Оснащенное гиперполосное распределение аффинного пространства

In this paper, we study a special class of hyperbands, i. e., a framed hyperband distribution. The study of hyperbands and their generalizations in spaces with different fundamental groups is of great interest in connec­tion with numerous applications in mathematics and physics. A special place is occupied by regular hyperstrips, for which the characteristic planes of families of principal tangent hyperplanes do not contain direc­tions tangent to the basal surface of the hyperstrip. In this paper, we use the method of external differential forms of E. Cartan and the group-theoretic method of G. F. Laptev. We consider a regular hyperband distribution of an affine space equipped with a field of conjugate planes with respect to an asymptotic bundle of tensors of the basic surface. The definition of the studied hy­perband distribution in the affine space with respect to the 1st order frame is given and the existence theorem is proved. A sequence of fundamental geometric objects of the 1st and 2nd order of a hyperband distribution equip­ped with a field of conjugate planes is constructed. Fields of qua­sitensors are constructed that define the fields of normals of the first kind of the distribution of the characteristics of the hyperband distribution. In a differential neighborhood of the 2nd order, the fields of Transon normals of the 1st and 2nd kind are constructed. The conditions for the coincidence of the Transon normal and the Blaschke normal are found.

Some affine version of the Bäcklund theorem

“The joy that engineers and mathematicians have come together.”¹

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

Inversion of Abel’s Integral Equation by Means of Orthogonal Polynomials

In the affine space the fundamental-group connection in the bundle associated with a surface as a manifold of tangent planes is investigated. The principal bundle contains a quotient bundle of tangent frames, the typical fiber of which is a linear group acting in a centered tangent plane and a quotient bundle of normal frames, the typical fiber of which is a linear group acting inefficiently in a normal quotient space. The curvature object of the fundamental-group connection is a tensor that contains two primary subtensors tangent and normal linear connections. The tensor of non-absolute parallel transference is constructed. Two envelopment of the connection object is obtained. Analytic and geometric conditions of coincidence of two types of envelopment are found. The covariant derivatives of equipping quasitensor form a tensor. The alternations of the covariant derivatives of the objects of the affine and linear connections of the first type are equal to the corresponding components of the curvature tensor and for the second type they vanish.

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

For convex hypersurfaces in the affine space \({\mathbb {A}}^{n+1}\) (\(n\ge 2\)), A.-M. Li introduced the notion of \(\alpha \)-normal field as a generalization of the affine normal field. By studying a Monge–Ampère equation with gradient blowup boundary condition, we show that regular domains in \({\mathbb {A}}^{n+1}\), defined with respect to a proper convex cone and satisfying some regularity assumption if \(n\ge 3\), are foliated by complete convex hypersurfaces with constant Gauss–Kronecker curvature relative to the Li-normalization. When \(n=2\), a key feature is that no regularity assumption is required, and the result extends our recent work about the \(\alpha =1\) case.

It is applicability alone which elevates arithmetic from a game to the rank of a science . Gottlob Frege (1848–1925) The applications of mathematics in the various branches of science give rise to an interesting philosophical problem: why should physical scientists find that they are unable to even state their theories without the resources of abstract mathematical theories? Moreover, the formulation of physical theories in the language of mathematics often leads to new physical predictions, which were quite unexpected on purely physical grounds. How can turning to the abstract representations of mathematics – far from physical applications – so often turn out to be just what is required in representing and understanding physical systems? In this chapter we will consider the problem of the applicability of mathematics and discuss the prospects for a solution. The unreasonable effectiveness of mathematics The Nobel-Prize-winning Hungarian physicist Eugene Wigner (1902–95) remarked in a famous paper that [t]he miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. (Wigner 1960, p. 14) This much-cited passage, however, does not make it clear exactly what the problem is. After all, in order for us to do science we need some language or other. Why shouldn't mathematics be that language? But what is worrying Wigner is something deeper than this. He is concerned about a mismatch between the methodologies of mathematics and physics – physics is driven by empirical evidence and mathematics is in some sense disconnected from the world – yet mathematics still turns out to be just what the physicist needs.

