig.2.Demonstration of image synthesis approach for cross-scale calibration.Images in the same row have the same MSE.Images in the same column have distortions only in one speciﬁc scale.Each subject was asked to select a set of images (one from each scale),having equal quality.As an example,one subject chose the marked images.For a given original 8bits/pixel gray scale test image,we syn-thesize a table of distorted images (as exempliﬁed by Fig.2),where each entry in the table is an image that is associated witha speciﬁc distortion level (deﬁned by MSE)and a speciﬁc scale.Each of the distorted image is created using an iterative procedure,where the initial image is generated by randomly adding white Gaussian noise to the original image and the iterative process em-ploys a constrained gradient descent algorithm to search for the worst images in terms of SSIM measure while constraining MSE to be ﬁxed and restricting the distortions to occur only in the spec-iﬁed scale.We use 5scales and 12distortion levels (range from 23to 214)in our experiment,resulting in a total of 60images,as demonstrated in Fig.2.Although the images at each row has the same MSE with respect to the original image,their visual quality is signiﬁcantly different.Thus the distortions at different scales are of very different importance in terms of perceived image quality.We employ 10original 64×64images with different types of con-tent (human faces,natural scenes,plants,man-made objects,etc.)in our experiment to create 10sets of distorted images (a total of 600distorted images).We gathered data for 8subjects,including one of the authors.The other subjects have general knowledge of human vision but did not know the detailed purpose of the study.Each subject was shown the 10sets of test images,one set at a time.The viewing dis-tance was ﬁxed to 32pixels per degree of visual angle.The subject was asked to compare the quality of the images across scales and detect one image from each of the ﬁve scales (shown as columns in Fig.2)that the subject believes having the same quality.For example,one subject chose the images marked in Fig.2to have equal quality.The positions of the selected images in each scale were recorded and averaged over all test images and all subjects.In general,the subjects agreed with each other on each image more than they agreed with themselves across different images.These test results were normalized (sum to one)and used to calculate the exponents in Eq.(7).The resulting parameters we obtained are β1=γ1=0.0448,β2=γ2=0.2856,β3=γ3=0.3001,β4=γ4=0.2363,and α5=β5=γ5=0.1333,respectively.4.TEST RESULTSWe test a number of image quality assessment algorithms using the LIVE database (available at [13]),which includes 344JPEG and JPEG2000compressed images (typically 768×512or similar size).The bit rate ranges from 0.028to 3.150bits/pixel,which allows the test images to cover a wide quality range,from in-distinguishable from the original image to highly distorted.The mean opinion score (MOS)of each image is obtained by averag-ing 13∼25subjective scores given by a group of human observers.Eight image quality assessment models are being compared,in-cluding PSNR,the Sarnoff model (JNDmetrix 8.0[14]),single-scale SSIM index with M equals 1to 5,and the proposed multi-scale SSIM index approach.The scatter plots of MOS versus model predictions are shown in Fig.3,where each point represents one test image,with its vertical and horizontal axes representing its MOS and the given objective quality score,respectively.To provide quantitative per-formance evaluation,we use the logistic function adopted in the video quality experts group (VQEG)Phase I FR-TV test [15]to provide a non-linear mapping between the objective and subjective scores.After the non-linear mapping,the linear correlation coef-ﬁcient (CC),the mean absolute error (MAE),and the root mean squared error (RMS)between the subjective and objective scores are calculated as measures of prediction accuracy .The prediction consistency is quantiﬁed using the outlier ratio (OR),which is de-Table1.Performance comparison of image quality assessment models on LIVE JPEG/JPEG2000database[13].SS-SSIM: single-scale SSIM;MS-SSIM:multi-scale SSIM;CC:non-linear regression correlation coefﬁcient;ROCC:Spearman rank-order correlation coefﬁcient;MAE:mean absolute error;RMS:root mean squared error;OR:outlier ratio.Model CC ROCC MAE RMS OR(%)PSNR0.9050.901 6.538.4515.7Sarnoff0.9560.947 4.66 5.81 3.20 SS-SSIM(M=1)0.9490.945 4.96 6.25 6.98 SS-SSIM(M=2)0.9630.959 4.21 5.38 2.62 SS-SSIM(M=3)0.9580.956 4.53 5.67 2.91 SS-SSIM(M=4)0.9480.946 4.99 6.31 5.81 SS-SSIM(M=5)0.9380.936 5.55 6.887.85 MS-SSIM0.9690.966 3.86 4.91 1.16ﬁned as the percentage of the number of predictions outside the range of±2times of the standard deviations.Finally,the predic-tion monotonicity is measured using the Spearman rank-order cor-relation coefﬁcient(ROCC).Readers can refer to[15]for a more detailed descriptions of these measures.The evaluation results for all the models being compared are given in Table1.From both the scatter plots and the quantitative evaluation re-sults,we see that the performance of single-scale SSIM model varies with scales and the best performance is given by the case of M=2.It can also be observed that the single-scale model tends to supply higher scores with the increase of scales.This is not surprising because image coding techniques such as JPEG and JPEG2000usually compressﬁne-scale details to a much higher degree than coarse-scale structures,and thus the distorted image “looks”more similar to the original image if evaluated at larger scales.Finally,for every one of the objective evaluation criteria, multi-scale SSIM model outperforms all the other models,includ-ing the best single-scale SSIM model,suggesting a meaningful balance between scales.5.DISCUSSIONSWe propose a multi-scale structural similarity approach for image quality assessment,which provides moreﬂexibility than single-scale approach in incorporating the variations of image resolution and viewing conditions.Experiments show that with an appropri-ate parameter settings,the multi-scale method outperforms the best single-scale SSIM model as well as state-of-the-art image quality metrics.In the development of top-down image quality models(such as structural similarity based algorithms),one of the most challeng-ing problems is to calibrate the model parameters,which are rather “abstract”and cannot be directly derived from simple-stimulus subjective experiments as in the bottom-up models.In this pa-per,we used an image synthesis approach to calibrate the param-eters that deﬁne the relative importance between scales.The im-provement from single-scale to multi-scale methods observed in our tests suggests the usefulness of this novel approach.However, this approach is still rather crude.We are working on developing it into a more systematic approach that can potentially be employed in a much broader range of applications.6.REFERENCES[1] A.M.Eskicioglu and P.S.Fisher,“Image quality mea-sures and their performance,”IEEE munications, vol.43,pp.2959–2965,Dec.1995.[2]T.N.Pappas and R.J.Safranek,“Perceptual criteria for im-age quality evaluation,”in Handbook of Image and Video Proc.(A.Bovik,ed.),Academic Press,2000.[3]Z.Wang and A.C.Bovik,“A universal image quality in-dex,”IEEE Signal Processing Letters,vol.9,pp.81–84,Mar.2002.[4]Z.Wang,H.R.Sheikh,and A.C.Bovik,“Objective videoquality assessment,”in The Handbook of Video Databases: Design and Applications(B.Furht and O.Marques,eds.), pp.1041–1078,CRC Press,Sept.2003.[5]Z.Wang,A.C.Bovik,H.R.Sheikh,and E.P.Simon-celli,“Image quality assessment:From error measurement to structural similarity,”IEEE Trans.Image Processing,vol.13, Jan.2004.[6]Z.Wang,L.Lu,and A.C.Bovik,“Video quality assessmentbased on structural distortion measurement,”Signal Process-ing:Image Communication,special issue on objective video quality metrics,vol.19,Jan.2004.[7] B.A.Wandell,Foundations of Vision.Sinauer Associates,Inc.,1995.[8]O.D.Faugeras and W.K.Pratt,“Decorrelation methods oftexture feature extraction,”IEEE Pat.Anal.Mach.Intell., vol.2,no.4,pp.323–332,1980.[9] A.Gagalowicz,“A new method for textureﬁelds synthesis:Some applications to the study of human vision,”IEEE Pat.Anal.Mach.Intell.,vol.3,no.5,pp.520–533,1981. [10] D.Heeger and J.Bergen,“Pyramid-based texture analy-sis/synthesis,”in Proc.ACM SIGGRAPH,pp.229–238,As-sociation for Computing Machinery,August1995.[11]J.Portilla and E.P.Simoncelli,“A parametric texture modelbased on joint statistics of complex wavelet coefﬁcients,”Int’l J Computer Vision,vol.40,pp.49–71,Dec2000. [12]P.C.Teo and D.J.Heeger,“Perceptual image distortion,”inProc.SPIE,vol.2179,pp.127–141,1994.[13]H.R.Sheikh,Z.Wang, A. C.Bovik,and L.K.Cormack,“Image and video quality assessment re-search at LIVE,”/ research/quality/.[14]Sarnoff Corporation,“JNDmetrix Technology,”http:///products_services/video_vision/jndmetrix/.[15]VQEG,“Final report from the video quality experts groupon the validation of objective models of video quality assess-ment,”Mar.2000./.PSNRM O SSarnoffM O S(a)(b)Single−scale SSIM (M=1)M O SSingle−scale SSIM (M=2)M O S(c)(d)Single−scale SSIM (M=3)M O SSingle−scale SSIM (M=4)M O S(e)(f)Single−scale SSIM (M=5)M O SMulti−scale SSIMM O S(g)(h)Fig.3.Scatter plots of MOS versus model predictions.Each sample point represents one test image in the LIVE JPEG/JPEG2000image database [13].(a)PSNR;(b)Sarnoff model;(c)-(g)single-scale SSIM method for M =1,2,3,4and 5,respectively;(h)multi-scale SSIM method.。

