CC BY
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k k k
Step 4. After k-th level of subdivision we project (by a subroutine, say, “Projection”) a mesh {b .. } onto the surface S. Let a .. = P(b ..) ,
У У У
k
where P is a projection, a .. G S.
Step 5. Now one should verify a condition 3). We suggest here to use a distance d x between a barycenter of corresponding triangle and
S (instead of d), and verify a condition dx<£ /2. Let the subroutine be called “Distance”. For the mesh to be conforming, we can use a
method from [7] for dividing big triangles into several smaller ones.
References
1. “Fuji technical research” company. Private communications, Tokyo, 2000.
2. Gerald Farm “Curves and surfaces for CAGD”. Academic press, 1993.
3. Ichiro Hagiwara. Private communications, Tokyo Institute of Technology, 2000.
4. Ho-Le K. “Finite element mesh generation methods: review and classification”. Computer-Aided Design, 20:27-38, 1988
5. K.-J. Bathe “Finite Element Procedures”. Prentice-Hall, 1996
6. “Subdivision for Modeling and Animation”. SIGGRAPH 99 Course Notes.
7. Rivara M.C. “Algorithms for refining triangular grids suitable for adaptive and multi-grid techniques”. Int. J. Numer. Meth. Eng. Vol 20 (1984) pp. 745-756.
8. Dmitry Berzin "Finite element mesh generation using subdivision technique" // Research Journal of International Studies, №8 (27) 2014, p. 6
9. Dmitry Berzin "Finite element automatic mesh generation using Modified Butterfly subdivision scheme" // Research Journal of International Studies, №8 (27) 2014, p. 8
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации, Москва О ТОПОЛОГИЧЕСКИХ МЕТОДАХ В ПРОСТРАНСТВЕННОМ МОДЕЛИРОВАНИИ
Аннотация
Существует множество хороших работ, касающихся геометрических методов пространственного моделирования. Но очень немногие исследования затрагивали топологический аспект. В настоящей работе мы предлагаем краткий обзор базовых топологических концепций в пространственном моделировании.
Ключевые слова: симплициальные комплексы, гомотопия, клеточные пространства, гомеоморфизм.
Berzin D.V.
PhD, Associate Professor, Financial University under the Government of the Russian Federation, Moscow ON TOPOLOGICAL METHODS IN SHAPE MODELING
Abstract
There are many excellent research results on geometrical shape modeling. As for the topological level of the modeling, limited researches have been conducted. In this paper we overview some basic concepts of topology, which can be applied to shape modeling.
Keywords: simplicial complex, cell space, homeomorphism
1. Introduction
The notions of topological space and homeomorphism are ones of fundamental in mathematics. Roughly speaking, homeomorphisms describe objects deformations, and the concept of homeomorphism is useful for discovering important properties of objects, that are not changing under such deformations. These properties are called topological, in contrast to metrical ones, which are connected with distances between points, angles between lines and so on. For example, cube and tetrahedron are different from metrical point of view, but they are homeomorphic. Subtle metric properties are not important for many problems, and it is needed to reveal “rough” topological properties. Topology (in particular, homotopy topology) is an important branch of mathematics (e.g., see [1], [2]). Its basic concepts, useful for shape modeling, are homeomorphism, homotopy, simplicial complex, cell space.
It is known, that the concept of manifold is a fundamental in geometry. Structure and properties of smooth manifolds give us a good basis for computer-aided geometrical design (CAGD) and computer graphics (CG) methods. For example, Gaussian curvature and other invariants are of a great importance in geometric modeling. It appears that sometimes a structure of a smooth manifold is not sufficient, so the notion of simplicial complex or cellular space is necessary. Simplicial complex can be regarded as a triangulated object (manifold or nonmanifold). A cellular space is an object, constructed from some primitives - cells, and can be regarded as generalization of the notion “smooth manifold”.
