The solution conformation of triarylmethyl radicals
希尔伯特的23个数学问题

BULLETIN(New Series)OF THEAMERICAN MATHEMATICAL SOCIETYVolume37,Number4,Pages407–436S0273-0979(00)00881-8Article electronically published on June26,2000MATHEMATICAL PROBLEMSDAVID HILBERTLecture delivered before the International Congress of Mathematicians at Paris in1900.Who of us would not be glad to lift the veil behind which the future lies hidden;to cast a glance at the next advances of our science and at the secrets of its development during future centuries?What particular goals will there be toward which the leading mathematical spirits of coming generations will strive?What new methods and new facts in the wide and richfield of mathematical thought will the new centuries disclose?History teaches the continuity of the development of science.We know that every age has its own problems,which the following age either solves or casts aside as profitless and replaces by new ones.If we would obtain an idea of the probable development of mathematical knowledge in the immediate future,we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future.To such a review of problems the present day,lying at the meeting of the centuries,seems to me well adapted.For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future.The deep significance of certain problems for the advance of mathematical science in general and the important rˆo le which they play in the work of the individual investigator are not to be denied.As long as a branch of science offers an abundance of problems,so long is it alive;a lack of problems foreshadows extinction or the cessation of independent development.Just as every human undertaking pursues certain objects,so also mathematical research requires its problems.It is by the solution of problems that the investigator tests the temper of his steel;hefinds new methods and new outlooks,and gains a wider and freer horizon.It is difficult and often impossible to judge the value of a problem correctly in advance;for thefinal award depends upon the grain which science obtains from the problem.Nevertheless we can ask whether there are general criteria which mark a good mathematical problem.An old French mathematician said:“A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to thefirst man whom you meet on the street.”This clearness and ease of comprehension,here insisted on for a mathematical theory,I should still more demand for a mathematical problem if it is to be perfect;for what is clear and easily comprehended attracts,the complicated repels us.Moreover a mathematical problem should be difficult in order to entice us,yet not completely inaccessible,lest it mock at our efforts.It should be to us a guide408DA VID HILBERTpost on the mazy paths to hidden truths,and ultimately a reminder of our pleasure in the successful solution.The mathematicians of past centuries were accustomed to devote themselves to the solution of difficult particular problems with passionate zeal.They knew the value of difficult problems.I remind you only of the“problem of the line of quickest descent,”proposed by John Bernoulli.Experience teaches,explains Bernoulli in the public announcement of this problem,that lofty minds are led to strive for the advance of science by nothing more than by laying before them difficult and at the same time useful problems,and he therefore hopes to earn the thanks of the mathematical world by following the example of men like Mersenne,Pascal, Fermat,Viviani and others and laying before the distinguished analysts of his time a problem by which,as a touchstone,they may test the value of their methods and measure their strength.The calculus of variations owes its origin to this problem of Bernoulli and to similar problems.Fermat had asserted,as is well known,that the diophantine equationx n+y n=z n(x,y and z integers)is unsolvable—except in certain self-evident cases.The attempt to prove this impossibility offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science.For Kummer,incited by Fermat’s problem,was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circularfield into ideal prime factors—a law which to-day in its generalization to any algebraicfield by Dedekind and Kronecker,stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions.To speak of a very different region of research,I remind you of the problem of three bodies.The fruitful methods and the far-reaching principles which Poincar´e has brought into celestial mechanics and which are to-day recognized and applied in practical astronomy are due to the circumstance that he undertook to treat anew that difficult problem and to approach nearer a solution.The two last mentioned problems—that of Fermat and the problem of the three bodies—seem to us almost like opposite poles—the former a free invention of pure reason,belonging to the region of abstract number theory,the latter forced upon us by astronomy and necessary to an understanding of the simplest fundamental phenomena of nature.But it often happens also that the same special problemfinds application in the most unlike branches of mathematical knowledge.So,for example,the problem of the shortest line plays a chief and historically important part in the foundations of geometry,in the theory of curved lines and surfaces,in mechanics and in the calculus of variations.And how convincingly has F.Klein,in his work on the icosahedron,pictured the significance which attaches to the problem of the regular polyhedra in elementary geometry,in group theory,in the theory of equations and in that of linear differential equations.In order to throw light on the importance of certain problems,I may also refer to Weierstrass,who spoke of it as his happy fortune that he found at the outset of his scientific career a problem so important as Jacobi’s problem of inversion on which to work.MATHEMATICAL PROBLEMS409 Having now recalled to mind the general importance of problems in mathematics, let us turn to the question from what sources this science derives its problems. Surely thefirst and oldest problems in every branch of mathematics spring from experience and are suggested by the world of external phenomena.Even the rules of calculation with integers must have been discovered in this fashion in a lower stage of human civilization,just as the child of to-day learns the application of these laws by empirical methods.The same is true of thefirst problems of geometry, the problems bequeathed us by antiquity,such as the duplication of the cube, the squaring of the circle;also the oldest problems in the theory of the solution of numerical equations,in the theory of curves and the differential and integral calculus,in the calculus of variations,the theory of Fourier series and the theory of potential—to say noting of the further abundance of problems properly belonging to mechanics,astronomy and physics.But,in the further development of a branch of mathematics,the human mind, encouraged by the success of its solutions,becomes conscious of its independence. It evolves from itself alone,often without appreciable influence from without,by means of logical combination,generalization,specialization,by separating and col-lecting ideas in fortunate ways,new and fruitful problems,and appears then itself as the real questioner.Thus arose the problem of prime numbers and the other problems of number theory,Galois’s theory of equations,the theory of algebraic invariants,the theory of abelian and automorphic functions;indeed almost all the nicer questions of modern arithmetic and function theory arise in this way.In the meantime,while the creative power of pure reason is at work,the outer world again comes into play,forces upon us new questions from actual experience, opens up new branches of mathematics,and while we seek to conquer these new fields of knowledge for the realm of pure thought,we oftenfind the answers to old unsolved problems and thus at the same time advance most successfully the old theories.And it seems to me that the numerous and surprising analogies and that apparently prearranged harmony which the mathematician so often perceives in the questions,methods and ideas of the various branches of his science,have their origin in this ever-recurring interplay between thought and experience.It remains to discuss briefly what general requirements may be justly laid down for the solution of a mathematical problem.I should sayfirst of all,this:that it shall be possible to establish the correctness of the solution by means of afinite number of steps based upon afinite number of hypotheses which are implied in the statement of the problem and which must always be exactly formulated.This requirement of logical deduction by means of afinite number of processes is sim-ply the requirement of rigor in reasoning.Indeed the requirement of rigor,which has become proverbial in mathematics,corresponds to a universal philosophical necessity of our understanding;and,on the other hand,only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect.A new problem,especially when it comes from the world of outer experience,is like a young twig,which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem,the established achievements of our mathematical science.Besides it is an error to believe that rigor in the proof is the enemy of simplic-ity.On the contrary wefind it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended.The410DA VID HILBERTvery effort for rigor forces us tofind out simpler methods of proof.It also fre-quently leads the way to methods which are more capable of development than the old methods of less rigor.Thus the theory of algebraic curves experienced a considerable simplification and attained greater unity by means of the more rigor-ous function-theoretical methods and the consistent introduction of transcendental devices.Further,the proof that the power series permits the application of the four elementary arithmetical operations a well as the term by term differentiation and integration,and the recognition of the utility of the power series depending upon this proof contributed materially to the simplification of all analysis,particularly of the theory of elimination and the theory of differential equations,and also of the existence proofs demanded in those theories.But the most striking example for my statement is the calculus of variations.The treatment of thefirst and second variations of definite integrals required in part extremely complicated calculations, and the processes applied by the old mathematicians had not the needful rigor. Weierstrass showed us the way to a new and sure foundation of the calculus of variations.By the examples of the simple and double integral I will show briefly,at the close of my lecture,how this way leads at once to a surprising simplification of the calculus of variations.For in the demonstration of the necessary and sufficient criteria for the occurrence of a maximum and minimum,the calculation of the sec-ond variation and in part,indeed,the wearisome reasoning connected with thefirst variation may be completely dispensed with—to say nothing of the advance which is involved in the removal of the restriction to variations for which the differential coefficients of the function vary but slightly.While insisting on rigor in the proof as a requirement for a perfect solution of a problem,I should like,on the other hand,to oppose the opinion that only the concepts of analysis,or even those of arithmetic alone,are susceptible of a fully rigorous treatment.This opinion,occasionally advocated by eminent men,I con-sider entirely erroneous.Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry,mechanics and physics,to a stoppage of theflow of new material from the outside world,and finally,indeed,as a last consequence,to the rejection of the ideas of the continuum and of the irrational number.But what an important nerve,vital to mathematical science,would be cut by the extirpation of geometry and mathematical physics! On the contrary I think that wherever,from the side of the theory of knowledge or in geometry,or from the theories of natural or physical science,mathematical ideas come up,the problem arises for mathematical science to investigate the principles underlying these ideas and so to establish them upon a simple and complete system of axioms,that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts.To new concepts correspond,necessarily,new signs.These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts.So the geometricalfigures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians.Who does not always use along with the double inequality a>b>c the picture of three points following one another on a straight line as the geometrical picture of the idea “between”?Who does not make use of drawings of segments and rectangles enclosed in one another,when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation?Who could dispense with thefigure of the triangle,the circle with its center,or with the crossMATHEMATICAL PROBLEMS411 of three perpendicular axes?Or who would give up the representation of the vector field,or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry,in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?The arithmetical symbols are written diagrams and the geometricalfigures are graphic formulas;and no mathematician could spare these graphic formulas,any more than in calculation the insertion and removal of parentheses or the use of other analytical signs.The use of geometrical signs as a means of strict proof presupposes the exact knowledge and complete mastery of the axioms which underlie thosefigures;and in order that these geometricalfigures may be incorporated in the general treasure of mathematical signs,there is necessary a rigorous axiomatic investigation of their conceptual content.Just as in adding two numbers,one must place the digits under each other in the right order,so that only the rules of calculation,i.e.,the axioms of arithmetic,determine the correct use of the digits,so the use of geometrical signs is determined by the axioms of geometrical concepts and their combinations.The agreement between geometrical and arithmetical thought is shown also in that we do not habitually follow the chain of reasoning back to the axioms in arithmetical,any more than in geometrical discussions.On the contrary we ap-ply,especially infirst attacking a problem,a rapid,unconscious,not absolutely sure combination,trusting to a certain arithmetical feeling for the behavior of the arithmetical symbols,which we could dispense with as little in arithmetic as with the geometrical imagination in geometry.As an example of an arithmetical theory operating rigorously with geometrical ideas and signs,I may mention Minkowski’s work,Die Geometrie der Zahlen.1Some remarks upon the difficulties which mathematical problems may offer,and the means of surmounting them,may be in place here.If we do not succeed in solving a mathematical problem,the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. Afterfinding this standpoint,not only is this problem frequently more accessible to our investigation,but at the same time we come into possession of a method which is applicable also to related problems.The introduction of complex paths of integration by Cauchy and of the notion of the ideals in number theory by Kummer may serve as examples.This way forfinding general methods is certainly the most practicable and the most certain;for he who seeks for methods without having a definite problem in mind seeks for the most part in vain.In dealing with mathematical problems,specialization plays,as I believe,a still more important part than generalization.Perhaps in most cases where we seek in vain the answer to a question,the cause of the failure lies in the fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved.All depends,then,onfinding out these easier problems,and on solving them by means of devices as perfect as possible and of concepts capable of generalization. This rule is one of the most important leers for overcoming mathematical difficulties and it seems to me that it is used almost always,though perhaps unconsciously.412DA VID HILBERTOccasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense,and for this reason do not succeed.The problem then arises:to show the impossibility of the solution under the given hypotheses,or in the sense contemplated.Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right triangle is irrational.In later mathematics,the question as to the impossibility of certain solutions plays a pre¨e minent part,and we perceive in this way that old and difficult problems,such as the proof of the axiom of parallels,the squaring of the circle,or the solution of equations of thefifth degree by radicals havefinally found fully satisfactory and rigorous solutions,although in another sense than that originally intended.It is probably this important fact along with other philosophical reasons that gives rise to the conviction(which every mathematician shares,but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement,either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts.Take any definite unsolved problem,such as the question as to the irrationality of the Euler-Mascheroni constant C,or the existence of an infinite number of prime numbers of the form2n+1.However unapproachable these problems may seem to us and however helpless we stand before them,we have,nevertheless,thefirm conviction that their solution must follow by afinite number of purely logical processes.Is this axiom of the solvability of every problem a peculiarity characteristic of mathematical thought alone,or is it possibly a general law inherent in the nature of the mind,that all questions which it asks must be answerable?For in other sciences also one meets old problems which have been settled in a manner most satisfactory and most useful to science by the proof of their impossibility.I instance the problem of perpetual motion.After seeking in vain for the construction of a perpetual motion machine,the relations were investigated which must subsist between the forces of nature if such a machine is to be impossible;2and this inverted question led to the discovery of the law of the conservation of energy,which,again,explained the impossibility of perpetual motion in the sense originally intended.This conviction of the solvability of every mathematical problem is a powerful incentive to the worker.We hear within us the perpetual call:There is the problem. Seek its solution.You canfind it by pure reason,for in mathematics there is no ignorabimus.The supply of problems in mathematics is inexhaustible,and as soon as one problem is solved numerous others come forth in its place.Permit me in the fol-lowing,tentatively as it were,to mention particular definite problems,drawn from various branches of mathematics,from the discussion of which an advancement of science may be expected.Let us look at the principles of analysis and geometry.The most suggestive and notable achievements of the last century in thisfield are,as it seems to me,the arithmetical formulation of the concept of the continuum in the works of Cauchy, Bolzano and Cantor,and the discovery of non-euclidean geometry by Gauss,Bolyai,MATHEMATICAL PROBLEMS413 and Lobachevsky.I thereforefirst direct your attention to some problems belonging to thesefields.1.Cantor’s problem of the cardinal number of the continuumTwo systems,i.e.,two assemblages of ordinary real numbers or points,are said to be(according to Cantor)equivalent or of equal cardinal number,if they can be brought into a relation to one another such that to every number of the one assemblage corresponds one and only one definite number of the other.The inves-tigations of Cantor on such assemblages of points suggest a very plausible theorem, which nevertheless,in spite of the most strenuous efforts,no one has succeeded in proving.This is the theorem:Every system of infinitely many real numbers,i.e.,every assemblage of numbers (or points),is either equivalent to the assemblage of natural integers,1,2,3,...or to the assemblage of all real numbers and therefore to the continuum,that is,to the points of a line;as regards equivalence there are,therefore,only two assemblages of numbers,the countable assemblage and the continuum.From this theorem it would follow at once that the continuum has the next cardinal number beyond that of the countable assemblage;the proof of this theorem would,therefore,form a new bridge between the countable assemblage and the continuum.Let me mention another very remarkable statement of Cantor’s which stands in the closest connection with the theorem mentioned and which,perhaps,offers the key to its proof.Any system of real numbers is said to be ordered,if for every two numbers of the system it is determined which one is the earlier and which the later, and if at the same time this determination is of such a kind that,if a is before b and b is before c,then a always comes before c.The natural arrangement of numbers of a system is defined to be that in which the smaller precedes the larger.But there are,as is easily seen,infinitely many other ways in which the numbers of a system may be arranged.If we think of a definite arrangement of numbers and select from them a particular system of these numbers,a so-called partial system or assemblage,this partial system will also prove to be ordered.Now Cantor considers a particular kind of ordered assemblage which he designates as a well ordered assemblage and which is characterized in this way,that not only in the assemblage itself but also in every partial assemblage there exists afirst number.The system of integers1,2,3,...in their natural order is evidently a well ordered assemblage.On the other hand the system of all real numbers,i.e.,the continuum in its natural order,is evidently not well ordered.For,if we think of the points of a segment of a straight line,with its initial point excluded,as our partial assemblage,it will have nofirst element.The question now arises whether the totality of all numbers may not be arranged in another manner so that every partial assemblage may have afirst element,i.e., whether the continuum cannot be considered as a well ordered assemblage—a ques-tion which Cantor thinks must be answered in the affirmative.It appears to me most desirable to obtain a direct proof of this remarkable statement of Cantor’s, perhaps by actually giving an arrangement of numbers such that in every partial system afirst number can be pointed out.414DA VID HILBERT2.The compatibility of the arithmetical axiomsWhen we are engaged in