How to use equations in a sentence

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Sentence Examples The genetic programming example above yields large equations that become impractical or too biased to the past. Recycled Malthusianism has long been used as an excuse for simplifying complex social problems into tidy biological equations. It revels in those multiple personal equations a bhakta can draw up with his god, as parent, friend or beloved. The walls were papered with graphs, equations, posters of graph, calculator schematics, and some diagrams I couldn't even make out. The digital currency is mined using specialized super computers which discover them by solving highly complex mathematical equations. It is generally regarded as a study of the iteration of maps, of time evolution of differential equations, and of group actions on manifolds. Or should they legitimately be applied only to continuous curves susceptible of being expressed by algebraic or transcendental equations? However he made many contributions to number theory and to the theory of equations. Goldbach also studied infinite sums, the theory of curves and the theory of equations. His mathematical work covered Cartesian geometry and the theory of equations. Use a black marker to outline the shape and to write scientific or mathematical equations on the bag. To calculate the concentrations of all microspecies at any pH we need to solve the following system of equations. In addition to his work on set theory, Cohen has worked on differential equations and harmonic analysis. We use equations derived from the blob theory to calculate the blob size and the theta temperature of star polymers. Bent over their computers, thirty savants were absorbed in equations of the ninety-fifth degree. Sometimes this general theory is contemplated or worked out in terms of mathematical equations. He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations. The first phase of the process is to search for equations that satisfy certain mathematical properties. The mathematical equations of Newton or Einstein make no explicit mention of direction in space. He did important work on differential equations and applications to mathematical physics. Show More Sentences He solved cubic equations by extending an algorithm for finding cube roots. However he also worked on differential equations, matrices and other topics in algebra, continued fractions, geometry and number theory. In his lectures Sylow explained Abel's and Galois's work on algebraic equations. A system of three equations was derived using distances from three adjacent line segments drawn between vertices. This paper contained equations which Laplace stated were important in mechanics and physical astronomy. In the field of mathematics he worked on probability, recurring decimals and the theory of equations. Weil's work on polynomial equations led to questions on what properties of a geometric object can be determined purely algebraically. Jain gives us interesting figures of these family structures that show the mesh of power equations at a glance. Omar Khayyam gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. Using this approximation in Maxwell's equations we see that light travels along null geodesics. He began to study the Galois theory of equations and from there he was led to study composition series of groups. From my mother's end, I could hear her gabbing about something to do with math equations. In 1823 Abel published papers on functional equations and integrals in a new scientific journal started up by Hansteen. You can always check by calculating the fugacity for yourself, using the equations in Pitzer and Sterner. Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work. Like Mazur, Pesic starts at the beginning with the irrationality of square roots and proceeds to the solution of cubic and quartic equations. Time as the fourth dimension is basically just a mathematical gimmick to make some relativity equations easier to work with. His work led him to study the acceleration of convergence of Fourier series and the approximate solutions to differential equations. We shall comment further below on this important work, covering topics in the theory of equations, number theory and geometry. Einstein introduced the cosmological constant to his equations of general relativity because he believed the universe was static. Show More Sentences Thus, equations for isotropic mixtures of phase domains are not applicable. Also in 1887 Voigt first wrote down the transformations and showed that certain equations were invariant under these transformations. Hippocrates' book also included geometrical solutions to quadratic equations and included early methods of integration. The only way to compute complete trajectories was to integrate the differential equations numerically. Also in this letter he asked his advice on integrating equations he had obtained from theoretically modelling bombs dropping. His first mathematical research was on analysis, in particular concentrating on integrals and solutions of differential equations. Some of them are about the theory of equations, others about integral functions. He developed the relation between the algebra of matrices and integral equations as well as studying boundary value problems. He was particularly interested in the courses in complex variable, integral equations and differential equations. Under his guidance she worked on integral equations studying infinite dimensional linear spaces. The path took around two weeks to construct and is inlaid with roman numerals, equations, a clock and a mathematical maze. He worked on computational mathematics, developing general methods for solving the equations of mathematical physics by numerical means. He succeeded in characterising those differential equations the solutions of which have no essential singularity in the extended complex plane. The same notions can be extended to polynomial equations involving complex