3rd-Week

第三周课堂练习Practice 1: Insert the article a or an where necessary1.Helium is gas which consists of one proton and one neutron.bor union is organization of workers formed to improve their economic status and working conditions.3.White dwarf is star that is unusually faint given its extreme temperature.4.Rice is cereal grain that usually requires subtropical climate and abundance of moisture for growth.5.Transduction is technique is which genes are inserted into host cell by means of viral infection.6.Heat is form of energy which can be transmitted through solid and liquid media by conduction.Practice 2: Edit the following by reducing the relative clauses where possible1.Aluminum is a lightweight metal that is often used for high-tension power transmission.2.Heat is a form of energy which can be transmitted through solid and liquid media by conduction.3. A brake is a device that is capable of slowing the motion of a mechanism.4. A dome is generally a hemispherical roof which is on top of a circular, square, or other-shaped space.5.Snow is a form of precipitation which results from the sublimation of water vapor into solid crystals attemperatures below 0 degrees centigrade.6．An antigen is a substance which causesthe formation of antibodies, the body’snatural response to foreign substances.7. A piccolo is a small flute that is pitched an octave higher than a standard fulte.8. An oocyte is a cell which undergoes meiosis to produce an ovum or egg.9. A catalyst is a substance that can speed up the rate of a chemical reaction without changing its own structure.10. A black hole is a celestial body which has approximately the same mass as the sun and a gravitational radius of about 3 km.Practice 3: Complete the following definitions by inserting an appropriate preposition1. A thermometer is an instrument ___ which temperature can be measured.2.Photosynthesis is process ___ which sunlight is used to manufacture carbohydrates from water and carbondioxide.3. A credit bureau is an organization ___ which businesses can apply for financial information on potentialcustomers.4.An anhydride is a compound __ which the elements of water have been removed.5.An eclipse is a celestial event ___ which one body, such as star, is covered by another, such as a planet.6.An axis is an imaginary line __ which a body is said to rotate.Practice 4: Put the sentences into a correct order.a.The term itself comes from the ancient Greek word palindromes meaning “running back again.”b.Another good and more rece nt example is “If I had a HiFi.”c.Some very common English words are palindromes, such as pop, dad, and noon.d. A palindrome is a word or phrase that results in the same sequence of letters no matter whether it is read from leftto right or from right to left.e.One of the classic long palindromes is “A man, a plan, a canal, Panama.”f.Long palindromes are very hard to construct, and some word puzzles spend immense amounts of time trying toproduce good example.Practice 5: Below you will find three pairs of sentences, each consisting of a definition and a generalization. When would it be better to begin a text with the first sentence in each pair rather than the second?A: 1. Russian is the first language of about 150 million inhabitants of the former Soviet Union.2. Russian is a language belonging to the West Slavic subgroup of the Indo-European language family.B: 1. A catalyst is a substance which increases the rate of a chemical reaction.2. Catalyst technology has progressed quickly as researchers better understand the complex interactions of molecules.Practice 6: Label the four parts of this problem-solution text.International students often study English for many years before going to an English speaking country to pursue a graduate degree. Their study of English usually focuses on grammar and reading, with little attention paid to speaking, writing, and listening(Belcher 1994.) Despite their many years of English instruction, after arriving in an English speaking country, many international students understandably find that their interactive skills are weak. In particular, they often experience difficulty actually using their English, which can result in frustration and misunderstanding. Recent research has shown that one way to overcome this frustration is to arrange a language exchange with a native speaker of English who wants to learn a foreign language(Brennan 1991). In this arrangement, the two partners exchange their knowledge of their native languages, thus providing a comfortable learning environment. Language exchanges can be one of the best ways to enhance one’s language skills, because they are done on a one-to-one basis.Practice 7: Here is another passage with the same structure. Read it and answer the questions that follow.①Madagascar has one of the world’s oldest systems of natural reserves. ②This system, established during the early 1900s, was designed to protect lemurs and other animal species unique to the island. ③However, due to severe economic hardship, this island country lacks the funds to properly mange the reserve; as a result , many species risk extinction. ④One recent solution to this problem has been offered by the international community. ⑤If Madagascar begins to better protect its reserves, its foreign debt will be reduced. ⑥This incentive should lead to some level of improvement. (based on information from Scientific American, January 1993)1.For what type of audience was this written?2.What assumptions does the author make about the audience’s knowledge background?3.What is the author’s purpose?4.What does this problem in sentence 4 refer to?5.What does this incentive in sentence 6 refer to?6.What does the author think of the solution?7.If the writer had thought that the solution would not work, what might he have written for the last sentence? In such acase, would this last sentence be enough to complete the text? If not, what would need to be added?Practice 8: write your own problem-solution text. Suggested topics:Private carsHigh educationPopulation explosionEnvironmental pollutionOne related to your own field of studyPractice 9: choose a verb from the list that reduces the informality of each sentence.assist reduce create investigate raise establish increase determine fluctuate eliminate1.Expert Systems can help out the user in the diagnosis of problems.2.This program was set up to improve access to medical care.3.Research expenditures have gone up to nearly $350 million.4.The use of optical character readers(OCRs) should cut down the number of problems with the U.S. mail service.5.Researchers have found out that this drug has serious side effects.6.Building a nuclear power plant will not get rid of the energy problem completely.7.Researchers have been looking into this problem for 15 years now.8.This issue was brought up during the investigation.9.Engineers can come up with better designs using CAD.10.The emission levels have been going up and down.Practice 10: Which of the italicized words would be more suitable for an academic paper?1.The government has made good/considerable progress in solving environmental problems.2.We got/obtained encouraging results.3.The results of a lot of/ numerous different projects have been pretty good/encouraging.4. A loss of jobs is one of the things that will happen /consequences if the process is automated.Practice 11: Consider the following passages. Underline the parts in passage b that differ from passage a. Why does b have better “flow” than a?a: Lasers have found widespread application in medicine. Lasers play an important role in the treatment of eye disease and the prevention of blindness. The eye is ideally suited for laser surgery. Most of the eye tissue is transparent. The frequency and focus of the laser beam can be adjusted according to the absorption of the tissue. The beam “cuts” inside the eye with minimal damage to the surrounding tissue –even the tissue between the laser and the incision. Lasers are effective in treating some causes of blindness. Other treatments are not. The interaction between laser light and eye tissue is not fully understood.b: Lasers have found widespread application in medicine. For example, (Lasers) they play an important role in the treatment of eye disease and the prevention of blindness. The eye is edally suited for laser surgery because most of the eye tissue is transparent. Because of this transparency, the frequency and focus of the laser beam can be adjusted according to the absorption of the tissue so that the beam “cuts” inside the eye with minimal damage to the surrounding tissue – even the tissue between the laser and the incision. Lasers are also effective than other methods in treating some causes of blindness. However, the interaction between laser light and eye tissue is not fully understood.Practice 12：Edit the following passage by adding semicolons or commas where necessary.Aluminum alloys are now more important in the automobile industry than ever before. The government is pressuring the industry to produce cars of high quality and with high fuel efficiency hence car makers are replacing traditional iron-based alloys with aluminum alloys. Aluminum alloy parts are typically one-third to one-half the weight of those made with steel as a result cars with all aluminum parts use approximately 50% less fuel than those with steel components. Although most aluminum alloys are soft they can have a higher tensile strength than steel. Adequate alloy and solution treatment can increase their tensile strength thus resulting in a vehicle with good impact capacity.。

