


Summary:
We choose some special unit vectors
n_{1},...,n_{5}
in R^{3} and denote by
L ⊂ R^{5}
the set of all points
(L_{1},...,L_{5}) ∈ R^{5}
with the following property:
there exists a compact convex polytope
P ⊂ R^{3}
such that the vectors
n_{1},...,n_{5}
(and no other vector) are unit outward normals to the faces of
P and the perimeter of the face with the outward normal
n_{k}
is equal to L_{k} for all k=1,...,5.
Our main result reads that L is not
a locallyanalytic set, i.e., we prove that, for some point
(L_{1},...,L_{5}) ∈ L,
it is not possible to find a neighborhood
U ⊂ R^{5}
and an analytic set
A ⊂ R^{5}
such that
L ∩ U = A ∩ U.
We interpret this result as an obstacle for finding an
existence theorem for a compact convex polytope with prescribed
directions and perimeters of the faces.
Summary:
We prove that the answer to the question of the title is `as many times as you want.' More precisely, given any constant c>0, we construct two oblique triangular bipyramids, P and Q, such that P is convex, Q is nonconvex and intrinsically isometric to P, and vol Q > c vol P > 0.
Summary:
The van der Waerden's theorem reads that an equilateral
pentagon in Euclidean 3space E^{3} with all diagonals of
the same length is necessarily planar and its vertex set
coincides with the vertex set of some convex regular pentagon.
We prove the following manydimensional analogue of this
theorem: for n ≥ 2, every ndimensional crosspolytope in
E^{2n2} with all diagonals of the same length and all edges
of the same length necessarily lies in E^{n} and hence is
a convex regular crosspolytope. We also apply our theorem
to the study of twodistance preserving mappings of Euclidean
spaces.
Summary:
We study an equation Qu=g, where Q is a continuous
quadratic operator acting from one normed space to another
normed space.
Obviously, if u is a solution of such equation then
u is also a solution.
We find conditions implying that there are no other solutions
and apply them to the study of the Dirichlet boundary
value problem for the partial differential equation
uΔ u =g.
Summary:
We construct an example of a closed nondegenerate nonflexible
polyhedron P in Euclidean 3space, which is the limit
of a sequence of nondegenerate polyhedra which have
the same combinatorial structure as P has and are not
isometrically equivalent to P.
From this we conclude that the set of all flexible
nondegenerate polyhedra polyhedra which have the
same combinatorial structure as P has is not algebraic.
Summary:
We prove that, both in the Lobachevskij and spherical 3spaces,
there exist nonconvex compact boundaryfree polyhedral surfaces
without selfintersections which admit nontrivial continuous
deformations preserving all dihedral angles and study
properties of such polyhedral surfaces.
In particular, we prove that the volume of the domain,
bounded by such a polyhedral surface, is necessarily
constant during such a deformation while, for some
families of polyhedral surfaces, the surface area,
the total mean curvature, and the Gauss curvature
of some vertices are nonconstant during deformations
that preserve the dihedral angles.
Moreover, we prove that, in the both spaces,
there exist tilings that possess nontrivial
deformations preserving the dihedral angles of every tile
in the course of deformation.
Summary:
In 1973, J. Cheeger and J. Simons raised the following
question that still remains open
and is known as the Rational Simplex Problem:
Given a geodesic simplex in the spherical 3space
so that all of its interior dihedral angles
are rational multiples of π, is it true that
its volume is a rational multiple of the volume of the
3sphere?
We propose an analytical approach to the
Rational Simplex Problem by deriving a function f(t),
defined as an integral of an elementary function, such
that if there is a rational t,
close enough to zero,
such that the value f(t) is an irrational number then
the answer to the Rational Simplex Problem is negative.
Summary:
This is a survey paper on various results relates to the
following theorem first proved by A.D. Alexandrov:
Let S be an analytic
convex spherehomeomorphic surface in R^{3}
and let k_{1}(x)≤ k_{2}(x)
be its principal curvatures at the point x.
If the inequalities
k_{1}(x)≤ k≤ k_{2}(x)
hold true with some constant k for all x∈ S
then S is a sphere. The
imphases is on a result of Y. MartinezMaure who first proved that the
above statement is not valid for convex C^{2}surfaces. For convenience of
the reader, in addendum we give a Russian translation of that paper by
Y. MartinezMaure originally published in French in C. R. Acad. Sci.,
Paris, Ser. I, Math. 332 (2001), 4144.
Summary:
We construct a flexible (non immersed) suspension with a hexagonal equator in Euclidean 3space
and study its properties related to the Strong Bellows Conjecture which reads as follows:
if an immersed polyhedron P in the Euclidean 3space is obtained
from another immersed polyhedron Q by a continuous flex then
P and Q are scissors congruent.
Summary:
In the year 1968 N.V. Efimov has proven the following remarkable theorem:
Let f:R^{2}→R^{2}∈C^{1} be such that det f(x)<0
for all x∈R^{2} and let there exist
a function a=a(x)>0 and constants C_{1}≥0, C_{2}≥0
such that the inequalities
1/a(x)1/a(y)≤C_{1} xy+C_{2}
and
det f'(x)≥a(x)>curl f(x)+a^{2}(x)
hold true for all x, y∈R^{2}.
Then f(R^{2}) is a convex domain and f maps R^{2} onto f(R^{2}) homeomorhically.
Here curl f(x) stands for the curl of f at x∈R^{2}.
This article is an overview of analogues of this theorem, its generalizations
and applications in the theory of surfaces, theory of functions, as well as in the study
of the Jacobian conjecture and global asymptotic stability of dynamical systems.