Triangulations of subdivided manifolds (and non-manifolds) are used extensively in solid modeling. Paoluzzi et al. [3] provide an overview of related work and analyze the benefits of representing such triangulations using simplicial complexes. Bertolotto et al. [4] present hierarchical simplicial representations for subdivided objects, but these do not support changes of topological type. Polyhedra can also be represented using more general representations. The simplicial set representation of Lang and Lienhardt [5] generalizes simplicial complexes to allow incomplete and degenerate simplices. Cell complexes (i.e. cell spaces), formed by subdividing manifolds into non-simplicial cells, can be represented using radial edge structure [6] or the cell tuple structure [7]. Kunii et al. [8-14] and [23] used so-called cellular approach for shape modeling. The authors give a definition of homotopy and cellular space, and examples of cellular decompositions of geometric objects together with corresponding attaching maps as well. The progressive simplicial complex (PSC) representation for geometrical objects was described in [15]. The PSC representation expresses an arbitrary triangulated model M (e.g. any dimension, non-orientable, nonmanifold) as the result of successive refinements applied to the base model M1 that always consists of a single vertex. Thus both geometric and topological complexities are recovered progressively. Combinatorial and topological properties of meshed solids were also used in [16]. In [17] the authors use the following modeling pipeline: state space - configuration space - image space. The state space is represented by a data structure that is topologically general and computationally practical: the simplicial complex. Topological approaches are used also in [18]. The authors proposed a novel technique, called Topology Matching, in which similarity between polyhedral models is quickly, accurately, and automatically calculated by comparing Multiresolutional Reeb Graphs (MRGs). By the way, Prof. Kunii and his followers use widely Reeb Graph representation of objects in their research [19-22].
2. Homotopy equivalence
We will start with a definition of a topological space. This concept is a basic one in topology. But this object is too general. Almost always mathematics deals with spaces with additional structures. Firstly, there are analytical structures: differential, Riemannian, symplectic, and so on. They are very natural. Secondly, combinatorial structures can be provided. One decomposes a space into similar parts and investigates how they are situated to each other. Important combinatorial structures are simplicial and cellular ones.
A set X is called a topological space, if a set of sets {X a}aGi is determined, where X a are subsets of X (called “open” sets), and I is an arbitrary set. In addition, three axioms hold:
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Empty set and the set X are open Any union of open sets is open Intersection of any two open sets is open.
Set V <A X is called closed, if its complement to X is open. Let X, Y be two topological spaces. Map f : X ^ Y is called continuous,
if for each open subset U <A Y the inverse image f 1 (U) is open in X. Let X, Y be two topological spaces. Map f : X ^ Y is called
homeomorphism, if it is a continuous one-to-one correspondence and the inverse map f 1 is continuous as well. Two spaces are homeomorphic, if there is a homeomorphism between them.
We need often to restrict such wide classes of mathematical objects. For this aim additional separability axiom is used in the next definition. Topological space X is Hausdorff space, if for each couple of points x,y £ X there exist two corresponding open neighborhoods
U,V in X, which intersection is empty: U О V = 0 , x £ X, y £ Y. Of course, our usual Euclidian space R3 is Hausdorff one.
Let f 0: X ^ Y and ^ : X ^ Y be two continuous maps between topological spaces X and Y. These maps are called homotopically
equivalent (or homotopic) if there exists a family , for 0 < t < 1, of continuous (with respect to t and x £ X simultaneously) maps:
Ф: X ^ Y ,
and satisfying ф0 (x) = f0(x) , ф1 (x) = f1 (x) . The family of maps is called a homotopy between X and Y; it can also be regarded as a
continuous map
F(x,t) : X X [0,1] ^ Y .
In words, two maps are homotopic, if we can go from one to another by means of a continuous deformation with parameter t£ [0,1]. Two topological spaces X and Y are called homotopically equivalent if there are continuous maps f : X ^ Y and g : Y ^ X
such that the composition fg : Y ^ Y is homotopic to the identical map id: Y ^ Y and the composition gf: X ^ X is homotopic to the identity map id: X ^ X .
