... very hard work in preparing the solutionsto these problems. A further and very substantial addition to this editionof Volume 5 is the inclusion of solutionstothe problems which appear in ChemicalEngineering, ... in order to washthesandtothesameextent?Solution The problem involves a mass balance around the two stages. If x kg/s salt is in the underflow discharge from stage 1, then:salt in feed to stage ... When the liquid flow is stopped the particles settle to form asegregated two-layer bed. The liquid flow is then started again. When the velocityis such that the larger particles are at their...
... pertinent The best choice is A. The words in the first half of the sentence that are especiallyrelated to those to be filled in in the second half are is concerned only and unique to the period. The ... is to be insipid; both terms refer tothe common-place or trite.33. D. To be ethereal is to be empyreal; both terms refer tothe heavenly or un-earthly. Similarly, to be obsequious is to be ... brought into accordE. chronological The prefix syn- means with or together. The root chron refers to time. So synchro-nized means occurring at the same time or occurring together. The best oppositewould...
... individuals in a database tends to increase the strength of the evidence against the suspect.As pointed out at the end of the previous section, the logic of the strengthened evidence against a matching ... not the sourceof the crime stain, the probability that it willnevertheless be found to correspond depends on the rarity of the characteristic. Therefore, node X2depends on the node γ . The ... the probability that the suspect’s profile corresponds to that of the crimestain and that none of the other n − 1 members on the database correspond, given that the suspect is the source of the...
... orbits in the lunar theory. Detailed descriptions are left toSection 2 and 3. Our maintheorem (Theorem 1) shows the existence of many periodic and quasi-periodicretrograde solutionstothe three-body ... on the existence ofcertain types of solutionstothe planar three-body problem with various choicesof masses. There is a natural way of classifying orbits by their topologicaltypes in the ... direction perpendicular tothe plane (see Chenciner [5], [6]), among the discoveries for the N-body problem, none of the new solutions constructedby variational methods can totally discard this...
... (s)Time (s)k=2pi/ k=2pi/ xxWavenumberWavenumberWavelengthWavelength Solutions tothe Solutions tothe Acoustic Wave Acoustic Wave EquationEquation rSpherical Wave in Spherical ... Green’s function, asymptotic Green’s function.3. Green’s function, asymptotic Green’s function. Plane Wave SolutionPlane Wave SolutionPlug (2) into (1) Plug (2) into (1) P P P = P = ... kr(x,z)xzrkO= |r||k| cos(O)= constantAny pt along line phase is cnst k is Perpendicular to Wavefrontk is Perpendicular to Wavefront Energy of an Acoustic WaveEnergy of an Acoustic WaveWork Performed:...
... principles. But thesolutionsto problems that we seek are then deduced from these axioms, with the assistance of little or no data. Information reenters the picture in the fi nal stage of the scientifi ... But not too simple.—EinsteinPessimism is ubiquitous throughout the Western world today. Whether today’s anxiety stems from the inability of politicians in Washington, DC, to fund the U.S. ... is because they are much more vulnerable to infl uences from the extreme fl ank of their own party than tothe rhetoric of the opposite party. This development in turn has widened the gulf between...
... Thus the bad square must have an S on one sideand an empty square on the other side. An S played there mustalso give an unstable board, so there must be another S on the other side of the empty ... +1, then using the vertical symmetry repeatedly, we see that all entries of the centerrow are +1. Otherwise, the entries of the center row, startingfrom the center entry and working toward either ... vectors. In Figure 4.2, set A to be the origin.Let each lowercase letter denote the vector from A tothe pointlabeled with the corresponding uppercase letter. Let x and y 62MOP 2004 - 2005Solutions...
... family of α-n-normson x corresponding tothe 2-fuzzy n-norm on X. 4. On the Mazur–Ulam problem In this section, we give a new generalization of the Mazur–Ulam the- orem when X is a 2-fuzzy n-normed ... when the n-isometry mapped to a linear n-normed space isaffine. They also obtain extensions of Rassias andˇSemrl’s theorem [18].Moslehian and Sadeghi [19] investigated the Mazur–Ulam theorem ... innon-archimedean spaces. Choy et al. [20] proved the Mazur–Ulam the- orem for the interior preserving mappings in linear 2-normed spaces.They also proved the theorem on non-Archimedean 2-normed spacesover...
... introduce the obstacl e problem about the nonhomogeneousA-harmonic equation. Then, we prove the existence and uniqueness of solutions to the nonhomogeneousA-harmonic equation and the obstacle problem. Keywords: ... the obstacle problem. In the mean time, we s howsome properties of their solutions. Then, we pro ve the existence and uniqueness of solutions tothe Dirichlet problem for the nonhomogeneousA-harmonic ... operator and f is a function s atisfying some a ssump-tions given in the next section. We give the definition of solutionstothe nonhomoge-neousA-harmonic equation and the obstacle problem. ...
... L1-Caratheodorytypefunctionhavebeenextensivelystudied. A very popular technique to obtain the existence and multiplicity of positive solutions totheproblem is Krasnosel’skii’ s fixed point theorem ... function may changessign. The aim is to prove the existence of positive solutionstothe problem. 2 PreliminariesConsider the periodic boundary value problem u+ ρ2u = e(t), 0 < t ... for the problem (10) and therefore, (10) have at least one positive solution from Theorem 3.Acknowledgements The authors are very grateful tothe anonymous referee whose careful reading of the manuscript...
... fractional derivative and the fixed point theorems which will be used in Section 3 to establish the existence of positive solutions. To conclude the paper, the feasibilityof some of the results is illustrated ... contraction. Therefore, the proof is complete with the help of Lemmas 3.1 and 2.5. The following result can be proved in the same spirit as that for Theorem 3.4. 8 Boundary Value ProblemsTheorem ... from 0, 1 ìN to 0, . The purpose is to establish sufficientconditions on the existence of positive solutionsto 1.1 by using some fixed point theoremsand some properties of the Green function....
... nontrivial solutionsto 1.1 together with their sign property.Remark 3.7. The condition k ≥ 2inTheorem 1.3 is necessary to obtain three or more nontrivial solutions to 1.1.Infact,ifk 1, then ... the negative critical points and no other critical points, thendJ, Σ1, 0 −1. 3.49 Boundary Value Problems 17Now, we can give the proof of Theorem 1.3.Proof of Theorem 1.3. The ... well.Here, by using critical point theory again,aswellasLyapunov-Schmidtreductionmethod and degree theory, a sharp condition to guarantee the existence of five or six solutions together with their sign...
... My∗.1.7 The problem 1.7 is called system variational inclusion problem denoted bySVIE, A, M.b Further, if x∗ y∗in theproblem 1.7, then theproblem 1.7 is reduced to findx∗∈ ... nonexpansive retraction from E onto C.In order to find a solution of the variational inequality 1.18, the authors proved the following theorem in the framework of Banach spaces.Theorem AIT see 24. ... by EPF. The generalized mixed equilibrium problemsinclude fixed point problems, variational inequality problems, optimization problems, Nashequilibrium problems, and the equilibrium problem...