investigating the foundations of a science,we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science.The axioms so set up are at the same time the definitions of those elementary ideas;and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of afinite number of logical steps.Upon closer consideration the question arises:Whether,in any way,certain statements of single axioms depend upon one another,and whether the axioms may not therefore contain certain parts in common,which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms:To prove that they are not contradictory,that is,that afinite number of logical steps based upon them can never lead to contradictory results.In geometry,the proof of the compatibility of the axioms can be effected by constructing a suitablefield of numbers,such that analogous relations between the numbers of thisfield correspond to the geometrical axioms.Any contradiction in the deductions from the geometrical axioms must thereupon be recognizable in the arithmetic of thisfield of numbers.In this way the desired proof for the compatibility of the geometrical axioms is made to depend upon the theorem of the compatibility of the arithmetical axioms.On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.The axioms of arithmetic are essentially nothing else than the known rules of calculation,with the addition of the axiom of continuity.I recently collected them3and in so doing replaced the axiom of continuity by two simpler axioms,namely,the well-known axiom of Archimedes,and a new axiom essentially as follows:that numbers form a system of things which is capable of no further extension,as long as all the other axioms hold(axiom of completeness).I am convinced that it must be possible tofind a direct proof for the compatibility of the arithmetical axioms,by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.To show the significance of the problem from another point of view,I add the following observation:If contradictory attributes be assigned to a concept,I say, that mathematically the concept does not exist.So,for example,a real number whose square is−1does not exist mathematically.But if it can be proved that the attributes assigned to the concept can never lead to a contradiction by the application of afinite number of logical processes,I say that the mathematical existence of the concept(for example,of a number or a function which satisfies certain conditions)is thereby proved.In the case before us,where we are concerned with the axioms of real numbers in arithmetic,the proof of the compatibility of the axioms is at the same time the proof of the mathematical existence of the complete system of real numbers or of the continuum.Indeed,when the proof for the compatibility of the axioms shall be fully accomplished,the doubts which have been expressed occasionally as to the existence of the complete system of real numbers will become totally groundless.The totality of real numbers,i.e., the continuum according to the point of view just indicated,is not the totality of。
simultaneous equation method

Simultaneous Equation MethodIntroductionIn mathematics, simultaneous equations play a crucial role in solving real-world problems and modeling various phenomena. The simultaneous equation method is a powerful technique used to find solutions for a system of equations. This method involves solving multiple equations together to determine the values of unknown variables. In this article, we will explore the simultaneous equation method in detail and discuss its applications.Understanding Simultaneous EquationsDefinitionSimultaneous equations, also known as a system of equations, are a set of equations that share the same variables. The solutions of these equations simultaneously satisfy each equation in the system. The general form of simultaneous equations can be written as:a1x + b1y = c1a2x + b2y = c2Here, x and y are the variables, while a1, a2, b1, b2, c1, and c2 are constants.Types of Simultaneous EquationsSimultaneous equations can be classified into three types based on the number of solutions they have:1.Consistent Equations: These equations have a unique solution,meaning there is a specific set of values for the variables thatsatisfy all the equations in the system.2.Inconsistent Equations: This type of system has no solution. Theequations are contradictory and cannot be satisfied simultaneously.3.Dependent Equations: In this case, the system has infinitely manysolutions. The equations are dependent on each other and represent the same line or plane in geometric terms.To solve simultaneous equations, we employ various methods, with the simultaneous equation method being one of the most commonly used techniques.The Simultaneous Equation MethodThe simultaneous equation method involves manipulating and combining the given equations to eliminate one variable at a time. By eliminating one variable, we can reduce the system to a single equation with one variable, making it easier to find the solution.ProcedureThe general procedure for solving simultaneous equations using the simultaneous equation method is as follows:1.Identify the unknow n variables. Let’s assume we have n variables.2.Write down the given equations.3.Choose two equations and eliminate one variable by employingsuitable techniques such as substitution or elimination.4.Repeat step 3 until you have a single equation with one variable.5.Solve the single equation to determine the value of the variable.6.Substitute the found value back into the other equations to obtainthe values of the remaining variables.7.Verify the solution by substituting the found values into all theoriginal equations. The values should satisfy each equation.If the system is inconsistent or dependent, the simultaneous equation method will also lead to appropriate conclusions.Applications of Simultaneous Equation MethodThe simultaneous equation method finds applications in numerous fields, including:EngineeringSimultaneous equations are widely used in engineering to model and solve various problems. Engineers employ this method to determine unknown quantities in electrical circuits, structural analysis, fluid mechanics, and many other fields.EconomicsIn economics, simultaneous equations help analyze the relationship between different economic variables. These equations assist in studying market equilibrium, economic growth, and other economic phenomena.PhysicsSimultaneous equations are a fundamental tool in physics for solving complex problems involving multiple variables. They are used in areas such as classical mechanics, electromagnetism, and quantum mechanics.OptimizationThe simultaneous equation method is utilized in optimization techniques to find the optimal solution of a system subject to certain constraints. This is applicable in operations research, logistics, and resource allocation problems.ConclusionThe simultaneous equation method is an essential mathematical technique for solving systems of equations. By employing this method, we can find the values of unknown variables and understand the relationships between different equations. The applications of this method span across various fields, making it a valuable tool in problem-solving and modeling real-world situations. So, the simultaneous equation method continues to be akey topic in mathematics and its practical applications in diverse disciplines.。
化学化工英语试题及答案

化学化工英语试题及答案一、选择题(每题2分,共20分)1. Which of the following is a chemical element?A. WaterB. OxygenC. HydrogenD. Carbon答案:B, C, D2. The chemical formula for table salt is:A. NaOHB. NaClC. HClD. NaHCO3答案:B3. What is the process called when a substance changes from a solid to a liquid?A. SublimationB. VaporizationC. MeltingD. Condensation答案:C4. In the periodic table, which group contains alkali metals?A. Group 1B. Group 2C. Group 17D. Group 18答案:A5. What is the name of the process where a substance decomposes into two or more substances due to heat?A. CombustionB. OxidationC. ReductionD. Decomposition答案:D6. Which of the following is a physical property of a substance?A. ColorB. TasteC. SolubilityD. Reactivity答案:A7. What is the term for a compound that releases hydrogen ions (H+) when dissolved in water?A. BaseB. AcidC. SaltD. Neutral答案:B8. The law of conservation of mass states that in a chemical reaction:A. Mass is lostB. Mass is gainedC. Mass remains constantD. Mass can be converted into energy答案:C9. Which of the following is a type of chemical bond?A. Ionic bondB. Covalent bondC. Hydrogen bondD. All of the above答案:D10. What is the name of the process where a substance absorbs energy and changes from a liquid to a gas?A. MeltingB. VaporizationC. SublimationD. Condensation答案:B二、填空题(每题2分,共20分)1. The symbol for the element iron is ________.答案:Fe2. The pH scale ranges from ________ to ________.答案:0 to 143. A compound that produces a basic solution when dissolvedin water is called a ________.答案:base4. The smallest particle of an element that retains its chemical properties is called a ________.答案:atom5. The process of separating a mixture into its individual components is known as ________.答案:separation6. The study of the composition, structure, and properties of matter is called ________.答案:chemistry7. The process of a substance changing from a gas to a liquid is called ________.答案:condensation8. A(n) ________ reaction is a type of chemical reactionwhere two or more substances combine to form a single product. 答案:synthesis9. The volume of a gas at constant temperature and pressureis directly proportional to the number of ________.答案:moles10. The process of converting a solid directly into a gas without passing through the liquid phase is known as ________. 答案:sublimation三、简答题(每题10分,共30分)1. Explain what is meant by the term "stoichiometry" in chemistry.答案:Stoichiometry is the calculation of the relative quantities of reactants and products in a chemical reaction.It is based on the law of conservation of mass and involvesthe use of balanced chemical equations and the molar massesof substances to determine the amounts of reactants needed to produce a certain amount of product or the amounts ofproducts formed from a given amount of reactant.2. Describe the difference between a physical change and a chemical change.答案:A physical change is a change in the state or form of a substance without altering its chemical composition. Examples include melting, freezing, and boiling. A chemical change, on the other hand, involves a change in the chemical composition of a substance, resulting in the formation of new substances. Examples include combustion and rusting.3. What are the three main types of chemical bonds, and givean example of each.答案:The three main types of chemical bonds are ionic bonds, covalent bonds, and metallic bonds. An ionic bond is formed when electrons are transferred from one atom to another, resulting in the formation of oppositely charged ions. An example is the bond between sodium (Na) and chloride (Cl) in table salt (NaCl). A covalent bond is formed when two atoms share electrons, as seen in water (H2O) where hydrogen atoms share electrons with oxygen. Metallic bonds occur in metals, where a "sea" of delocalized electrons is shared among positively charged metal ions, as in sodium metal。
伊藤引理

where
u(XT , T ) − u(X0, 0) = ∆uk ,
tk <T
∆uk = u(Xk + ∆Xk, tk + ∆t) − u(Xk, tk) , ∆Xk = Xk+1 − Xk .