numbers. Books 1-3 contain linear or quadratic indeterminate equations, many of them simultaneous. He considers problems of indeterminate equations of the first degree and trigonometric formulas. Problems of this type which are found in the manuscript are examined in and some of these lead to indeterminate equations. Total inapplicability of such equations for direct numerical calculations is proven analytically by comparison with the analytical solution. Their presence is implicitly accounted for in the equations describing complex reaction mechanisms. Families of curves arise, for example, in the solutions to differential equations with a free parameter. Show More Sentences The equations coincided with the experimentally revealed correlations in the present study, and also with observations from other studies. What was most perplexing was that in using these subtle and imaginary numbers it was possible to solve cubic equations. He studied time dependent hyperbolic partial differential equations and also began to work on the Cauchy problem. In another set of models, called hydromagnetic models, a velocity field, is derived by appropriately coupling the dynamo equations. In particular he applied his methods to equations resulting from electromagnetics, then later to those arising from hydrodynamics. In 1904 he extended Whittaker's solution of the potential and wave equation by definite integrals to more general partial differential equations. During the years 1892 to 1894 Volterra published papers on partial differential equations, particularly the equation of cylindrical waves. He generalised Monge's idea of the characteristic of a partial differential equation from 2nd order equations to nth order equations. I've long since forgotten quadratic equations and how to calculate the circumference of a circle. Memorising the dates and events in your history books and the complex equations in chemistry would just be child's play. For example, the equations describing ordinary dynamics do not depend on where the origin of the co-ordinate system is. The transient equations for the second orders of the identity coefficients are too complicated to solve. In the study of differential equations his methods of lowering the order of an equation and separating variables were important. Linear equations can't easily capture magnetic field helicity, a property that prevents much of the field's energy from dissipating as heat. It was heavy duty mathematics at that, involving an obscure technique known as stochastic differential equations. We apply one-dimensional, cylindrical symmetrical mass transport equations for a single refilling vessel. Beurling worked on the theory of generalized functions, differential equations, harmonic analysis, Dirichlet series and potential theory. The Einstein equation can be derived as the solution to Euler-Lagrange equations that represent the stationary point, or extremum, of the action. For problems like these, you want to add the two equations together in such a way as to cancel out either the x or y terms. This field obeys the expected equations of motion for an electromagnetic vector potential in four spacetime dimensions. Show More Sentences The distribution of probability amplitudes obeys wave-like equations, the amplitude changing in space-time like a wave does. These laws are very precise and can be expressed by exact mathematical equations. Fortunately it is possible to write computer programmes which can find numerical solutions to these equations. Other topics he worked on include algebraic geometry, number theory and integral equations. Mason's research interests lay in differential equations, the calculus of variations and electromagnetic theory. Pure mathematics became Clebsch's main research topic when he began to study the calculus of variations and partial differential equations. He worked on the four colour problem and also published books on calculus, differential equations, complex variable and Fourier series. He became a physics major but differential equations and calculus just didn't excite him. Simion was soon teaching college-level courses such as multivariate calculus and differential equations to the most advanced math students. Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics. In 1865 Fuchs studied nth order linear ordinary differential equations with complex functions as coefficients. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. Below are the equations you can use to predict how much weight you can squat for 10 reps. Norman decided to shift his field from gap and density theorems to non-linear differential equations, both ordinary and partial. It was then that he became aware of the mysteries underlying the subject of non-linear partial differential equations. This approximation was adopted in the model because of the mathematical difficulties in solving non-linear systems of equations. Currently, the feeding value of hay is established using regression equations based on chemical constituents. These are equations involving the rate of change of quantities either in time, or in space, or in both, and are a part of calculus. At the same time, he couldn't abide facile equations between criminal desperadoes and the legalized murder machinery of a state. Find the 2 roots and a continued fraction for a root of these quadratic equations. Show More Sentences Historically, imaginary numbers first came to light when trying to solve cubic equations, rather than quadratics. Although Galois had used groups extensively throughout his paper on equations, he had not given a definition. The study of integral equations was driven by the study of electrostatics and potential theory. The idea of dimensional analysis is that in all equations, the units on the left hand side must match those on the right hand side. Ramanujan was shown how to solve cubic equations in 1902 and he went on to find his own method to solve the quartic. He undertook a large-scale work on generalised