武汉大学研究生英语课文原文 Stumbling Blocks in Intercultural

武汉大学研究生英语课文原文 Stumbling Blocks inInterculturalStumbling Blocks inIntercultural Communication1. Why is it that contact with persons from other cultures is so often frustrating and fraught with misunderstanding? Good intentions, the use of what one considers to be a friendly approach and even the possibility of mutual benefits don't seem to be sufficient to ensure success-to many people's surprise. Sometimes rejection occurs just because the group to which a person belongs is\changes in the international scene to take a look at some of the reasons for the disappointing results of attempts at communication. They are actually stumbling blocks in intercultural communication.2. Assumption of similaritiesOne answer to the question of why misunderstanding or rejection happens might be that many of us naivety assume there are sufficient similarities among peoples of the world to enable us tosuccessfully exchange information or feelings, solve problems ofmutual concerns, cement business relationships, or just make the kind of impression we wish to make. The tendency for all peopleto reproduce, group into families or societies, develop a language, and adapt to their environment is particularly deceiving because it leads to the expectation that the forms of these behaviors and the attitudes and values surrounding them will also besimilar. It's comforting to believe that\we'reall alike,\a determined search for proof of this leads to disappointment.3. Promising are the cross-cultural studies seeking to supportDarwin's theory that facial expressions are universal and researchers found that theparticular visible pattern on the face, the combination of muscles contracted for anger, fear, surprise, sadness, disgust, and happiness is the same for allmembers of ourspecies, but this seems helpful until it is realized that a person’s cultu ral upbringing determines whether or notthat emotionwill be displayed or suppressed, as well as on which occasions and to what degree. The situations that bring about the emotional feeling also differ from culture to culture, for example, the death of a loved one may be a cause forjoy, sorrow, or some other emotion, depending upon the accepted cultural belief.4. Since there seems to be no universals of “human nature” that can be used as a basis for automatic understanding, we must treat each encounter as an individual case, searching for whatever perceptions and communication means are held in common and proceed from there. If we realize that we are all culture bound andculturally modified, we will accept the fact that, being unlike, wedo not reallyknow what someone else\5. The aura of similarity is a serious stumbling block to successful intercultural communication. A look-alike facade is deceiving when representatives from contrasting cultures meet, eachwearing Western dress, speaking English, and using similar greeting rituals. It is like assuming that New York, Tokyo, and Tehran are all alike because each has the appearance of a moderncity. Without being alert to possible differences and the needto learn new rules for functioning, persons going from one city tothe other will be in immediatetrouble, even when acting simple roles such as pedestrian and driver.6. Theconfidence that goes with the myth of similarity is much more comfortable than the assumption of differences, the latter requiring tentative assumptions and behaviors and a willingness toaccept the anxietyof\knowing.\with the assumptionof differences, however, can reactionsandinterpretationsbeadjustedtofit\happening.\someone is likely tomisread signs and judge the sceneethnocentrically.7. The stumbling block of assumedsimilarity is a “troublem,”as one English learner expressed it, not only for the foreigner but for the people in the host country with whom the internationalvisitor comes into contact. The native inhabitants are likely to be lulled into the expectation that, since the foreign person isdressed appropriately and speak some of the language, heor she will also have similar nonverbal codes, thoughts and feelings. Thus, nodding, smiling, and affirmative comment will probably be confidentlyinterpreted as meaning that they have informed,helped, and pleased the newcomer. It is likely, however, that theforeigner actually understood very little of the verbal and nonverbal content and was merely indicating polite interest or trying not to embarrass himself or herself of the host with verbalized questions. 8. Language DifferenceThe second stumbling block is so obvious that it hardly needs mentioning--language. Vocabulary, syntax, idioms, slang, dialects, and so on all cause difficulty, but the person strugglingwith a different language is at least aware of being in this kind oftrouble. A worse language problem is thetenacity with whichsomeone will cling to just one meaning of a word or phrase in thenew language, regardless of connotation or context. The variations in possible meaning, especially if inflection and tonal qualities are added, are so difficult to cope with that they are oftenwaved aside. The reason this problem is worse than simple struggling to translate foreign words is because each person thinks heor she understands. The nationwide misinterpretation of Khrushchev's sentence \example.Even\causetrouble.WhenaJapanesehears,\you have some tea?\or she listens to the literalmeaning of the sentence and answers,\that he orshe wants some.\host or hostess that there may be a misunderstanding. Also, in some culture, it is polite to refuse the first or second offer of refreshment. Many foreign guests have gone hungrybec ause their US host or hostess never presented the thirdoffer―another case of “no\9. Nonverbal misinterpretationsLearning the language, which most visitors to foreign countries consider their only barrier to understanding, is actually onlythe beginning. The third stumbling block is nonverbal misinterpretations. People from differentcultures inhabit different sensory realities. They see, hear, feel, and smell only that which has some meaning or importance for them. They abstractwhatever fits into their personal world of recognition and then interpret itthrough the frame of reference of their own culture. For example, a Saudi Arab would nonverbally signal that he liked a girl by smoothing back his hair, which to an American girl would be justa common nervous gesture signifying noting.10. The lack of comprehension ofnonverbal signs and symbols that are easyto observe--such as gestures, postures, and otherbody movements--is a definite communication barrier. But it is possible to learn the meanings of these observable messages, usually in informal rather than formal ways. It is more difficult tonote correctly the unspoken codes of the other culture that are further from awareness, such as the handling of time and spatialrelationships and subtle signs of respect of formality. 11. The Presence of Preconceptions and StereotypesThe fourth stumbling block is the presence of preconceptions and stereotypes. If the label\inscrutable\thus we explain the Japanese constantand inappropriate smile. The stereotype that Arabs are\”may cause the US students to keep their distance or even alert authorities when an animated and noisy group from the Middle Eastgathers. A professor who expects everyone from Indonesia,Mexico, and many other countries to%unfairly interpret a hesitationor request from an international student as amove to manipulate preferential treatment.12. Stereotypes help do what Ernest Becker says the anxiety--prone human race must do--reduce the threat of the unknownby making theworld predictable. Indeed, this is one of the basicfunctions of culture:to lay out a predictable world in which the individual is firmly oriented. Stereotypes are overgeneralized, secondhand beliefs that provide conceptual bases from which to\whether or not they are accurate or fit the circumstances. In a foreign land theiruse increases our feeling of security and is psychologically necessary to the degree that we cannot tolerate ambiguity orthe senseof helplessness resulting from inability to understand and dealwith people and situations beyond our comprehension.13. Stereotypes are stumbling blocks for communicators becausetheyinterfere with objective viewing of stimuli--the sensitivesearch for cues to guide the imagination toward the other persons' reality. Stereotypes are not easy to overcome in ourselves or tocorrect in others, even with thepresentation of evidence. Theypersist because they are firmly established as myths or truisms byone's own national culture and because they sometimesrationalize prejudices. They are also sustained and fed by the tendency toperceive selectively only those pieces of new information that correspond to the image held.14. ThePractice ofImmediate EvaluationThe fifth stumbling block and another deterrent to understanding between persons of differing cultures or ethnic groups isthe tendency to evaluate, to approve or disapprove, the statements and actions of the other person or group. Rather than to try to comprehend completely the thoughts and feelings expressedfrom the world view of the other, we assume our own culture or way of life always seems right, proper, and natural. This biasprevents the open-minded attention needed to look at the attitudes and behavior patterns fromthe other’s point of view.15. Fresh from a conference in Tokyo where Japanese professorshad emphasized the preference the people of Japan for simple natural settings感谢您的阅读，祝您生活愉快。