The Taylor expansion is
∆uk
=
∂x uk ∆Xk
+
1 2
∂x2uk
(∆Xk )2
+
3
≤ C∆t 2 .
2
up the terms (∆X)2 and adding up the terms v(Xt)∆t have the same limit as ∆t → 0. Both of these arguments use ideas from Lesson 3 and Assignment 3.
There also is an application of Borel Cantelli to show that the arguments are
∂t uk ∆t
+ O(|∆Xk|3) + O(∆t |∆Xk|) + O(∆t2) .
We sum over k. On the left side we get u(XT , t) − u(X0, 0). There are six sums on the right to consider.
常德“PEP”24年小学6年级上册J卷英语第6单元真题试卷

常德“PEP”24年小学6年级上册英语第6单元真题试卷考试时间:100分钟(总分:140)A卷考试人:_________题号一二三四五总分得分一、综合题(共计100题)1、听力题:The __________ can indicate areas of potential geological hazards.2、What do we call the part of the plant that grows below the ground?A. LeafB. StemC. RootD. Flower答案: C3、填空题:My mom encourages me to follow my __________ (热情).4、What is 2 + 2?A. 3B. 4C. 5D. 6答案:B5、听力题:The chemical formula for barium sulfate is _____.6、What do we call the process of converting a solid directly to gas?A. MeltingB. FreezingC. SublimationD. Evaporation答案:CSome _______ can be toxic to pets.8、填空题:The ________ is very friendly and enjoys company.9、What is a baby chicken called?A. DucklingB. GoslingC. ChickD. Lamb答案:C10、填空题:My pet dog is very __________. (可爱)11、听力题:The cat is ___ (chasing/hiding) from the dog.12、选择题:What is the name of the famous American author known for his short stories?A. Edgar Allan PoeB. Mark TwainC. Nathaniel HawthorneD. Ernest Hemingway13、听力题:The main component of enzymes is _____.14、What is the capital of Kenya?A. NairobiB. KampalaC. Addis AbabaD. Dar es Salaam答案:A. Nairobi15、选择题:What is the capital of Chile?A. SantiagoB. ValparaísoC. ConcepciónD. La Serena16、填空题:The sunflowers turn towards the ______.The ________ is a lovely flower.18、填空题:My favorite animal is a _______ (狐狸).19、What is the main purpose of a compass?A. Measure temperatureB. Tell timeC. Show directionD. Measure weight答案:C20、How many legs does a spider have?A. SixB. EightC. TenD. Twelve答案:B21、填空题:I planted some ______ (种子) in a pot. I hope they grow into beautiful ______ (花).22、What is the name of the famous playwright known for his tragedies?A. William ShakespeareB. Anton ChekhovC. Tennessee WilliamsD. Arthur Miller答案: A23、填空题:My dog always _______ (跟随) me on walks.24、填空题:The _______ (猴子) swings on vines.25、填空题:My favorite thing to do is ________ (玩游戏) with friends.26、听力题:The main component of essential oils is _____.27、听力题:The __________ is critical for sustaining life on earth.28、What do we call the annual celebration of a person's birth?A. AnniversaryB. HolidayC. BirthdayD. Festival答案:C29、听力题:The kitten is very ______ (curious).30、填空题:The ______ (小鱼儿) swims gracefully through the coral reefs.31、What is the name of the famous American musician known for "Counting Stars"?A. OneRepublicB. Maroon 5C. ColdplayD. Imagine Dragons答案:A32、What is the name of the famous bear from the jungle?A. BalooB. PaddingtonC. YogiD. Winnie答案:A33、听力题:In a reaction, the limiting reagent is the reactant that is completely _____ first.34、听力题:Chemical bonds can be ionic or ______.35、听力题:The dog is ________ in the grass.36、填空题:The _____ (clover) is a small plant with three leaves.37、What do you call the frozen form of water?A. SteamB. IceC. SnowD. Rain答案:BWhich animal has a pouch for carrying its babies?A. DogB. KangarooC. CatD. Elephant39、What is the name of the longest river in Africa?A. NileB. CongoC. ZambeziD. Niger答案:A40、Which of these is a vegetable?A. AppleB. CarrotC. BananaD. Grape41、What is the smallest continent?A. AfricaB. AsiaC. AustraliaD. Europe答案: C42、听力题:The sun is shining ________ today.43、Which instrument has keys and is played by pressing them?A. GuitarB. DrumsC. PianoD. Violin44、填空题:The ________ was a major turning point in the American Civil War.45、What is the capital of Afghanistan?A. KabulB. HeratC. KandaharD. Mazar-i-SharifI saw a ________ dancing in the yard.47、听力题:The ice cream is _____ melting. (slowly)48、What do we call the act of expressing opinions in writing?A. CommentaryB. EditorialC. ReviewD. All of the Above答案:D49、选择题:What do we call the place where we keep our money?A. BankB. StoreC. LibraryD. School50、听力题:The stars are ________ at night.51、填空题:In my opinion, being _______ (形容词) is a valuable trait in life. It helps us connect with others.52、填空题:The _____ (植物标本) helps preserve plant species for study.53、选择题:How many continents are there in the world?A. 5B. 6C. 7D. 854、What do we call the process of a seed developing into a new plant?A. GerminationB. PhotosynthesisC. PollinationD. Fertilization答案:A55、听力题:She is _____ (cooking) breakfast.What do you use to see in the dark?A. CandleB. FlashlightC. LampD. Mirror57、填空题:The dolphin is very _______ (聪明) and playful.58、听力题:The ________ (mountain) is difficult to climb.59、What do we call a plant that grows in water?A. TerrestrialB. AquaticC. MarineD. Hydroponic答案: B60、What do you call the largest land animal?A. RhinoB. ElephantC. HippoD. Giraffe答案:B61、Which shape has four equal sides?A. RectangleB. TriangleC. SquareD. Circle答案:C62、填空题:The zebra has black and white _______ (条纹).63、听力题:The _______ of a sound can be influenced by the environment.64、填空题:I hope to discover new passions as I continue to _______ (成长) and learn.65、填空题:The sun is ________ (明亮) today.I can ______ my homework. (finish)67、What do we call the small, sweet fruit that grows on a vine?A. RaspberryB. StrawberryC. GrapeD. Blueberry答案:C68、填空题:The garden has many _______ that blossom beautifully all year long.69、听力题:The process of electroplating deposits a layer of ______.70、填空题:The _____ (植物生命周期) includes several stages.71、填空题:I am learning to speak ________ (法语) because I want to visit ________ (法国).72、 gather ______ (坚果) for winter. 填空题:Squirrel73、What do you call a baby lion?A. CubB. PupC. CalfD. Kit答案:A74、填空题:A parakeet can have many different ______ (颜色).75、What is the name of the famous mountain range in South America?A. AndesB. RockiesC. HimalayasD. Alps答案:A76、听力题:The chemical formula for cyclopentane is ______.77、What do we call the study of the interaction between organisms and their environment?A. EcologyB. BiologyC. ZoologyD. Botany78、填空题:The ________ (城市发展) requires careful planning.79、Where do fish live?A. TreesB. MountainsC. WaterD. Grass答案:C80、What is the opposite of empty?A. FullB. HeavyC. LightD. Open答案:A81、听力题:The capital of Pakistan is _______.82、听力题:The flowers bloom in the _____ (spring).83、填空题:The __________ (历史学家) study past events and societies.84、n Conquest changed the course of ________ (英国历史). 填空题:The Nort85、填空题:The ancient Romans built _____ to celebrate their victories.86、听力题:The chemical symbol for tellurium is _______.87、What is the name of the famous artist known for his abstract paintings?A. PicassoB. Van GoghC. Da VinciD. Monet答案:A88、填空题:My _____ (邻居) is very friendly.89、听力题:She is _____ (playing/reading) a book right now.90、What is the opposite of tall?A. ShortB. LongC. HighD. Low答案:A91、填空题:The _______ (蝙蝠) hangs upside down.92、填空题:The tarantula is a type of ________________ (蜘蛛).93、What do we call a large body of freshwater?A. OceanB. SeaC. LakeD. River答案:C94、听力题:The Amazon __________ is the longest river in South America.95、填空题:The _____ (小鸟) made a nest in our backyard. It has three eggs.小鸟在我们后院筑了一个巢,它有三个蛋。
四川大学化工考研884复试面试英语题库翻译原文