differential equations in functional derivatives. Duhamel worked on partial differential equations and applied his methods to the theory of heat, to rational mechanics and to acoustics. The relationships between changing magnetic and electric fields are summarized in the well-known Maxwell's equations. In 1845 Wantzel, continuing his researches into equations, gave a new proof of the impossibility of solving all algebraic equations by radicals. Many of these mathematicians turned to other topics such as topology, differential equations, and functions of a complex variable. He extended the applications of the operational method to linear ordinary differential equations with variable coefficients. These equations always seem to be out of whack, but it takes time to withdraw from the hungry ghosts within and give more to the right people. In order to reconcile Maxwell's equations with Newtonian gravity, with Newtonian mechanics, we had to replace Newton with special relativity. The error terms from the regression equations are modeled as random variables with a probability distribution. It won't be the central government because power equations make a rap on the knuckles impossible. The whole situation is made more difficult by the complexity of the cost equations. By generalizing the Onsager transport model we derived a set of equations that rationalize all pertinent observations. Equivalents can be used to simplify balancing chemical equations for many reactions. The chemical equations for some reactions may have a lone reactant or a single product. The somatotype provides total information which is more useful than calculating particular measurements from predicting equations. Show More Sentences He then wrote on algebraic functions, differential equations and complex analysis. The sections on equations cover quadratic equations where he discusses two solutions. Our corrections are based on recasting the integral equations into a hierarchy of simpler integral equations that can be solved analytically. It is shown that Mie's solution to Maxwell's equations no longer holds for the analysis of resonance of a plasmonic metal nanosphere. Its used in all sorts of things in calculus where recursion is necessary, like differential equations. This leads to 5 linear equations in 5 unknowns and he refers the reader to an appendix containing Cramer's rule for their solution. This is a word problem whose solution includes solving a system of two linear equations. If you offer no resistance to your own magick, Carroll's equations work out in your favor. X was popularised as the unknown in maths when Descartes' printer ran out of Ys and Zs for the mathematician's equations. A memorable presentation will rarely be an unpunctuated sequence of equations or an uninflected recitation of sources of systematic error. Among his many mathematical achievements can be included profound discoveries in logic, algebra and differential equations. Omar Khayyam, known chiefly in Europe as a poet, combined trigonometry and approximation theory to solve algebraic equations using geometry. He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy. Arabic mathematicians learned to manipulate polynomials, to solve certain algebraic equations, and more. Gauss gave a systematic method for solving such equations which is precisely Gaussian elimination on the coefficient matrix. Her mother always reminded her of how she was doing long division when she was three and complex algebraic equations by the time she was five. Naimark's first work for his candidate's thesis was on the separation of roots of algebraic equations. It was a matter of estimating eigenvalues of non-self-adjoint differential equations and matrices. After writing a thesis on algebraic functions and equations, he worked on space curves. The first person known to have solved cubic equations algebraically was del Ferro but he told nobody of his achievement. Show More Sentences Lagrange's main object was to find out why cubic and quartic equations could be solved algebraically. These methods were, to a large extent, motivated by attempts to solve equations algebraically. This allowed him to begin seeing that he could recover Newton's equations when gravitational fields were weak. The mutual inductance appears in the circuit equations for both the primary and secondary circuits of the transformer. It is important to appreciate that equations cannot be used to predict the lower yield stress unless the resultant grain size is known. A diagrammatic illustration of the reactions described in equations 1 through 4, as they occur in muscle, is shown below. With a multiallelic locus, there are more independent equations than the parameters to be estimated. Distraction was based on arithmetic equations displayed for 10 seconds in a corner of the screen. Generalizing, we might say that just as points underdetermine equations, so facts underdetermine theories. The residual covariance matrix of the share equations will be singular, and thus one equation must be omitted when estimating the system. Rate equations in a simplified form are given by JONG 2002, where R is a diagonal matrix and G is a vector of nonlinear functions. The equations were also estimated with ordinary least squares regression, and very similar results were obtained. Those equations and present particle physics say that matter and antimatter are equivalent. His work is almost exclusively on calculus, in particular differential equations and functions of a real variable. Solving the equations of the best response functions gives expressions for equilibrium outputs. Ten years later Leibniz, independently, used determinants to solve simultaneous equations although Seki's version was the more general. These are called linear equations where A and B are, in general, any real numbers. Essentially linear equations are solved by making two guesses at the solution, then computing the correct answer from the two errors. He worked on numerical