There are several well-known examples of homotopically equivalent (but not homeomorphic) spaces: Euclidian space R N and a point;
a Mobius strip (non-orientable surface) and a circle; a sphere with three holes and bouquet of two circles S1 VS1 (i.e. two circles, intersecting at one common point); a torus with a hole and a bouquet of two circles; a circle and an annulus. The last case is illustrated in
fig. 1. We may choose f : S 1 ^ S1 as identical map, let h be a compression along radii, let g be a composition g=f 1 h. Then fg and gf are homotopy equivalent to corresponding identical maps.
*•
Fig. 1 - Circle and annulus are not homeomorphic, but they are homotopic
An example of not homotopical manifolds: sphere and torus. Deformations (morphings) of real objects often can be considered as homeomorphic or homotopy deformations, and in animation problems the parameter t can be regarded as time.
3. Simplicial complexes
In this paper we will consider only finite simplicial complexes. An n-simplex О n = (p 0p 1 ■ ■ pn ) is the convex hull of n+1 points p0 ,p1,... ,p in general positions in N-dimensional Euclidian space (N > n). Thus a 0-simplex is a point, a 1 -simplex is a segment, a 2-segment
is a triangle, and a 3-simplex is a tetrahedron. The points p 0 ,p1 ,.,p are the vertices of the simplex. We will always consider О with an orientation, which can be specified by ordering the vertices; an even permutation of the vertices specifies the same orientation, while an odd permutation specifies the opposite orientation. An m-dimensional face or subsimplex of О n is the complex hull of an m-point subset of { p 0 ,p1 ,.,p }. The zero-dimensional faces are the vertices; the one-dimensional faces are called edges. A simplicial complex is a finite
collection K of simplexes of satisfying the following condition: intersection of two simplexes in K is either empty or a face of both (see fig. 2):
K= U (U of ) , where J is some subset of the set {0,1,. ,N}.
f£j i£lq
8
Fig. 2 - Legal ways in which the simplexes of a simplicial complex can meet
The whole set of points K is called a polyhedron. The k-skeleton of K is the subcomplex consisting of simplexes of dimension k or lower. Some generalization is possible here: we can also think of simplicial complex as being made up of “curvilinear simplexes”, i.e. objects, homeomorphic to simplexes (see also fig.3). Let us consider a formal finite linear combination of simplexes:
c = Z a7i,
where the a i are integers.
The set C (K) of such linear combinations is the free abelian group generated by q-simplexes of K; it is a direct sum of copies of Z, one for each 7,q . The elements of C (K) are called q-chains; thus C (K) is the q-dimensional chain group of K.
By definition, the (q-l)-dimensional faces of a q-simplex 7q =(p0pi ■ ■ p q ) have an induced orientation given by
(-1)’( po - pi_ipi+i - pq X for i = 0 ..., q .
The boundary of a q-simplex is the sum of all its (q-l)-dimensional faces, taken with the induced orientation:
q
dq ( po- p q )= Z (-1) ' ( po-p i_ip i+1-p q X
i=1
Here are some examples, see fig.3.
A one-simplex 7l = (p op i) is given by its two vertices. Its boundary is 3 (pоpi) = pi- po.
2
The boundary of a two-simplex 7 ( p о pip 2 ) is what you get by going around the triangle in the order po ^ pi ^ p2 ^ po:
3( popip2) = ( popi) - ( pop2) + (pip2) = ( popi) + (pip2) + (p2po).
The boundary of a zero-simplex is 0.
4. Cellular spaces
Another way to calculate homology groups is cellular decomposition of the object. Cellular space (or CW-complex, which is the same) is Hausdorff topological space, represented as a union of non-intersecting sets (called cells)
X= U (U <)
q&J i&Ig
where q is a dimension of a cell eq ; i is its number; Iis some set, corresponding to dimension q ; J is a subset of the set of integer
i q
non-negative numbers.