1.When a fluid flows through a duct or over a surface, the velocity over a place at right angles tothe stream is not normally uniform. The variation of velocity can be shown by the use ofstreamlines which are lines so drawn that the velocity vector is always tangential to them. The flow rate between any two streamlines is always the same. Constant velocity over a cross-section is shown by equidistant streamlines and increase in velocity by closer spacing ofstreamlines.2.It is found the heat transfer rate per unit area q is dependent on those physical propertieswhich affect flow pattern. The thermal properties of the fluid and the velocity of flow the fluid over the surface, the temperature difference t and a factor deternming the natural circulation effect cause by expansion of the fluid on heating.3.The oil distilled in a very large steel tower, the technical names of which is the fractionatingtower. The tower is thirty to fifty meters tall and its diameter is one to three meters. It isdivided into chambers, each of which contains a layer of trays. There are holes in the trays. The chanbers are at different heights, and temperature at each height is different.4. A group of operation for separating the components of mixture is based on the transfer ofmaterial from one homogeneous phase to anther. Unlike purely mechanical separation, these methods utilize differences in vapor pressure or solubility, not density or particle size. The driving force for transfer is a conoantration differce to or a concentration gradient, much as a temperature difference or a temperature gradient provides the driving force heat transfer.5.The three most important characteristics of an individual particle are its composition, its sizeand its shape. Composition determine such properties as density and conductivity, provided that the particle is completely uniform. A particle shape may be regular such as spherical or cubic, or it may be irregular as, for example, with a piece of broken glass.6.homogeneous catalysis is the industrical oxo process for manufacturing normal iosobuty(异丁醛)。
1 Introduction to the Galois Theory of Linear Differential Equations

1.2 What is a Linear Differential Equation? I will develop an algebraic setting for the study of linear differential equations. Although there are many interesting questions concerning differential equations in characteristic p [MvdP03a, MvdP03b, dP95, dP96], we will restrict ourselves throughout this paper, without further mention, to fields of characteristic 0. I begin with some basic definitions.
宁波2024年05版小学六年级下册T卷英语上册试卷(含答案)

宁波2024年05版小学六年级下册英语上册试卷(含答案)考试时间:80分钟(总分:120)A卷考试人:_________题号一二三四五总分得分一、综合题(共计100题)1、填空题:I can ______ (定期) reflect on my progress.2、填空题:The turtle swims slowly in the _______ (水).3、填空题:A __________ (催化循环) enhances reaction efficiency in chemical processing.4、What do you call a person who plays a musical instrument?A. MusicianB. SingerC. ComposerD. Conductor答案:A5、听力题:The chemical formula for calcium hydroxide is _______.6、How do you say "hand" in Spanish?A. ManoB. MainC. HandD. Maño7、听力题:The main gas in the air we breathe is _____.8、填空题:The ________ was a significant period in the evolution of human rights.The ________ (jacket) keeps me warm.10、填空题:Certain plants can ______ (抵抗) pests naturally.11、听力题:A _______ can be a beautiful centerpiece for a table.12、听力题:We eat ______ (snacks) during recess.13、听力题:A liquid that can dissolve a solute is called a _______.14、What is 3 + 5?A. 6B. 7C. 8D. 9答案:C15、填空题:Julius Caesar was a famous Roman _______. (统治者)16、听力题:A solution that contains the maximum amount of solute is _____ (saturated).17、What is the name of the famous ancient city in Iraq?A. BabylonB. NinevehC. UrD. All of the above18、听力题:The Magna Carta was signed in _______.19、What do we call the process of a caterpillar becoming a butterfly?A. MetamorphosisB. TransformationC. EvolutionD. Development答案:A20、填空题:I can ______ (表达) my thoughts clearly.The chemical symbol for rubidium is __________.22、听力题:The study of chemicals and their reactions is known as _______.23、What is the name of the famous explorer who sailed across the ocean in 1492?A. Vasco da GamaB. Ferdinand MagellanC. Christopher ColumbusD. Marco Polo答案:C24、What is the name of the famous ancient city in Jordan?A. PetraB. BabylonC. AthensD. Rome答案:A25、填空题:I want to grow _____ (蔬菜) this year.26、填空题:We can play with a ________ outside.27、选择题:What do you call the part of the plant that absorbs water?A. LeafB. StemC. RootD. Flower28、听力题:We enjoy going to the ___. (beach) every summer.29、听力题:A mixture that contains two or more phases is called a ______.30、What is the capital of Canada?A. TorontoB. OttawaC. VancouverD. MontrealI like to _______ (与朋友一起)去健身房。
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The solution conformation of triarylmethyl radicalsM.K.Bowman a,*,C.Mailer b ,H.J.Halpern baStructural Biology and Microimaging,Pacific Northwest National Laboratory,Richland,WA 99352-0999,USA bDepartment of Radiation and Cellular Oncology,Center for EPR Imaging In Vivo Physiology,Chicago,IL 60637,USAReceived 8June 2004;revised 21October 2004Available online 21November 2004AbstractHyperfine coupling tensors to 1H,2H,and natural abundance 13C were measured using X-band pulsed electron nuclear double resonance (ENDOR)spectroscopy for two triarylmethyl (trityl)radicals used in electron paramagnetic resonance imaging and oximetry:methyl tris(8-carboxy-2,2,6,6-tetramethyl-benzo[1,2d :4,5-d 0]bis(1,3)dithiol-4-yl)and methyl tris(8-carboxy-2,2,6,6-tetra-methyl(-d 3)-benzo[1,2d :4,5-d 0]bis(1,3)dithiol-4-yl).Quantum chemical calculations using density functional theory predict a structure that reproduces the experimentally determined hyperfine tensors.The radicals are propeller-shaped with the three aryl rings nearly mutually orthogonal.The central carbon atom carrying most of the unpaired electron spin density is surrounded by the sulfur atoms in the radical and is completely shielded from solvent.This structure explains features of the electron spin relaxation of these radicals and suggests ways in which the radicals can be chemically modified to improve their characteristics for imaging and oximetry.Ó2004Elsevier Inc.All rights reserved.Keywords:Trityl radical;ENDOR;Pulsed EPR1.IntroductionThe trityl family of triarylmethyl radicals has recently come into use in a number of magnetic resonance appli-cations because of their favorable relaxation and spec-tral properties.These radicals,as typified by I ,(see Scheme 1),methyl tris(8-carboxy-2,2,6,6-tetramethyl-benzo[1,2d :4,5-d 0]bis(1,3)dithiol-4-yl),have a single,nar-row electron paramagnetic resonance (EPR)line even at high fields and have long relaxation times in fluid solu-tion,which make them useful electron spins for some classes of dynamic nuclear polarization measurements [1–4].They also have good chemical stability and are not generally broadened by interaction with proteins and other biological molecules,making them attractive probes for biological EPR imaging and EPR oximetry applications [5–11].The trityl family has two major sites for chemical var-iation that can be used to adjust their chemical or spec-troscopic properties for specific uses:the para position on the phenyl rings or the methylene bridge in the five-membered rings.An improved knowledge of the electronic and physical structure of the trityl radicals in solution would help guide development of improved trityl radicals and would aid in understanding the re-sponse of current trityl radicals to molecular oxygen and to other paramagnetic species,and the transfer of polarization to solvent nuclei in liquid phase dynamic nuclear polarization (DNP).The chemical structure of I is usually drawn as if the molecules were planar.How-ever,the substituted phenyl rings are far too bulky to al-low three coplanar phenyl rings around the central carbon.The rings must twist substantially out of plane,decreasing conjugation of the p -electron system.The conformation adopted by I in solution will have a large effect on the unpaired spin density distribution and on the ability of dissolved species such as molecular oxygen and paramagnetic metal ions to interact with the/locate/jmrJournal of Magnetic Resonance 172(2005)254–267*Corresponding author.Fax:+15093762303.E-mail address:michael.bowman@ (M.K.Bowman).unpaired electron spin and to cause relaxation or broad-ening of the EPR spectral lines.In addition,fluctuationsin the conformation and transient hydrogen bondsformed with solvent have the potential to modulatethe unpaired spin distribution and contribute to intrinsicEPR linewidth.The conformation