人体结构学 Human Structure学习通章节答案期末考试题库2023年

人体结构学 Human Structure学习通超星课后章节答案期末考试题库2023年1.Which bones belong to the shoulder girdle?答案:Scapula###Clavicle2.The paired cerebral bones are答案:parietal bone###temporal bone3.Shoulder joint is formed by答案:head of humerus###glenoid cavity of scapula4.Please deseribe the location and openings of the paranasal sinuses答案:答5.Please describe the formation, main structures and communications of the middle Cranial fossa.答案:答6.Please describe the joints of the vertebral bodies.答案:答7.Please describe the joints of the vertebral arches.答案:答8.Please describe the composition, characteristics and movements of theshoulder joint.答案:答9.Which bone belongs to the long bone?答案:Femur10.Which bones belong to the irregular bone?答案:Vertebra###Sphenoid bone11.The blood- testis barrier does NOT include the答案:gap junction between adjacent spermatogomia12.Which of the following description is true about the primordial follicles答案:The primordial follicle consists of a primary oocyte and a layer of flat follicle cells.13.(英文答题,第一空填1个单词,第二空3个单词)The axial bone contains fromup downwards _____ and_____.答案:skull###bonesoftrunk14.About the component of nephron, the correct option is答案:renal corpuscle, proximal tubules, distal tubules and thin segment15.About the features of proximal tubule, the WRONG option is答案:The cytoplasm of epithelial cell is weakly basophilic.16. A patient presents in your office after having a positive result on a homepregnancy test. Her menstrual cycle has always been the classic 28-day cycle discussed in textbooks, with ovulation occurring on the 14th day following the start of menstruation. Her menstrual period began on August 19.2019.You estimate her EDD to be答案:on May 26, 202017.Which of the following is NOT considered one of the fetal membranes答案:buccopharyngeal membrane18.Which bone does not form the anterior cranial fossa?答案:Temporal bone19.Which bone forms both the middle and posterior cranial fossa?答案:Temporal bone20.(英文答题,第一空填1个单词,第二空1个单词,第三空2个单词)Thesternum consists from up downwards of_____ , _______ and ______ .答案:manubrium###body###xiphoidprocess21.Of the following statements about epididymis, the WRONG option is答案:The ductus epididymis is lined with a simple columnar epithelium22.Of the following statements about trachea, the WRONG option is答案:The adventitia is constructed of the elastic cartilage rings.23.The interalveolar septum does NOT contain答案:ciliated cell24.All the following cells are included in the spermatogenic epithelium, EXCEPTthe答案:Leydig cells25.Please describe the general features of the vertebrae.答案:答26.Which bone forms the posteroinferior part of the bony nasal septum?答案:Vomer27.Drawing pictures of Thoracic vertebra from anterior and lateral view.答案:答28.Which bone does not form the thoracic Cage?答案:Sacrum29.About the scapula, which of the statements is not true?答案:It has three borders, three angles and three surfaces.30.About the component of the renal corpuscle, the WRONG option is答案:At the vascular pole, the efferent arteriole enters the glomerulus.31.Of the following statements about podocyte, the correct option is答案:They form the visceral layer of the Bowman's capsule.32.An infant is born with a sacrococcygeal teratoma. Biopsy(组织活检) andhistologic analysis reveal that it contains intestinal epithelia, cardiac muscle, cartilage, and integument tissue. You counsel the mother that the tumor is benign（良性的）and recommend surgical removal. This tumor was caused by which developmental anomaly?答案:Failure of primitive streak regression33.Which of the following structure is NOT included in the secondary follicle?答案:secondary oocyte34.All the following are from mesoderm EXCEPT the答案:spinal cord35.Which of the following descriptions is NOT true about the corpus luteum?答案:The corpus luteum continues to produce estrogen and progesterone during the whole process of pregnancy.36.Of the following statements about the alveolus of lung, the WRONG option is答案:It opens on the wall of terminal bronchioles.37.Which of the following descriptions is NOT true about the secretory phase ofa menstrual cycle?答案:The basal layer of endometrium becomes thicker .38.Of the following statements about Leydig cells, the correct option is答案:It secretes testosterone.39.Of the following statements about macula densa, the WRONG option is答案:It is derived from smooth muscle fibers of afferent arteriole.40.Of the following statements concerning terminal bronchioles, the WRONGoption is答案:They have some mixed gland.41.Of the following options, the blood-air barrier does NOT contain答案:typeⅡalveolar cells42.All the following cells are included in the spermatogenic cells, EXCEPT答案:Sertoli cells43.Which of the following does not belong to the joints of the vertebral arches?答案:Anterior longitudinal ligament44.Human chorionic gonadotropin is produced by the答案:syncytiotrophoblast45.Which of the followings is not enclosed in the articular capsule of shoulderjoint ?答案:Tendon of the short head of the biceps46.The pathway connecting the infratemporal fossa with the orbit is答案:inferior orbital fissure47.When does ovulation occur in a menstrual cycle?答案:the 14th day。