methods for solving systems of linear equations and eigenvalue problems. Systems of linear equations were studied in the context of solving number problems. Show More Sentences The estimators are thus computed solving the linear equations obtained deriving equation for the unknowns. With these assumptions, we can write the revenue management problem as a series of linear equations. There is, however, work in progress concerning the numerical solution of linear equations with several unknowns using electrical circuits. What Yang's method essentially reduces to is finding the determinant of the matrix of coefficients of the system of equations. It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. Von Koch's first results were on infinitely many linear equations in infinitely many unknowns. Retention equations have been developed for humans and may be appropriately applied to model depuration for some other mammals. The field equations governing the propagation and interaction of these particles are different. Are there any equations that come out of linguistics that should be included in my hypothetical course? We solve numerically a system of two simultaneous equations, with two unknowns, as follows. The flows can also be turbulent, which means there is little hope of solving the necessary equations. The Mathematics Computation subtest assesses skills in computing with whole numbers, fractions, mixed numbers, decimals, and algebraic equations. Clearly the method will not work if the stand of plants is clonally propagated because the equations will not solve. The extent of this distortion of space and time is described by a set of equations that are now known as Lorentz transformations. Servois worked in projective geometry, functional equations and complex numbers. He also worked on asymptotic analysis, fractional integration and singular partial differential equations. Throughout the module, students develop, graph, and solve linear equations verbally, tabularly, graphically, and symbolically. The existence of tachyons is allowed by the equations of Einstein's special theory of relativity. The first person to claim that equations of degree 5 could not be solved algebraically was Ruffini. In contrast, the off-diagonal elements, representing autocorrelation across equations, are generally negative, although small in value. Show More Sentences Their main research interests are in probability theory, partial differential equations and biomathematics. Omar Khayyam combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. An avalanche roaring down a mountainside may seem to be wildly out of control, but actually it is governed by certain equations. In particular he worked on Galois theory, ideals and equations of the fifth degree. The radian measure makes many trigonometric equations simpler and so it is the preferred unit of measuring angles in mathematics. The algebraic solution of general equations of degree greater than four is always impossible. According to the equations, black holes appeared to be emitting radiation. He showed how to find integrals of a general system of partial differential equations by using sequential complete systems instead of passing to Jacobian systems. After subsequent work failed to solve equations of higher degree, Lagrange undertook an analysis in 1770 to explain why the methods for cubics and quartics are successful. Pacioli does not discuss cubic equations but does discuss quartics. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions. I don't care about rationalising denominators with surds in them, and I don't care that by using maths induction you can prove many equations to be right or wrong. Satellites normally orbit planets in Keplerian orbits, named after Johannes Kepler, who wrote the mathematical equations that describe how satellites move. They can solve a variety of two-step problems using variables, identify equivalent algebraic expressions, and solve linear equations and inequalities. Paniker not only played upon the abstraction of the words and scripts but also incorporated mathematical formulae, algebraic equations and diagrams of horoscopes. It has proved to be the computational method of choice for symbolic manipulation in algebraic geometry, differential equations, and combinatorics. This is specious quantification, whether or not differential equations have been incompetently applied. Ideally, the inverse problem consists of N independent linear equations. Geometrically the resolvent enables us to resolve the whole spread represented by any given set of algebraic equations into definite irreducible spreads. Linear functions of the input and key make poor choices for encryption functions because such systems can always be broken by solving small sets of linear equations. Show More Sentences Euler's theorem is far from the whole story for face vectors of polytopes, but it is not difficult to show that it is the whole story of linear equations on face numbers. Topics include analytic solutions of linear equations, equilibria, linearization, stability, phase portraits, bifurcations, simulations, and modeling methodology. They cover discrete and continuous time equations, linear models and linearization, qualitative analysis and phase space, bifurcations, and delay equations. Its development depends on the linearization of the coupled electro-hydrodynamic governing equations, which is not strictly valid except at low fields. The equations of general relativity unambiguously predict event horizons forming if mass is sufficiently concentrated. Again, the agreement between the full wave and quasi-static calculations is due to the use of equations 4 and 5 to find the surface impedance of the lossy semiconductor layer. Working with his graduate student Carl Brans, Dicke incorporated Mach's principle into an alternate theory of gravity by adding an extra term to Einstein's equations. Just as the apothecary and avoirdupois systems of measurement have faded from our texts so