of free radicals in solution is oftenprobed experimentally using proton hyperfine interac-tions to map out the unpaired spin density distributionin the radical.The trityl radicals of most interest forEPR oximetry are purposely designed to minimize oreliminate hyperfine couplings to protons,leaving fewsites on I that can be probed.An alternative probe ofthe unpaired electron spin density is provided by the car-bons which form most of I.This requires measurementof the hyperfine couplings of13C,the only stable carbonisotope with a nuclear spin,either by EPR or by electronnuclear double resonance(ENDOR).Although the nat-ural abundance of the13C isotope is only1.11%,thereare two factors that make it practical to measure the 13C hyperfine interactions of the trityl radical in natural abundance samples.Thefirst factor is that I has manycarbon atoms in it arranged in a few chemically equiva-lent groups with degeneracies of3,6or12.Thus,a sub-stantial fraction,about36%,of radicals in the sample isnaturallyÔlabeledÕwith13C.The second favorable factorfor the measurement of13C ENDOR is the fact that the 13C hyperfine splittings for several of the carbon posi-tions are much larger than the EPR linewidth of theÔun-labeledÕradical.Measurements can be cleanly made of only the naturallyÔlabeledÕI without interference from ÔunlabeledÕI.These factors allow a detailed experimental characterization of the electronic structure of I based on both the proton and carbon hyperfine interactions.We report here on an EPR,ENDOR,and quantum chemical study of the simple,symmetric trityl radical I.The results indicate a single structure in both liquid and frozen solutions in which the three phenyl rings are twisted51°out of plane so that the rings are almost mutually perpendicular,making an angle of84.5°to each other.This conformation shields the central carbon atom bearing the majority of the unpaired electron spin density from the solvent and may be responsible for ren-dering I relatively insensitive to most interactions with its environment.2.Experimental2.1.EPR and ENDORCW EPR spectra were obtained with a spectrometer operating at255MHz.The bridge has a reference arm and uses quadrature RF detection[12].One channel de-tects the modulated RF EPR signal that is passed to the phase sensitive detector,and the other is used in a DC Pound automatic frequency control(AFC)system[13]. One required extra feature of the bridge is a correction for frequency drift of the EPR resonator.The resonator drifts in frequency by tens of kHz in the course of a mea-surement.Because resonance position changes by0.1l T for a2.8kHz frequency change,such a slow frequency drift would cause broadening in the recorded EPR spec-tra and would prevent accurate analysis of the hyperfine splittings.The frequency drift correction operates by measuring the moment-to-moment RF frequency of the250MHz source oscillator as it is adjusted by the AFC.These frequency data are used to correct for the shifts in resonance position.A one loop-one gap resonator with a diameter of 19mm and length15mm was used.The computer-con-trolled DC magnetfield has a high absolute stability of 0.05l T which represents a fractional stability of6ppm relative to the center magneticfield of only9mT.The $5kHz Zeeman modulation is produced by a Helm-holtz coil pair with less than0.02l T(20ppm/cm)vari-ation over a1cm diameter region along the resonator axis(RF B1field direction).For accurate simulations of the spectra careful calibration of the modulation amplitude was done.Pulsed EPR measurements were performed at X-band on a Bruker ESP380E EPR spectrometer with DICE ENDOR accessory and a Flexline CFG-935he-liumflow cryostat and Flexline ENDOR resonator. Microwave frequency was measured with an EIP575B microwave counter.A20W ENI320L RF amplifier was used for Mims ENDOR measurements and a 500W ENI A-500amplifier was used forcoherencetransfer ENDOR of protons based on the Mims EN-DOR sequence and using time proportional phase incre-mentation(TPPI)as described by Ho¨fer[14]to separate the higher multiple quantum orders.Pulsed EPR mea-surements of frozen solutions were made with the expec-tation that the hyperfine anisotropy could provide additional checks on the accuracy of the quantum chem-ical calculations.The echo induced EPR spectrum[15]was measured with a p/2–s–p–s-detect sequence where p/2and p indi-cate the turning angles of the respective microwave pulses separated by a time s.A simple phase cycle was used to remove unwanted signals and to correct baseline offset.The phase of thefirst pulse was set to0°and180°on successive scans that were then subtracted from each other.The Mims ENDOR spectrum[16]was measured with the sequence p/2–s–p/2–RF(T)–p/2–s-detect.The RF frequency was swept and the RF pulse length is T. The phase of the second microwave pulse was set to0°and180°on successive scans that were then subtracted from each other.The electron nuclear quadruple resonance(ENQOR) spectrum[17]was measured with a variation of the Mims ENDOR sequence,that is,p/2–s–p/2–RF1–RF2–p–RF20–RF10–p/2–s-detect.Phase cycling of the second microwave pulse was used as for Mims ENDOR. The RF pulses have equal length.RF1and RF10indi-cate the same frequency which is different from that of RF2and RF20.The amplitude of one of the RF pulses in each pair RF n and RF n0is always zero.Additional phase cycling was superimposed on that of the micro-wave pulses depending on whether RF1and RF2have the same or different amplitudes.For coherence transfer ENDOR[14,16],the single RF pulse was replaced by a pair of high-power RF pulses with turning angles of p/2for the protons.The frequency of the RF pulses was the same but their rela-tive phase was incremented with the time between the RF pulses at a rate of1.44°/ns.The total time between the second and third microwave pulses was kept con-stant and the same phase cycling used as for Mims EN-DOR.First RF pulse produces proton coherences which then evolve and are converted by the second RF pulse back to population differences for detection by the elec-tron spin echo.The TPPI produces a phase modulation that is proportional to that multiple quantum order of the proton coherence.Fourier transformation of such a coherence transfer ENDOR signal produces an EN-DOR spectrum with the different multiple quantum or-ders offset in frequency.The Davies ENDOR spectrum[16]was measured with the sequence p–RF(T)–p/2–s–p–s-detect.The RF frequency was swept and the RF pulse length is T. The phase of the second microwave pulse was set to0°and180°on successive scans that were then subtracted from each other.In the pulsed ENDOR measurements,the pulse widths,delays,amplitudes,and the magneticfield were adjusted to emphasize a particular feature in the EN-DOR spectrum.That feature was then used to determine one of the many hyperfine tensor elements.The hyperfine tensors were constructed and assigned to specific posi-tions in I on the basis of the tensors obtained from the quantum chemical calculations and the experimental EPR spectra.Precisefitting of experimental ENDOR spectra is not practical because it requires more informa-tion concerning the range of conformations of I present in the frozen solution samples and the consequent range of hyperfine tensor elements than is available.putationalCalculations of the hyperfine couplings were per-formed using the Gaussian98suite of programs[18]. The calculations were carried out on an8CPU Silicon Graphics Origin2000computer.The structure of I was optimized without the use of symmetry using density functional theory with the B3LYP functional,first with a6-21G basis set and then with the6-31G basis set. The optimized6-31G structure was then used for calcu-lations with other basis sets.The structure of I was mod-eled starting with the carboxyl groups ionized and three H3O+in order to have a neutral system and to avoid hav-ing naked counter ions.The optimized