Let M be a smooth surface in real affine 3-space. Consider the pairs of points of this surface at which the tangent planes are parallel and in particular the chords (infinite lines) joining these pairs. We study in detail and classify the singularities of the envelope of these chords, that is a (singular) surface tangent to all of them. This is called the Centre Symmetry Set (CSS) of M. The study is local in character and is based upon a more general investigation by the authors of the n-dimensional case. The construction of the CSS is affinely invariant and generalises the focal set of a surface in euclidean space and the affine focal set of a surface in affine space. Many standard and some unusual singularities occur in a natural way as the singularities of the CSS. There are illustrations of the various cases by means of concrete examples.

In this paper we shall deduce some results concerning the theory of surfaces in affine space. Some of these results may be regarded as analogues of the projective theory of surfaces immersed in ordinary space. Others seem to be properties of surfaces immersed in affine space. In what follows we shall use the same iotations as in Blaschke's Vorlesungen jiber Differentialgeometrie II. First, we define the canonical quadric Q at an ordinary point P of a surface ar in affine space by means of the Bompiani-Klouboucek's asymptotic osculating quadrics. Using this quadric Q we give a simple geometrical interpretation of the Pick invariant J and the Gaussian curvature S of the fundamental quadric form = 2Fdudv of ai. In particular if S =0, Q is a paraboloid, and conversely. Next we study the Moutard quadrics Q. (u) and Q. (v) belonging to the tangent t. and the asymptotic ruled surfaces R (u) and R(v) generated by the asymptotic uand v-tangents along the vand u-curves respectively. Using these two quadrics Q, (u) and Q. (v) we give a characteristic property of the tangents of Segre. We have shown that the loci of the diameters dn (u) anid dn (v) of Qn (u) and Q. (i) are two quadric cones r2 ( and r2(v) passing respectively through the asymptotic uand v-tangents and having the affine surface normal as a common generator. The cones r2 (u) and r2(V) intersect the quadric of Lie in the asymptotic tangents and in a pair of non-composite twisted cubics C3 (u) and C3 (v). The tangent plane Xr of cr at P cuts the tangent surface of U3(u) and C3 (V) in a pair of parabolas, mutually intersecting, besides the point P, and at three other points lying on the tangents of Segre. This property of the Segre tangents is similar to properties demonstrated by Cech ' and Su.2 Analogous to a theorem of Transon we prove that the loci of the affine

This study is primarily concerned with the presentation of a review of a collection (which could be regarded as a set) of linear and nonlinear singular integro-differential equations with Cauchy kernels, all of which arise from practical applications in Applied Mathematics and Mathematical Physics. The main objective of this review is to provide numerical analysts and researchers interested in algorithm development with model problems of genuine scientific interest on which to test their algorithms. Brief details of the methodology of derivation of the equations are provided and, where possible, existence, uniqueness and asymptotic results are discussed. References are also given to other studies that have dealt with similar problems. The importance of carrying out the necessary mathematical analysis is emphasized for one class of problems where it is shown that the solution abruptly ceases to exist as a parameter is varied. It is further shown that developing asymptotic estimates for the behavior of the solutions is very often a crucial component in the design of effective numerical methods. The importance of regularization is discussed for a class of problems, specific conclusions are drawn and recommendations are discussed. An appendix contains further related problems that may be used for further comparison purposes.

This volume contains the abstracts of more than fifty oral presentations to be given at the International Conference Emerging Trends in Applied Mathematics and Mechanics (ETAMM 2024) hosted by the School of Nautical Studies and Marine Engineering of the University of A Coru˜na, Spain. It is the third edition of a series of conferences in Applied Mathematics. Previous editions took place in June 2016 at the University of Perpignan Via Domitia, France (ETAMM 2016) and June 2018 at the Jagiellonian University in Krakow, Poland (ETAMM 2018). Unfortunatelly, the series was interrupted in 2020 by the eruption of the COVID-19 pandemic. The main aim of this series of conferences is to identify new trends in the domains of Applied Mathematics and Mechanics, and their Applications in Physics, Engineering Sciences, Biology and Economy and provide an opportunity for the dissemination of important and recent results. The conference will bring together mathematicians, interdisciplinary scientists and engineers from all over the world. I hope that it will be a scientifically enriching and socially exciting event for all the participants. The conference will consist of ten invited lectures, four specialized minisymposia and more than twenty contributed talks, focused on recent results and future challenges in the field. Selected and peer reviewed papers will be included in a special issue of the Springer international revue SeMA Journal, which is the official journal of the Spanish Society of Applied Mathematics (SeMA).