In addition, the following conditions must be satisfied:
i. Each cell of zero dimension is a point of X. For every positive q the closure eq of each cell eq is the image of a closed q -dimensional ball Bq under some continuous map (called characteristic map)
Xq: Bq ^ eq,
where eq 7 X , and the restriction of this map to the open ball int Bq is a homeomorphism. Here ball Bq is an ordinary ball in q-dimensional Euclidian space R q :
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Bq = {||w|| < 1; w eRq }
2. (C-axiom - “closure finite”).
A boundary of each cell (with dimension more or equal than 1) is contained in a union of a finite number of cells of fewer dimensions.
3. (W-axiom - “weak topology”).
A subset K of X is closed if and only if the intersection K ^ eq is closed for every cell eq of space X.
It is easy to understand that for finite CW-complexes (i.e. consisting of finite number of cells) C-axiom and W-axiom are held automatically.
Consider an important example of a cellular subspace of X - its skeleton
skp x = и (и eq) ,
qeJ,q<p ieIq
where p is an integer non-negative number.
Thus, from the above definitions we can obtain the next procedure of cellular spaces constructing. Consider a restriction Fq of
the characteristic map %q to the boundary Sq 1 of the closed q-dimensional ball Bq , and after that consider continuous map from a union of spheres into a skeleton:
Fq: U ST' ^ sk,-,X ,
iej
so that Fiq is a restriction of Fq to the sphere Sq '. Thus,
Skq X = Skq—1 X U F1 ( U B )
ieIq
Continuous map Fq is called gluing map (or attaching map). Indeed, gluing map identifies each point x from Sq ' with its image
Fq (x) e skq—, X . So, we have a sequence
sk0 x ^ sk, x ^ ... ^ skq x ^ ... ^ x ,
which is called a filtration.
Roughly speaking, if we add a differential structure to a cellular space, we will obtain a smooth manifold. Differential geometry, an important branch of mathematics, describes various properties of manifolds Two-dimensional smooth manifold is called a surface. Fig.4 depicts the interaction among various spaces.
Fig. 4
In [19-22] so-called homotopy model based on Morse theory for smooth manifolds was constructed. The scheme vRmL data ^ Homotopy data ^ VRML data
is used there. Homotopy data consist of information about critical points of Morse function chosen as a height function (Reeb graph, see fig.5), cross sections, and other. Each critical point of Morse function [1] has its index (an integer non-negative number). For example, for peak points index X =2, for saddle points X =1, and for pit points X =0. Generalization of Reeb graph was used in [18] for shape recognition of geometrical objects.
peak
Fig. 5 - Reeb graph for torus 10
Consider a simple example - a torus T2 and a cellular decomposition of it. A torus is a smooth compact orientable 2-dimensional manifold with genus g = 1 (see fig.6).
One can easily distinguish two main circles: along and across a torus. On fig.6 they are intersecting at a point A. Indeed, torus is a direct product of these two circles
T2= S1 X S1.
Thus we are able to consider the following cellular decomposition: one 0-cell e° (that is a point A ); two 1-cells
2
intervals (that are main circles without point A ); and one 2-cell e - an open rectangle. As for skeletons, we have: sk0 T2 = e° is simply the point A; skl T2 = e° + ej1 + e1 is a bouquet S1 VS1 of two circles;
sk2 T2= e° + ej + e2 + e2 is the torus T2 itself.
As for characteristic and gluing maps, in this example we have:
e
and e2- open
^ e1, ^ e1
Fj1:SO ^ e0, F21:S20 ^ e0,
where Sj0 , S2 , are couples of points by definition, , В° are closed line segments. And
sk1 T2 = sk0 T' U F1 ( U Br) = S° V s1.
i=1
Indeed, the next general statement is held.
2
Theorem 2. A closed orientable surface M with genus g has a cell decomposition
M2= e° + (a1 + bj+... ag + bg)+e2,
where e° is a point on M 2 , and ai, b. are loops starting and ending at e° .
There is a close connection between cell decomposition and triangulation of a surface. Triangulation is a decomposition of a surface into a set of triangles in some reasonable way (i.e. in the way of simplicial decompositions, see [1]).
Theorem 3. Any smooth compact 2-dimensional manifold admits a triangulation.