structures have protonated carboxyl groups that are hydrogen bonded to a water molecule.The hyperfine tensors were calcu-lated using the EPR Properties keyword of Gaussian.ENDOR spectra were simulated using the program EPR-NMR(Department of Chemistry,University of Saskatchewan,Saskatoon,SK).3.Results3.1.Liquid phase CW EPRThe CW EPR spectrum of I is dominated by a single, narrow line.This is one of its attractions for magnetic resonance applications.However,there are a number of weakflanking lines from those radicals that contain a13C nucleus(natural abundance1.11%).With40car-bons in I,a random distribution would give64%of the trityl radicals with no13C,28%with one,6.3%with 2,and0.9%with3.Each equivalent carbon position produces a pair of lines with equal intensity split approximately symmetrically about the central EPR line and an intensity directly related to its degeneracy in I. The splitting directly gives the isotropic hyperfine inter-action after correction for second-order shifts that are noticeable in the255MHz EPR measurements.The line shape of the central line was carefullyfit using an anal-ysis program that completely corrects for Zeeman mod-256M.K.Bowman et al./Journal of Magnetic Resonance172(2005)254–267ulation frequency and amplitude and also models the re-solved and unresolved hyperfine patterns expected from the structure(Table1)[19].Equivalent results,Table1, were obtained in room temperature CW EPR measure-ments at255MHz on1mM deuterated I in water and at X-band on1.5mM deuterated I in perdeuterated meth-anol.The degeneracies listed in Table1are derived from the relative EPR line intensities.There appears to be suf-ficient conformational mobility of I to make the poten-tially inequivalent sets of methyl groups magnetically equivalent in solution.The13C hyperfine lines from central methyl carbon were not measured in the255MHz EPR spectra because narrow,high-resolutionfield sweeps were used to focus on the smaller splittings of the remaining carbons. Although13C hyperfine splittings were readily observed, there was no trace of hyperfine splittings from33S which has a natural abundance of0.75%and I=3/2.There should be two sets of33S hyperfine splittings from the 2,6-and the3,5-phenyl positions,respectively,each with a degeneracy of6.No such splittings were observed, most likely because the large quadrupole moment of 33S cause rapid nuclear relaxation and broaden the hyperfine lines beyond detection.3.2.Solid-state EPRThe CW EPR spectrum of frozen solutions of I are also dominated by a single,narrow line.The sharp13C hyperfine lines are replaced by the full hyperfine powder patterns for each13C position in the radical.Each one has the form of a high-field13C powder ENDOR spec-trum centered near the EPR frequency of I.The relative intensity of each hyperfine powder pattern is determined by the degeneracy of that carbon position in the radical and by the relative spectral dispersion or hyperfine anisotropy.Thus,the more anisotropic the coupling the weaker its intensity in the spectrum.Only a few of the larger hyperfine couplings were well resolved in either CW EPR or ESE detected EPR spectra,Fig.1,giving values in Table2.The spectra were consistent with roughly axial hyperfine tensors.There is one well-re-Table1Isotropic hyperfine couplings of I from liquid phase CW EPR measurements and DFT calculations with the6-311G**basis setNucleus A iso(MHz)Degeneracy Assignment 255MHz X-band Calculated13C67.161.601Central carbon13C32.0931.7À32.51±0.0431-Phenyl13C25.3025.425.66±0.1062,6-Phenyl13C9.41— 5.90±0.0434-Phenyl13C 6.60—À5.07±0.0563,5-Phenyl13C 3.57—À3.42±0.033Carboxyl13C a0.494±0.036—À0.51±0.016Methylene bridge 13C a0.066±0.001—0.083±0.0030.031±0.00112Methyl carbons1H a0.033±0.001(0.0062H)—Average:À0.050(1H)36Methyl protons1H——À0.11±0.0033Carboxyl protons b 33S——0.48±0.0362,6-Phenyl33S——À0.42±0.0163,5-PhenylEPR measurements determine only the absolute value of the hyperfine coupling constant while the DFT calculations also give the sign of the coupling.a Splittings were not resolved and couplings were obtained fromfitting the line shape of the central line.The values of the hyperfine couplings are highly correlated and the estimated uncertainties are more relevant to the second moment than for the individual couplings.The uncertainty in the resolved hyperfine couplings is a fraction of the$0.076MHz CW EPR linewidth of the deuterated radical.b Protons hydrogen bonded to the carbonyl oxygen have smaller,À0.01MHz,calculated isotropic hyperfinecouplings.solved shoulder in both the CW and ESE detected spec-tra that has small anisotropy and rather weak intensity considering its small anisotropy.The calculated isotropic hyperfine coupling of55.1MHz is quite different from those measured in the liquid phase and is consequently assigned to a trace impurity in the sample that is notice-able only because of its large,nearly isotropic coupling.Overlap,from the wings of the intense EPR line from radicals that contain no13C,obscures the region ex-pected to contain the features from the small13C hyper-fine tensor components.The largest splitting observed, A i=160.1MHz,must correspond to the largest isotro-pic coupling of67.1MHz because all other features in the solid-state EPR spectrum have splittings less than this isotropic coupling.The other two principal values of this hyperfine tensor would average20.6MHz and lie buried under the much more intense central portion of the solid-state EPR spectrum.3.3.Pulsed ENDOR3.3.1.Hydrogen Mims ENDORThe proton Mims ENDOR spectrum of I is narrow and contains partially resolved features that shift notice-ably as the RF pulse amplitude and length are varied. However,the shape of the Mims ENDOR spectrum does not depend on temperature.Spectra measured be-tween4.5and60K can be superimposed and show no indication that methyl group dynamics alters the spec-trum over this temperature range.In a typical spectrum,Fig.2,there is a set of broad lobes split by$800kHz and a pair of sharper peaks split by$300kHz.These lines may result either from two overlapping powder patterns with similar principal val-ues from the pair of slightly inequivalent methyl groups on eachfive-membered ring,or from a distribution of hyperfine couplings from methyl groups in slightly dif-ferent conformations.However,the variation in split-tings with RF pulse parameters strongly suggest that the spectra are modified by coherence splittings[20,21] or multiple quantum transient nutation effects[22].Each trityl radical contains36similar,if not exactly equiva-lent,protons,so that excitation of proton multiple quantum coherences by the72l s RF pulses used in Fig.2is a distinct possibility.An apparent pair of peaks with a splitting of $100kHz is actually a single featureless peak from residual protons in the solvent with aÔpolarization holeÕburned in the center.This hole is a well-characterized feature of Mims ENDOR spectra.Unlike real hyperfine splittings,the apparent splitting of this peak strongly de-pends on experimental parameters that determine the depth and width of theÔpolarization hole.