科学家最宝贵的品质考研英语作文

科学家最宝贵的品质考研英语作文全文共3篇示例，供读者参考篇1The Most Valuable Qualities of ScientistsAs a student aspiring to pursue a career in scientific research, I have often pondered the essential qualities that define a truly great scientist. While technical prowess and academic brilliance are undoubtedly crucial, I have come to realize that the most exceptional scientists possess a unique blend of intellectual and personal traits that transcend mere knowledge and skills. These qualities are the bedrock upon which groundbreaking discoveries and paradigm-shifting theories are built, and they are what separate the ordinary from the extraordinary in the realm of scientific exploration.Curiosity: The Catalyst for DiscoveryAt the core of every scientific endeavor lies an insatiable curiosity – a relentless desire to unravel the mysteries of the universe, to push the boundaries of human understanding, and to challenge the status quo. Great scientists are driven by an unquenchable thirst for knowledge, constantly questioning theworld around them and seeking answers to the most perplexing conundrums. From the pioneering work of Isaac Newton, whose curiosity about the nature of gravity revolutionized our understanding of the physical world, to the groundbreaking discoveries of Marie Curie, whose inquisitive spirit propelled her to uncover the secrets of radioactivity, curiosity has been the driving force behind some of the most profound scientific breakthroughs.Critical Thinking: The Cornerstone of Scientific ReasoningThe ability to think critically and analyze problems from multiple perspectives is an indispensable quality for any scientist worthy of the name. Great scientists do not merely accept established theories and dogmas at face value; rather, they scrutinize them with a critical eye, constantly questioning assumptions and seeking evidence to support or refute existing paradigms. This intellectual rigor and unwavering commitment to objective analysis have paved the way for countless scientific revolutions, from the Copernican revolution that shattered the geocentric model of the universe to the groundbreaking work of Charles Darwin, whose theory of evolution by natural selection challenged long-held beliefs about the origins of life.Perseverance: The Fuel for Scientific ProgressScientific inquiry is often a long and arduous journey, fraught with obstacles, setbacks, and failures. It is the perseverance and unwavering determination of great scientists that enable them to overcome these challenges and continue their quest for knowledge, undeterred by temporary setbacks or disheartening failures. The history of science is replete with examples of scientists who defied the odds and persevered against all odds, such as Albert Einstein, whose revolutionary theories of relativity were initially met with skepticism and resistance, or Rosalind Franklin, whose crucial contributions to the discovery of the double-helix structure of DNA were initially overlooked due to gender bias.Creativity: The Spark of InnovationWhile scientific progress is built upon a foundation of rigorous methodology and empirical evidence, true breakthroughs often require a spark of creativity – the ability to think outside the box, to challenge conventional wisdom, and to envision novel solutions to seemingly intractable problems. Great scientists possess an innate ability to synthesize disparate ideas and forge new connections, allowing them to approach challenges from unconventional angles and uncover insights that elude their more conventional counterparts. From thegroundbreaking work of Nikola Tesla, whose creative genius revolutionized the field of electrical engineering, to the visionary ideas of Stephen Hawking, whose theoretical contributions have reshaped our understanding of the cosmos, creativity has been the driving force behind many of the most profound scientific achievements.Collaboration and CommunicationWhile the image of the solitary genius toiling away in isolation may be a romantic one, the reality of modern scientific research is that it is a highly collaborative endeavor, requiring effective communication and teamwork among researchers from diverse backgrounds and disciplines. Great scientists must possess the ability to articulate their ideas clearly and persuasively, to listen to and incorporate the perspectives of others, and to foster an environment of open and constructive dialogue. By embracing collaboration and fostering an atmosphere of intellectual exchange, scientists can leverage the collective knowledge and expertise of their peers, accelerating the pace of discovery and amplifying the impact of their work.Ethical Integrity: The Moral Compass of ScienceScience is not merely a pursuit of knowledge; it is a discipline that carries immense ethical responsibilities and implications.Great scientists must possess a deep sense of ethical integrity, a commitment to upholding the highest standards of intellectual honesty, and an unwavering dedication to the responsible and ethical application of their discoveries. From the pioneering work of Jonas Salk, whose development of the polio vaccine saved countless lives, to the cautionary tale of the Manhattan Project, which highlighted the grave consequences of scientific knowledge unchecked by ethical considerations, the ethical conduct of scientists has profound implications for humanity and the world we inhabit.As I embark on my journey to become a scientist, I am acutely aware of the immense responsibility that comes with this noble pursuit. It is not merely a matter of acquiring knowledge and technical skills; it is a lifelong commitment to cultivating the qualities that define true greatness in the scientific篇2The Most Valuable Qualities of a ScientistAs a student aspiring to become a scientist, I've spent a lot of time contemplating what truly makes a great researcher. What qualities and traits are essential for unlocking the mysteries of the universe and driving scientific progress? Based on myobservations and experiences, I believe the most valuable qualities of a scientist are insatiable curiosity, relentless perseverance, and integrity grounded in humility.Insatiable CuriosityAt its core, science is fueled by an unquenchable thirst for knowledge and understanding. The greatest scientists are those who possess an insatiable curiosity that compels them to constantly question, explore, and push the boundaries of what is known. They are driven by a deep yearning to unravel the complexities of the natural world, to uncover the fundamental principles that govern the universe, and to unearth the solutions to problems that have long confounded humanity.This unwavering curiosity manifests itself in a willingness to challenge conventional wisdom, to embrace uncertainty, and to venture into uncharted territory. It is a quality that prevents scientists from becoming complacent or satisfied with the status quo. Instead, it propels them to continually ask probing questions, to seek out new perspectives, and to approach familiar concepts with fresh eyes.Truly curious scientists are not content with merely accumulating facts; they seek to understand the underlying mechanisms, the interconnections, and the broader implications.They are driven to explore the "why" and "how" behind phenomena, not just the "what." This insatiable curiosity is the spark that ignites scientific revolutions, propelling