The integer number E(M) = ^ (— 1)Ш 1 , where P. are critical points of Morse function, indP. are their indexes, C is the set of
critical points of this Morse function, is called Euler characteristic of a given manifold M.
Theorem 4. For a triangulation of a surface E (M) = V -E +F = 2-2g,
where V is a number of vertices, E is a number of edges, F is a number of faces, and g is a genus of the surface M.
We can define the notion of homology groups for cellular space in the way, similar to one for simplicial complexes. The (cell) boundary operator is a homomorphism
dq : C q (X) ^ C q—!(X ,
defined as follows. If e q is one of the generators of the q-chain group C (X), we have
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B e ' - Z [ e ' :e П = Г-
where the sum is over all (q-1)-cells of X, and the integer indexes [ eq :eq '] are introduced by degree of a map (e.g., see [']). Like its simplicial counterpart, the cell boundary operator satisfies
Bq Bq+1 = 0 for all q.
A q-chain is called a q-cycle if its boundary is 0. It is a q-boundary if it is a boundary of some (q+l)-chain. Thus, the group of q-cycles
Z q (X) = ker Bq ,
And the group of q-boundaries is
B q (X) = im Bq+! .
The q-th cell homology group is the quotient
H q (X)- Z q (X)/ B q (X) .
Theorem 5. If a topological space has two distinct cell decompositions, the cell homology groups are the same for both decompositions. The polyhedron K of a simplicial complex is a cell space, in an obvious way (fix a homeomorhism between a closed ball and a standard closed simplex in dimension k). Consequently, for any such K we can calculate its simplicial homology group and the cell homology group.
Theorem 6. If X is the polyhedron of a finite simplicial complex, the simplicial homology groups and the cell homology groups of X are isomorphic.
Thus, we know two ways of computing homology groups: if we have a cell decomposition, we can compute the cell homology,
and if we have a simplicial structure, we can compute the simplicial homology. The result is the same. If the numbers [ e q :e q 1] are known,
it is not difficult (easier, than in the case of simplicial complexes) to calculate cell homology groups, using the technique of homology exact pair sequence [1,2].
There are many profound theorems on cellular spaces. We just want to mention some of them.
Theorem 7. Given two homotopically equivalent gluing maps Fq and Gq ,q-1
F,G : U ST1 ^ sk,-1
X,
then spaces skq_ 1 X U f, < U Bq ) and skq_ 1 X U (U в ) are homotopically equivalent.
q
q
Theorem 8. Let f be a Morse function on a compact smooth manifold M. Then M is homotopically equivalent to a finite CW-complex containing one cell of dimension X for each critical point of f of index X .
Continuous map f : X ^ Y, where X and Y are cellular spaces, is called a cellular map, if
f ( skq X ) C skq Y
for each integer non-negative q.
Theorem 9 (about cellular approximation). If X and Y are cellular spaces and f is a continuous map f : X ^ Y
than cellular map g exists:
g : X ^ Y
that f and g are homotopically equivalent maps.
References
1. A. T. Fomenko, T. L. Kunii “Topological Modeling for Visualization”, Springer, 1998
2. A. T. Fomenko, D. B. Fuchs “Course of Homotopic Topology.” Kluwer Academic Publishers
3. Paoluzzi A. et al. Dimension-independent modeling with simplicial complexes. ACM Transactions on graphics 12, 1 (January 1993), 56-102
4. Bertolotto M. et al. Pyramidal simplicial complexes. In Solid Modeling’95, pp.153-162
5. Lang V., Lienhardt P. Geometric modeling with simplicial sets. In Pasific Graphics’95, pp.475-493
6. Weiler K. The radial edge structure: a topological representation for non-manifold geometric boundary modeling. In Geometric modeling for CAD applications. Elsevier Science Publish., 1988
7. Brisson E. Representation of d-dimentional geometric objects. PhD thesis, U. of Washington, 1990
8. T. L. Kunii “Valid Computational Shape Modeling: Design and Implementation.” World Scientific, December 1999
9. T. L. Kunii “Homotopy Modeling as World Modeling.” Proceedings of Computer Graphics International ’99, pp.130-141.
10. T. L. Kunii “Technological Impact of Modern Abstract Mathematics” Proceedings of Third Asian Technology Conference in Mathematics, 1998, pp.13-23.