should Young's, Cowling's, Fried's, and Clarke's equations fade into history. Mathematicians have developed algorithms for finding the roots of polynomial equations without having to plot graphs and locate where curves cross the x axis. Can the elusive art of matchmaking be reduced to equations and databases? He shook his head as he glossed over the mathematical equations again. So how do mathematicians solve these complicated differential equations? His doctoral dissertation was on the theory of the propeller which led to his developing a theory of partial differential equations of the second order. The protein concentration of the protein conjugate and the degree of labeling were calculated from the following equations according to the instructions of the manufacturer. Morse theory is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view. Young Thomasina turns out to have been a natural genius who died before she could finish her equations. He majored in mathematical physics, studying mind-bending theories of quantum mechanics and partial differential equations. Of course, that means the minuend of the equations does not exist. Most of his papers are solutions to problems posed by Euler on spherical geometry, trigonometry, series, differential geometry and differential equations. In dealing with simultaneous equations, Zhu certainly presented improvements, giving a method essentially equivalent to Gauss's pivotal condensation. Show More Sentences Because the issue is not how the simultaneous equations, partial derivatives or matrices are manipulated, but what the symbols in the various equations purport to represent. The simultaneous equations suggested in Equation 2 and developed at each road section were used to study the variation in PCU with influencing parameters. An illustration of the mathematical complexity of the problem is that the final stage involved the solution of over six million simultaneous equations. As there are only four boards in Northern Ireland, we used dummy variables in the single level regression equations to control for this rather than multilevel modelling. Table 6 shows actors' strategies transformed into Boolean equations. I am using the hydrodynamic equations for smectics A to predict the results of these experiments, which are being performed by a colleague, Professor Stephen Kevan. The properties of a gas are described by a series of equations known as the gas laws, these are Boyle's law, Charles's law, and the constant volume law. Also, when Dembski is wielding his equations, he gets to play the part of the hard scientist busily correcting the errors of those soft-headed biologists. Differential equations describe the relationship of causality. Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, spheres, and the like, but equations for human emotions. Commercially available spirometers include normative equations, and lung function is expressed as a percentage of the predicted normal down to 6 years of age. He also investigated mathematical questions which were related to these applications, in particular writing important works on non-linear integral equations. I stared at the page of equations I had just finished, looking at the neat rows of numbers and seeing nothing but a blur of meaningless squiggles. We must make the course accessible to students whose common background includes only the freshman and sophomore courses in calculus and differential equations. He has interests in differential equations, the calculus of variations, and functions of a real variable which he would work on for the rest of his life. Immediately after his return Bolza continued teaching and research, in particular on function theory, integral equations and the calculus of variations. Osgood's main work was on the convergence of sequences of continuous functions, solutions of differential equations, the calculus of variations and space filling curves. Yet, perhaps, a century from today, someone using Gates' equations will be able to build starships and make what we know as science fiction in 2001 an everyday reality. By adding simultaneous equations we may cancel out one of the unknowns. The predictor variables were entered in the same order for both equations. Show More Sentences His research was mostly on second order ordinary differential equations. After the model was selected, the AREG subprogram computed the equations. These equations will involve both a rate of change of the proportion of the population succumbing to disease, and some unknown parameters, which we will consider shortly. One puzzle, however, is that, according to Einstein's equations, the funnel of a black hole necessarily connects our universe with a parallel universe. Maybe one day it will be sufficient to think about the images involved in a mathematical idea or proof, and a computer will compute the underlying equations for us. Mathematicians find uses for complex numbers in solving equations. He did this by equating Coulomb's law to Newton's equations of motion. He continued to study functional analysis applications to integral equations producing a large number of papers on composition and permutable functions. He conjectured results about the number of solutions to polynomial equations over the integers using intuition on how algebraic topology should apply in this novel situation. He wrote on algebraic integrals of certain differential equations. Balanced chemical equations reflect the law of conservation of mass. The second version occurs as Corollary 2 to Proposition 7 and was thought of as a method of expanding solutions of fluxional equations in infinite series. The horribly complex equations cover two pages, with factors including angular momentum, frictional force, angular velocity, axis of symmetry and gyroscopic balance. In order for their to be a consistent quantum theory in this case, the curved space in which the string travels must be a solution to the Einstein equations. His interests in research relate to