humanity towards groundbreaking discoveries and paradigm shifts.Relentless PerseveranceScience is a pursuit that demands unwavering perseverance in the face of seemingly insurmountable obstacles and setbacks. The path to scientific breakthroughs is often paved with frustration, failure, and disappointment. Experiments may yield perplexing results, hypotheses may be disproven, and years of painstaking work may seemingly lead to dead ends.It is in these moments that a scientist's perseverance is truly tested. The great scientists are those who possess an unshakable determination to push forward, to learn from their mistakes, and to continue searching for answers with unwavering resolve. They understand that failure is not a defeat but rather an opportunity to gain valuable insights and refine their approach.This relentless perseverance is fueled by a combination of passion, resilience, and an uncompromising commitment to their goals. It enables scientists to weather the storms of doubt, criticism, and adversity that are inherent in the scientific endeavor. They possess the mental fortitude to endure longhours in the lab, to meticulously analyze data, and to tirelessly refine their theories and experiments until they achieve a breakthrough.Moreover, perseverance is not limited to the pursuit of a single objective; it extends to the broader quest for knowledge itself. Great scientists recognize that the pursuit of understanding is a lifelong journey, filled with detours, obstacles, and ever-evolving challenges. They embrace this journey with an unwavering spirit, driven by the knowledge that each discovery, no matter how small, contributes to the collective understanding of the world around us.Integrity Grounded in HumilityScience is a collaborative endeavor built upon a foundation of trust, transparency, and intellectual honesty. It is a pursuit that demands the highest levels of integrity, both in the conduct of research and in the communication of findings. Without integrity, the entire scientific enterprise would crumble, eroding the credibility and reliability upon which progress depends.At the heart of scientific integrity lies humility – a recognition that our understanding of the universe is inherently limited and that our theories and conclusions are subject to revision in the face of new evidence. Truly great scientists approach their workwith a sense of awe and respect for the complexities of the natural world, acknowledging that there is always more to learn and that their findings may one day be superseded or refined.This humility manifests itself in a willingness to scrutinize one's own work with a critical eye, to embrace constructive criticism, and to acknowledge the contributions and perspectives of others. It is a quality that safeguards against arrogance, dogmatism, and the temptation to manipulate or misrepresent data to fit preconceived notions.Moreover, integrity grounded in humility fosters an environment of collaboration and open discourse. Scientists who possess this quality are not threatened by differing viewpoints or contradictory evidence; instead, they welcome them as opportunities to deepen their understanding and refine their theories. They recognize that scientific progress is a collective endeavor, built upon the foundations laid by countless minds across generations and disciplines.ConclusionIn the ever-evolving landscape of scientific inquiry, the qualities of insatiable curiosity, relentless perseverance, and integrity grounded in humility are the bedrock upon which true progress is built. These traits transcend the boundaries of anyspecific discipline or field, serving as the driving force behind humanity's quest to understand the universe and unlock its secrets.As a student aspiring to join the ranks of the great scientists, I am deeply inspired by these qualities and strive to embody them in my own journey. It is my sincere hope that by cultivating these traits, I can contribute, however small, to the collective pursuit of knowledge that has defined humanity's greatest achievements and propelled us ever forward into the unknown.篇3The Most Valuable Qualities of ScientistsAs someone aspiring to become a scientist, I've spent a lot of time thinking about what truly makes a great researcher. Sure, intelligence and technical knowledge are important, but I don't think those are the most essential qualities. In my opinion, the best scientists possess an insatiable curiosity, unwavering perseverance, and a humble ability to admit what they don't know.First and foremost, an unquenchable thirst for knowledge is absolutely vital for any scientist worth their salt. The desire to constantly learn and understand the world around us is whatdrives scientific discovery and progress. The greatest minds in history like Newton, Einstein, and Curie all possessed an intense curiosity that propelled them to explore the boundaries of human understanding.A scientist without curiosity is like a sailor without a compass – aimlessly adrift with no sense of direction. The curious mind is always asking questions, poking and prodding at explanations, and refusing to simply accept things at face value. They approach every situation with a sense of wonder, determined to get to the root of how and why things work the way they do.I'll never forget in my freshman biology class when we learned about the incredible complexity of cellular biology and the millions of tiny molecular machines operating in perfect synchrony within every living cell. Instead of just memorizing the textbook diagrams, our professor (a brilliant and impassioned scientist himself) encouraged us to let our curiosity run wild. "Don't just accept what the book tells you, ask yourself how this all came to be! How did evolution arrange such an intricate dance of proteins and enzymes? What evolutionary mechanisms carved these nanoscopic factories from the chaos of random mutation?"His contagious curiosity ignited a fire within me that day that still burns bright. That's the attitude I aspire to – a childlike sense of wonder and unrelenting inquisitiveness about the natural world. Because at the end of the day, curiosity is the spark that has always driven humanity's scientific efforts to reveal the secrets of the cosmos.However, curiosity alone is not enough – it must be coupled with grit, perseverance, and a willingness to face failure head on. Science is an arduous endeavor, fraught with obstacles, dead ends, and disappointments. Every major breakthrough is built on the shoulders of those who stopped at nothing to turn their curiosity into tangible understanding.The path of least resistance is a fruitless one for scientists. Instead, they must forge ahead despite tremendous challenges and setbacks. It took Michael Faraday years of painstaking experimentation before he could prove the concept ofelectro-magnetic induction. Marie Curie labored for over four years, processing literal tons of radioactive material, to eventually discover the elements radium and polonium. Their brilliant minds were matched only by their steadfast determination.I've already experienced my share of frustrations and "failed" experiments during my undergraduate research projects. Weeks of diligently following protocols, only for the results to defy my expectations. Hours meticulously constructing a computer model, just to find a crippling flaw that sends me back to square。