11. T. L. Kunii, Y. Saito, M. Shiine “A Graphics Compiler for a 3-Dimensional Captured Image Database and Captured Image Reusability.” Proceedings of CAPTECH’98.
12. T. L. Kunii “Graphics with Shape Property Inheritance.” Proceedings of Pacific Graphics’98.
13. T. L. Kunii “The 3-rd Industrial Revolution through Integrated Intelligent Processing Systems.” Proceedings of IEEE First International Conference ICIPS’97, pp.1-6
14. K. Ohmori, T.L.Kunii “Shape Modeling Using Homotopy”, IEEE 2001.
15. J. Popovic, H.Hoppe. Progressive Simplicial Complexes. SIGGRAPH’1997
16. M.Muller-Hanneman. Hexahedral mesh generation by successive dual cycle elimination. Engineering with computers 15 (1999), pp. 269-279
17. T. Ngo et al. Accessible animation and customizable graphics via simplicial configuration modeling. SIGGRAPH’2000
18. Masaki Hilaga, Yoshihisa Shinagawa, Taku Kohmura, Tosiyasu L. Kunii (2001). Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes. SIGGRAPH’2001
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19. Y. Shinagawa, T. L. Kunii, Y. L. Kergosien “Surface Coding Based on Morse Theory.” IEEE Computer Graphics & Applications, 1991.
20. Y. Shinagawa, T. L. Kunii “Constructing a Reeb Graph Automatically from Cross Sections.” IEEE Computer Graphics & Applications, 1991.
21. M. Hilaga “Enhanced homotopy model for orientable closed surface”. Master thesis, Department of Information Science, Faculty of Science, University of Tokyo, 1999.
22. S. Noda “Development of CG systems fundamental technology using homotopy modeling”. Master thesis, Department of Mechanical Engineering and Science, Faculty of Engineering, Tokyo Institute of Technology, 2000 (In Japanese).
23. Dmitry Berzin "On homotopy and cellular approaches to shape modelling" // Research Journal of International Studies, №8 (27) 2014, p. 4
Берзин Д.В.
Кандидат физико-математических наук, доцент Финансового университета при Правительстве Российской Федерации, Москва ПРЕПОДАВАНИЕ ФИНАНСОВЫХ ВЫЧИСЛЕНИЙ В EXCEL НА АНГЛИЙСКОМ ЯЗЫКЕ
Аннотация
Данная статья основана на 6-летнем опыте преподавания на английском языке информационных дисциплин (в частности, финансовых вычислений в MS Excel) на Международном финансовом факультете Финансового университета при Правительстве Российской Федерации.
Ключевые слова: преподавание на английском языке, информатика для финансистов, MS Excel.
Berzin D.V.
PhD, Financial University under the Government of Russian Federation TEACHING MS EXCEL FINANCIAL COMPUTATIONS IN ENGLISH
Abstract
This article is based on 6 years experience of teaching IT disciplines (in particular, financial computations in MS Excel) in English at International Finance Faculty of the Financial University under the Government of the Russian Federation.
Keywords: teaching in English, IT for finance students, MS Excel.
В наше время постоянно усиливающейся глобализации и выхода российских компаний на мировой рынок, когда английский язык приобретает статус официального языка общения в деловом мире, обучение на английском языке приобретает особую значимость. Экономика и финансовая сфера здесь не является исключением. Неизбежное сближение российской и зарубежной систем финансовой отчетности, правил бухгалтерского учета, привлечение на российский рынок передовых форм и методов корпоративного управления, а также новых финансовых и инвестиционных инструментов представляет собой вызов образовательному сообществу России. В связи с этим, важное значение приобретает подготовка кадров, не только обладающих современными экономическими знаниями, но и способных применять их на практике в условиях глобализации, в том числе ведя документацию и общение на английском языке.