Previous article Next article Reduction of the Number of Variables in Systems of Partial Differential Equations, with Auxiliary ConditionsM. J. Moran and R. A. GaggioliM. J. Moran and R. A. Gaggiolihttps://doi.org/10.1137/0116018PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Garrett Birkhoff, Hydrodynamics: A study in logic, fact and similitude, Revised ed, Princeton Univ. Press, Princeton, N.J., 1960xi+184 pp. (1 plate) MR0122193 0095.20303 Google Scholar[2] A. J. A. Morgan, The reduction by one of the number of independent variables in some systems of partial differential equations, Quart. J. Math., Oxford Ser. (2), 3 (1952), 250–259 MR0056183 0047.33403 CrossrefGoogle Scholar[3] Aristotle D. Michal, Differential invariants and invariant partial differential equations under continuous transformation groups in normed linear spaces, Proc. Nat. Acad. Sci. U. S. A., 37 (1951), 623–627 MR0043376 0054.04904 CrossrefISIGoogle Scholar[4] M. J. Moran, Masters Thesis, A unification of dimensional and similarity analysis via group theory, Doctoral dissertation, University of Wisconsin, Madison, 1967 Google Scholar[5] R. Manohar, Some similarity solutions of partial differential equations of boundary layer, Tech. Summary Rep., 375, Mathematics Research Center, University of Wisconsin, Madison, 1963 Google Scholar[6] Arthur G. Hansen, Similarity analyses of boundary value problems in engineering, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964xiv+114 MR0178596 0137.22603 Google Scholar[7] Luther Pfahler Eisenhart, Continuous groups of transformations, Dover Publications Inc., New York, 1961ix+301 MR0124008 0096.02103 Google Scholar[8] G. F. D. Duff, Partial differential equations, Mathematical expositions no. 9, University of Toronto Press, Toronto, 1956x+248 MR0078550 0071.30903 Google Scholar[9] R. A. Gaggioli and , M. J. Moran, Group theoretic techniques for the similarity solution of systems of partial differential equations with auxiliary conditions, Tech. Summary Rep., 693, Mathematics Research Center, University of Wisconsin, Madison, 1966 Google Scholar[10] Hermann Schlichting, Boundary layer theory, Translated by J. Kestin. 4th ed. McGraw-Hill Series in Mechanical Engineering, McGraw-Hill Book Co., Inc., New York, 1960xx+647 MR0122222 0096.20105 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Group method analysis for blood‐Mn‐ZnFe 2 O 4 flow and heat transfer under ferrohydrodynamics through a stretched cylinderMathematical Methods in the Applied Sciences, Vol. 28 | 21 June 2022 Cross Ref Extension of Blasius Newtonian Boundary Layer to Blasius Non-Newtonian Boundary LayerMathematical Journal of Interdisciplinary Sciences, Vol. 9, No. 2 | 8 June 2021 Cross Ref Forward scattering for non-linear wave propagation in (3 + 1)-dimensional Jimbo-Miwa equation using singular manifold and group transformation methodsWaves in Random and Complex Media, Vol. 9 | 21 July 2020 Cross Ref Advanced Ground Truth Multimodal Imaging Using Time Reversal (TR) Based Nonlinear Elastic Wave Spectroscopy (NEWS): Medical Imaging Trends Versus Non-destructive Testing ApplicationsRecent Advances in 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Approximate integrals over bounded volumes with smooth boundaries