用英语介绍爱因斯坦

Part 4
General Theory
4 of Relativity and Later Years
General Theory of Relativity and Later Years
In 1915, Einstein published his general theory of relativity, which explained gravitation as a consequence of the curvature of spacetime. The theory revolutionized our unde rstanding of gravity and marked a turning point in Einstein's career. The following year, he was awarded the Nobel Prize in Physics for his work on the photoelectric effect
答辩人：XXX
Part 2
2
Studies and Work in Switzerland
and Germany
Studies and Work in Switzerland and Germany
In 1900, Einstein graduated from the University of Zurich with a teaching diploma in mathematics and physics. He began teaching mathematics and physics at a series of Swiss schools, including the Cantonal School i

同素异形体英语

同素异形体英语《同素异形体英语》I. IntroductionIsomers are molecules that have the same number of atoms of each element, but different arrangements of atoms. They can differ in the way the atoms are connected, the arrangement in space, or the shape of the molecule. For example, two molecules of ethanol, CH3CH2OH, can exist as two different isomers, as can the two molecules of propanoic acid, CH3CH2COOH.II. Types of IsomersA. Structural IsomersStructural isomers, also known as constitutional isomers, are molecules that have the same chemical formula but a different structure. They differ in the way thattheir atoms are connected, so their structural formulas are different. In the case of ethanol, there are two structural isomers, which are called 1-propanol and 2-propanol.B. StereoisomersStereoisomers are molecules that have the same molecular formula and the same structural formula, butdifferent arrangements in space. They can be divided into two main categories: geometric isomers and optical isomers. Geometric isomers differ in the orientation of functional groups relative to a plane in the molecule. Optical isomers differ in the arrangement of their atoms in space, with one form being the mirror image of the other.C. EnantiomersEnantiomers are a type of optical isomer that are mirror images of each other. They are sometimes also referred to as “handedness”, since they are like tworight or two left hands that are the exact opposite of each other. Examples of enantiomers include the molecules of lactic acid and sucrose.III. ConclusionIn conclusion, isomers are molecules that have the same chemical formula, but different structures or arrangements in space. They can be divided into three main categories: structural isomers, stereoisomers, and enantiomers. Understanding the different types of isomers is essential for visualizing and understanding the structure of molecules.。

黎曼猜想英语

黎曼猜想英语The Riemann Hypothesis, named after the 19th-century mathematician Bernhard Riemann, is one of the most profound and consequential conjectures in mathematics. It is concerned with the distribution of the zeros of the Riemann zeta function, a complex function denoted as $$\zeta(s)$$, where $$s$$ is a complex number. The hypothesis posits that all non-trivial zeros of this analytical function have their real parts equal to $$\frac{1}{2}$$.To understand the significance of this conjecture, one must delve into the realm of number theory and the distribution of prime numbers. Prime numbers are the building blocks of arithmetic, as every natural number greater than 1 is either a prime or can be factored into primes. The distribution of these primes, however, has puzzled mathematicians for centuries. The Riemann zeta function encodes information about the distribution of primes through its zeros, and thus, the Riemann Hypothesis is directly linked to understanding this distribution.The zeta function is defined for all complex numbers except for $$s = 1$$, where it has a simple pole. For values of $$s$$ with a real part greater than 1, it converges to a sum over the positive integers, as shown in the following equation:$$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$。