ÕThere is also a pair of wings that extend out beyond the lobes split by $750kHz mentioned earlier and are the tails of the sol-vent or matrix ENDOR line.This is clearly seen in a sample of perdeuterated I in H2O/glycerol,Fig.2,inset. The wings of the matrix line are enhanced by theÔs-sup-pression holesÕorÔblind spotsÕcharacteristic of Mims ENDOR[16,23].The ENDOR intensity is modulated by a factor roughly proportional to(1Àcos(2p A s)) where A is the hyperfine splitting and s is the separation between thefirst two microwave pulses in the Mims EN-DOR pulse sequence.In Fig.2,s=312ns and the rele-vant blind spots or holes appear for|A|=0or 3.20MHz,that is,for protons at m p and m p±1.6MHz,Table2Hyperfine tensors of I from solid-state EPR and ENDOR measurements and from DFT calculationsNucleus CW EPR hyperfinecoupling(MHz)ENDOR hyperfinecoupling(MHz)Gaussian-98hyperfine coupling(MHz)A i A^A i A^A1A2,A3Central13C160.1(20.6)——176.7 4.1,4.11-Phenyl13C39.629.540.327.8À42.1À28.5,À26.92,6-Phenyl13C37.717.036.718.037.020.8,19.14-Phenyl13C15.6(6.3)12.77.616.80.9,0.03,5-Phenyl13C——10.5 5.4À10.2À3.1,À1.9Carboxyl13C————À3.7À3.3,À3.2Bridging13C——À0.4aÀ0.8aÀ0.18À0.66,À0.69Methyl13C——0.160.00.240.26À0.05,À0.05,0.01,À0.05 Methyl1H—— 1.0bÀ0.5b0.4–1.1À0.2toÀ0.6Carboxyl1H c————0.62À0.45,À0.50Impurity64.450.5————The calculated values for the methyl protons show the range of values while the methyl13C couplings for axial and equatorial groups are reported on different lines.The experimental spectra did not allow resolution of the A^.The CW EPR values reported in parentheses were calculated from the measured A i and A iso but were not directly observed in the spectra.a The sign of the hyperfine couplings were determined by ENQOR relative to the methyl1H couplings which were assumed to be dominated by a nearly point–dipole interaction with spin density on the central carbon atom.b Extrema from features in the coherence transfer ENDOR spectrum.c The anisotropic part of the calculated hyperfine tensor from protons hydrogen bonded to the carbonyl oxygen is slightly smaller than for the –OH proton.258M.K.Bowman et al./Journal of Magnetic Resonance172(2005)254–267while the full intensity is observed only at m p±0.8MHz which is near the two edges of the spectrum in Fig.2. The practical consequence is the partial suppression of that part of the spectrum with small couplings and the apparentÔemphasisÕof the wings near m p±0.7MHz by comparison.The extreme ends of this matrix ENDOR line corre-spond to a hyperfine splitting of1.4MHz.If this is as-signed to the parallel principal value of a purely dipolar interaction between the closest solvent proton and the unpaired electron in the point–dipole approxi-mation,it would indicate a distance of closest approach of0.48nm.This would suggest steric shielding of the un-paired electron spin from the solvent in I.The deuterium ENDOR spectrum of the perdeuter-ated I in a protiated solvent,Fig.2,is much broader than would be expected simply by scaling the proton hyperfine couplings.The deuteron has a much smaller magnetic moment than the proton,which reduces the nuclear Zeeman frequency and the hyperfine couplings to15.35%those of the protons.Yet the scaled deuterium ENDOR spectrum is more than twice the width of the proton ENDOR spectrum and the lineshapes are quite different.The reason for this difference in spectra is that the deuteron has a nuclear spin I=1and consequently a nuclear quadrupole moment while the proton with I=1/2does not.The deuterium ENDOR spectrum is $200kHz wider than would be expected from scaling the proton ENDOR spectrum of I and indicates that the quadrupole splitting is characteristic of an immobi-lized C–D bond rather than a freely rotating methyl group.The deuterium ENDOR spectrum is approximately 10%of the intensity of the proton ENDOR spectrum. This decrease is caused in large part by theÔs-suppres-sion holeÕorÔblind spotÕfor very small hyperfine split-tings.The product A s,even with the slightly larger value of used to measure the deuterium ENDOR spec-trum,is only a quarter that for the corresponding splitting in the proton ENDOR spectrum,thus pro-ducing a substantial decrease in the intensity factor (1Àcos(2p A s)).The carboxyl groups are free to exchange with the solvent and will have the same isotopic composition as the solvent.No1H or2H ENDOR peaks were observed that could be distinguished from the solvent and as-signed to protons on the carboxyl groups of the trityl radical.Because of their low degeneracy and expected strong overlap with the methyl protons and solvent pro-tons,the carboxyl protons should not be easily observed.3.3.2.Proton coherence transfer ENDORThe best proton ENDOR spectrum was obtained using coherence transfer ENDOR with TPPI to separate out multiple quantum contributions.The p/2RF pulses were3l s wide which is too wide to completely excite the proton ENDOR spectrum,particularly in the higher multiple quantum orders.Therefore,a two-dimensional (2D)measurement was made in which the second dimension is the RF frequency.The coherence transfer ENDOR spectrum,Fig.3,was obtained as a skyline projection of the2D spectrum.The TPPI results in the single-quantum ENDOR spectrum appearing centered at4MHz,the double-quantum ENDOR spectrum at 8MHz,and the triple-quantum ENDOR spectrum at 12MHz as marked by the arrows.The spectrum shows spurious responses near11.25and15.25MHz appar-ently from coherent noise of the pulse programmer.The single-quantum ENDOR spectrum is similar to, but broader than,the Mims ENDOR spectrum.The outer lobes of the spectrum indicate a maximum hyper-fine splitting of about1MHz and the polarization hole produces a pair of inner lobes split by260kHz.The double-quantum ENDOR peak is weak and ratherfea-tureless and the triple-quantum peak is missing entirely. The absence of a triple-quantum ENDOR signal indi-cates that the protons in the methyl groups are non-de-generate and hence non-rotating.In this case,the single quantum and the Mims ENDOR spectra would be the superposition of spectra from each methyl conformer in the sample and the proton ENDOR spectrum couldnot be interpreted in terms of one or two hyperfine ten-sors for the methyl protons.3.3.3.13C ENDORThe ENDOR spectrum measured at the center of theintense EPR line,Fig.4,upper trace,shows weak EN-DOR signals near the13C nuclear Zeeman frequencycorresponding to small hyperfine couplings of less than0.9MHz.The relative signs of hyperfine couplings canbe determined using ENQOR spectroscopy.The EN-QOR spectrum,Fig.4,lower trace,reveals a correlationbetween the13C ENDOR and the proton ENDOR.Here the RF pump frequency of14.452MHz corre-sponds to the low-frequency lobe of the methyl protons.The13C ENDOR peak near 3.4MHz has negativeamplitude in this ENQOR spectrum while its partnernear 4.0MHz has positive amplitude.The ENQORspectrum inverts when the high-frequency lobe at14.996MHz is pumped,indicating that the$0.8MHz 13C hyperfine coupling and the largest hyperfine splitting of the methyl protons have opposite signs.The aniso-tropic part of the methyl proton hyperfine interactionis dominated by dipolar interaction with the large un-paired electron spin density at the central carbon atomas discussed later.Consequently,the largest hyperfinesplitting of the methyl protons is positive and the13Chyperfine coupling in Fig.4of$0.8MHz negative.Despite observations of large13C hyperfine couplings in CW EPR spectra,no large couplings were observed in ENDOR spectra measured at the center of the intense EPR line of I.This result should not be a surprise be-cause the vast majority of radicals containing13C are only singlyÔlabeled.ÕConsequently,the radicals that are resonant at the center of the EPR spectrum have only a small or no13C hyperfine splitting.The radicals with the large couplings are split out away from the cen-ter of the EPR spectrum so that the ENDOR spectrum depends on the position in the EPR spectrum at which it is measured.This phenomenon is well known in the fields of ENDOR and ESEEM asÔorientation selectionÕ[24].That is,measurements made in any part of an anisotropic EPR spectrum select out a subset of radicals whose orientations,hyperfine couplings,isotopic com-position,etc.satisfy the resonance condition for that specific EPRfield/frequency combination.Thus in order to observe the large hyperfine couplings in the ENDOR spectrum it is necessary to make the measurement in a part of the EPR spectrum where the observed radicals have a large hyperfine coupling.This is equally true for CW ENDOR whether in the liquid or frozen state. For the larger13C hyperfine couplings,solution EN-DOR has limited value because the hyperfine coupling must be determined in order tofind thefield position for the ENDORmeasurement.。