В 2008 году в Финансовом университете при Правительстве Российской Федерации заработал новый факультет -Международный финансовый (далее - сокращенно МФФ). Его главная особенность - преподавание всех предметов на английском языке. Данная статья основана на моем опыте преподавания на МФФ курсов "экономическая информатика", "профессиональные компьютерные программы" студентам бакалавриата. Стоит отметить, что часть студентов (как на первом, так и на четвертом курсах) практически не владеют русским языком. В частности, на МФФ обучаются студенты из Вьетнама, Монголии, Франции, Австрии и других стран дальнего зарубежья. По состоянию на 2014/2015 учебный год на первом курсе бакалавриата МФФ обучаются около 80 студентов (4 группы, из них две - "китайских", то есть с углубленным изучением китайского языка), а на четвертом курсе порядка 60 студентов (3 группы, из них одна - "китайская").
На основании результатов ЕГЭ отчетливо видно, что абитуриенты МФФ намного лучше подготовлены по английскому языку, чем по математике. Автор провел своего рода анонимный социологический опрос среди студентов бакалавриата, на основании которого тоже можно однозначно прийти к выводу, что главным стимулом при поступлении на МФФ является желание продолжать обучение на английском языке.
Усвоение информационных дисциплин на английском языке проявляется в проецировании значения термина через призму собственных свойств, установок. Барьер, который называется психологическим, и является наиболее труднопреодолимым в преподавании информатики на английском языке между личностями, т.к. у всех людей механизм “шифрования” и “дешифрования” разный и зависит от различных причин, одной из который является разная психология людей.
Поэтому актуальность коммуникативной проблемы в преподавании информатики на английском языке приобретает остроту. Эта проблема связана также с одной из проблем теории перевода, а именно со способами передачи эквивалентной лексики информатики, т.е. узкоспецифичной лексики. Она создает большое препятствие в усвоении материала. Решение данной проблемы видится в расширении фоновых знаний студентов и заучивании отдельного пласта узкоспецифических терминов. Фоновая лексика - это слова или выражения, имеющие дополнительное содержание и сопутствующие семантические или стилистические оттенки, которые накладываются на его основное значение, известное говорящим и слушающим, принадлежащим к информационной сфере. Поэтому важным этапом в обучении информатике является целенаправленное ознакомление студентов с соответствующими терминами перед каждым семинаром.
В начале каждого семинара или перед особо сложным материалом представляется значимым много внимания уделять элементам-терминам, узкоспецифичным выражениям. Терминологический аспект является неотъемлемой частью занятий по математике на английском языке, так как все более ощущается необходимость обучать смыслу понятий и терминов, этап усвоения правил и формул, т.е. сути того или иного явления.
Англоязычная математика и информатика имеет существенные отличия от «нашей» науки. Не вдаваясь в детали конкретных программ, отметим одну общую и самую главную особенность. Эта особенность связана с принципиальным различием менталитетов и проявляется в том, что англоязычная математика и информатика в гораздо большей степени нацелена на практические приложения. Не зря говорят, что основной целью западного образования является «know how», а российского -«know why». В результате многие наши студенты, умеющие преобразовывать громоздкие выражения с комплексными числами, обращать матрицы и решать системы линейных уравнений, оказываются бессильными уже в простейших комбинаторных, статистических или финансовых расчетах, путаются в графической информации, не могут формализовать и решить задачу, описанную в терминах конкретной житейской ситуации. А ведь всем этим вещам в западных школах учат чуть ли не с четвертого класса.
Другой факт состоит в том, что среди российских преподавателей мало кто знаком с данной спецификой. В России англоязычная математика и информатика является своего рода эксклюзивной услугой, и существуют профессионалы, работающие в этом направлении много лет, но в основном только на рынке репетиторских услуг, а также старших классах языковых средних школ.
Как уже отмечалось не раз, преподавание предметов на МФФ на английском языке имеет ряд неоспоримых преимуществ [1-
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