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n=1
1
∞
(1 − q n ),
and the Jacobian theta functions
∞
(1.3)
θ2 (q ) :=
n=−∞
q ( n+ 2 ) ,
1 2
Received by the editors April 2, 1991. 1980 Mathematics Subject Classiﬁcation (1985 Revision) Primary 33D10, Secondary 11B65, 11F11, 11F20, 33C05 Key words and phrases. Theta functions, q -series, eta function, modular forms, cubic modular equations, hypergeometric functions. Research of the ﬁrst and second author was partially supported by NSERC. Research completed while the third author was an NSERC International Fellow. 1
1
2
2
2
Letting k tend to inﬁnity gives 1 = (q 3 ; q 3 )∞ m Hence (q )3 ∞ = (q 3 ; q 3 )∞ m =
m1 ,m2
ω
m1 −m2
0 +m1 +m2 =0
q 2 (m0 +m1 +m2 ) . (q )3 ∞
1
2
2
2
ω m1 −m2 q 2 (m0 +m1 +m2 ) ω m1 −m2 q m1 +m1 m2 +m2
2π 3 (n
− m) q n −
2
+nm+m2
qn
3|n−m
2
+nm+m2
1 2
qn
2
+nm+m2
.
Now we write the ﬁrst sum by q (3p+m)
m p
2
+(3p+m)m+m2
= a(q 3 ),
SOME CUBIC MODULAR IDENTITIES OF RAMANUJAN
(1.6)
a(q ) :=
n,m=−∞ ∞
qn
2
+nm+m2
,
(1.7)
b(q ) :=
n,m=−∞ ∞
ω n−m q n q ( n+ 3 )
n,m=−∞
1 2
2
+nm+m2
,
(1.8)
c(q ) :=
1 1 2 +(n+ 1 3 )(m+ 3 )+(m+ 3 )
.
We will prove that a3 = b3 + c3 . Note that in (1.3) and (1.8) we take principal roots. 2. Results. Lemma 2.1. For |q | < 1 (i) (a) a(q ) = θ3 (q )θ3 (q 3 ) + θ2 (q )θ2 (q 3 ); 3 3 a(q 4 ) = 1 2 θ3 (q )θ3 (q ) + θ4 (q )θ4 (q ) ; 3 1 3 (ii) b(q ) = 2 a(q ) − 2 a(q ); 1 1 (iii) c(q ) = 2 a(q 3 ) − 1 2 a(q ). Proof. (i) In (1.6) write n2 + nm + m2 = (n + a(q ) =
∞ ∞
(2.2)
(−x; q )∞ =
n=0
(1 + xq ) =
k=0 ∞ n n=0 (1−aq ),
n
q (2) xk , (q )k
k
where as usual (a)∞ = (a; q )∞ := n = k=1 (1 − aq k−1 ). Observe that
(a)n = (a; q )n := (a; q )∞ /(aq n ; q )∞
(−x3 ; q 3 )∞ = (−x; q )∞ (−xω ; q )∞ (−xω 2 ; q )∞ , so that (2.2) gives
∞ k=0
k x3k q 3(2) = (q 3 ; q 3 )k
ω
n0 ,n1 ,n2 ≥0
n1 +2n2
n0 n1 n2 q ( 2 )+( 2 )+( 2 ) xn0 +n1 +n2 . (q )n0 (q )n1 (q )n2
m even m 2 2) 2 + 3( m 2 ) . Now
q ( n+ 2 )
m 2
2 +3( m 2 )
+
m odd
q ( n+ 2 )
m 2
2 +3( m 2 )
.
This identity is (i) (a). Identity (i) (b) follows from θ3 (q 4 ) + θ2 (q 4 ) = θ3 (q ) and θ3 (q 4 ) − θ2 (q 4 ) = θ4 (q ). (ii) We observe that b(q ) is real and that b(q ) = = 3 2 cos
3
as is apparent on substituting m − p for m. Identity (ii) follows. (iii) Since (ii) holds, it suﬃces to show that (2.1) c(q 3 ) = 1 (a(q ) − b(q )). 3
Trans. Amer. Math. Soc. 343 (1994), 35–47 SOME CUBIC MODULAR IDENTITIES OF RAMANUJAN
J. M. Borwein, P. B. Borwein and F. G. Garvan
Abstract. There is a beautiful cubic analogue of Jacobi’s fundamental theta 4 = θ 4 + θ 4 . It is function identity: θ3 4 2
We replace ni by mi + k for i = 0, 1, 2. Then
2
i=0
ni 2
1 = 2
2
m2 i +3
i=0
k 2
since
2 i=0
mi = 0. Hence 1 = 3 (q ; q 3 )k m ω
m1 −m2
0 +m1 +m2 =0
q 2 (m0 +m1 +m2 ) . (q )m0 +k (q )m1 +k (q )m2 +k
p,m
2
+3p+3pm+3m2 +3m+1
,
which, by (1.8), equals c(q 3 ). Thus, (2.1) has been proved. As is well known, [4, p.64], the theta functions possess simple product expansions. This is not true of a. It is true of b and c as our main preliminary result shows: Proposition 2.2. (i) (ii) b(q ) = η 3 (q ) = η (q 3 ) (1 − q n )3 , (1 − q 3n ) (1 − q 3n )3 . (1 − q n )
2 2
1
2
2
2
0 +m1 +m2 =0
as claimed. (ii) This may be established in a similar fashion. Alternatively, we may argue as follows. The previous identity combines with Lemma 2.1 (ii) to produce (2.3) 3a(q 3 ) − a(q ) η 3 (q ) = . η (q 3 ) 2
∞
2 2
qn
+nm+m
3
n,m=−∞
=

∞
ω n−m q n
∞
2
+nm+m
2
3
n,m=−∞
+ 3 .
q
1 )2 +(n+ 1 )(m+ 1 )+(m+ 1 )2 (n+ 3 3 3 3
n,m=−∞
Here ω = exp(2πi/3). In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.