finite geometries and the group theory related to them, to Cremona transformations related to the Galois theory of equations. His work in studying the trajectories of point masses on a surface led to certain non-linear differential equations whose solution also gave properties of geodesics. Rothman debunks the romantic notion of a doomed genius working feverishly by candlelight to commit his revolutionary theory of equations and groups to paper. By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day. His doctoral dissertation On definite integrals and functions with application in expansion of series was an early investigation of the theory of singular integral equations. The degree of the final equation resulting from any number of complete equations in the same number of unknowns, is equal to the product of the degrees of the equations. Show More Sentences He worked on how to derive class number relations from modular equations. I went back to my own locker and spent the next hour and a half working on a sheet of old Kinetics equations and graphing exponential functions to get me ready for class. From 1939 Titchmarsh concentrated on the theory of series expansions of eigenfunctions of differential equations, work which helped to resolve problems in quantum mechanics. In this thesis he studied the representation of arbitrary functions by the eigenfunctions of partial differential equations and other given sets of functions. He used differentials in his courses, which he supplemented with a special one on difference equations and numerical mathematics because engineers often work on discrete data. Esclangon elaborated a theory for these functions, studied their differentiation and integration, and examined the differential equations which allow them as coefficients. In the early 20th century, it was realized that the stability of atomic matter could not be explained using the Maxwell equations of classical electrodynamics. That conjecture offers a new way of expressing Diophantine problems, in effect translating an infinite number of Diophantine equations into a single mathematical statement. In addition Poinsot worked on number theory and on this topic he studied Diophantine equations, how to express numbers as the difference of two squares and primitive roots. His most important contributions were methods of elimination to produce from a set of simultaneous equations, a single resultant equation in one of the unknowns. What are the general solutions to cubic and quartic polynomial equations? One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of each other. Ptolemy wrote about how musical notes could be translated into mathematical equations and vice versa in Harmonics. Concentrations of chlorophyll a were calculated using the trichromatic equations of Jeffrey and Humphrey. The system dynamic mathematical model of the marine steam turbine can be defined by means of differential equations. And the chalkboards in the film with the field equations are legit. The use of scalar and vector potentials is now standard in the solution of Maxwell's equations. The gift equation deserves special attention among the land transfer equations for both paternal and maternal sibsets. To solve rational equations and simplify rational expressions it is necessary to apply generalised common fraction concepts. Show More Sentences The basis of digitalization of a chemosensor is the development of algebraic equations and their applications for proper microprocessor coding. It could only do some partial unifications, such as the geometry of conics and the theory of equations. In this Section we adonize the Lax equations of 89.1 into an infinite commuting hierarchy in the space of dressing variables. Four hundred years ago, Gauss taught calculus and differential equations at the University of Gottingen in Germany. Manjunatha, On some new modular equations and their applications to continued fractions, Int. Convex duality and control theory will help develop such tools, together with partial differential equations techniques. Such a solution exists subject to solvability conditions of integral equations. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences. Together with the equation defining Ka, there are now three equations in three unknowns. Chapters include sidebars and tips, illustrations and equations as well as checklists to use throughout the solarizing process. Students are assumed to have solid algebra skills for solving simple equations and a working knowledge of the Cartesian coordinate system. In addition, some typographical errors and misnumbered equations in the paper add to the reviewer's difficulties. He reduced all of the current knowledge into a linked set of differential equations with 20 equations in 20 variables. Using the system of equations in its final form derived here one can directly solve the vibration problem for a girder with any parameters. If we change the problem into system of equations, then we get inconsistent system of equations. Global regularity for the Yang-Mills equations on high dimensional Minkowski space. The relationships between differential operators in the two cartesian systems, fixed and mobile are presented in equations. The solutions of the field equations are the components of the metric tensor of spacetime. According to her Web site, her research focuses on calculus of variations and symmetry methods for differential equations. The questions are not just equations, but brain-teasers, mental slight-of-hands and real-life scenarios for different age levels. Find more words! 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Words that rhyme with equations What is the adverb for equations? What is the adjective for equations? What is the noun for equations? Use our Sentence Generator Nearby Words equation equating equates equated to equated equate equative equator Equator equatorial equatorially equatorward 9-letter Words Starting With e eq equ equa equat equati equatio equation