Summary:
We prove that the Dehn invariants of any Bricard octahedron remain constant during the flex and that the Strong Bellows Conjecture holds true for the Steffen flexible polyhedron.
MathSciNet review [MR2823098 (2012h:52032)]:
The Dehn invariant D(P) of a 3dimensional polyhedron
P can be defined as follows.
Let f:R→R be a
Qlinear function such that f(π)=0.
Then D(P)=∑_{l}lf(α_{l}),
where the sum runs over all edges l; α_{l}
is the dihedral angle of the two faces of P meeting at
l; and l is the length.
The Dehn invariant is additive under slicing, so D(P)=D(Q)
if P and Q are ``scissorsequivalent'', i.e., if
P can be sliced into finitely many pieces and reassembled
to form Q. Dehn showed that D_{f} is not
constant over all polyhedra, implying that not all polyhedra
are scissorsequivalent (this was Hilbert's third problem).
The Strong Bellows Conjecture states roughly that if two
polyhedra are related by a flex (i.e., by a motion that does
not change the congruence type of any face) then they are
scissorsequivalent. By a theorem of Dehn and Sydler/Jessen,
an equivalent statement is that the Dehn invariant remains
constant during the flex. The main result of the article
under review is that the Dehn invariant is constant for
flexes of certain selfintersecting octahedra called Bricard
octahedra. A consequence concerns the Steffen polyhedron,
which has 9 vertices and 14 faces, and is one of the
smallest known flexible polyhedra. It can be constructed by
gluing together a tetrahedron and two flexible (but
selfintersecting) octahedra first described by Bricard.
Therefore, a corollary of the main result is that the Steffen
polyhedron satisfies the Strong Bellows Conjecture, i.e.,
all flexes of the Steffen polyhedron are scissorsequivalent.
[Reviewed by Jeremy L. Martin]
Summary:
We show how the rotation and translation fields of a surface,
introduced by Gaston Darboux, may be used to obtain short
proofs of a wellknown theorem (asserting that the total
mean curvature of a surface is stationary under an
infinitesimal bending) and a new theorem (asserting that
every infinitesimal bending of any simply connected closed
surface S⊂R^{3} is orthogonal
to S at least at two points).
MathSciNet review [MR2795785 (2012f:53007)]:
Associated with (infinitesimal) isometric deformations of
surfaces in R^{3} are two vector fields, one called the rotation
field (also referred to as the Drehriss in the literature)
and the other called the translation field. In this paper,
in the case of compact surfaces, the author expresses the
variation of the total mean curvature under such infinitesimal
deformations in terms of a line integral of the rotation
field over the boundary of the surface. This leads to a very
simple proof of a result of F.J. Almgren Jr. and I. Rivin
that the total mean curvature variation is zero when the
surface is without boundary [see in
The Epstein birthday schrift , 121, Geom. Topol. Monogr., 1, Geom. Topol.
Publ., Coventry, 1998; MR1668323 (2000a:53133)].
Furthermore, when the surface is simply connected, it is
shown that, on the surface, a potential function, φ,
can be associated with the translation field and that the
variation field of the deformation is orthogonal to the
surface at the critical points of φ, of which there are
at least two. [Reviewed by Michael J. Clancy]
Summary:
Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove the both theorems. As a byproduct, we prove that the total mean curvature of any polyhedron in the Euclidean 3space is not an algebraic function of its edge lengths.
MathSciNet review [MR2665533 (2011f:52033)]:
In this paper, the author discusses the background to two
important results about closed polyhedral surfaces in
R^{3} which are flexible, in that their
shapes can be changed continuously solely through varying
the dihedral angles at their edges.
The total mean curvature of such a polyhedron P is
M(P)=(1/2)Σ_{l ∈ E}
l(πα(l)),
where E is the set of edges of P, l is
the length of l∈E and α(l) is
the dihedral angle of P at l, measured from
inside P.
Then, first, if {P_{t}} for 0≤t≤1
is a flex of P_{0}, then M(P_{t})
is independent of t. Second, the oriented volume of
P_{t} is constant in t.
Standing behind these results are two more general ones.
A vector field w on a polyhedron P which is
linear on each face of P is an infinitesimal flex if,
for any curve γ⊂P, the length of the curve
γ_{w}(t)={r+tw  r∈γ}
is stationary at t=0. The first result follows from:
if w is the infinitesimal flex on a closed oriented
polyhedron P and P(t)={r+tw 
r∈P}, then
(d/dt)_{t=0}M(P(t))=0;
this is due to J.R. Alexander
[Trans. Amer. Math. Soc. 288 (1985), no. 2, 661678;
MR0776397 (86c:52004)]. The second result is that, if
P_{K} is the set of all closed triangulated
polyhedra in R^{3} with prescribed combinatorial
structure K, then there exists a universal polynomial
p_{K} whose coefficients are universal
polynomials in the edge lengths of P∈P_{K},
such that the oriented volume of any such P is a root of
p_{K}; this was shown by I.K. Sabitov
[Discrete Comput. Geom. 20 (1998), no. 4, 405425; MR1651896
(2000f:52010)].
The first result is analytic in nature, while the second
is algebraic. Here it is shown that there is no crossover,
with an algebraic proof of the first or analytic proof of
the second. Thus there is a closed oriented polyhedron P,
with no planar star at any vertex, having some nonzero
infinitesimal flux. Further, there is such a polyhedron
(of a fixed combinatorial type) whose total mean curvature
is not an algebraic function of its edge lengths.
{For additional information pertaining to this item see
[V. Alexandrov, Aequationes Math. 81 (2011), no. 12,
199; MR2773100].} [Reviewed by P. McMullen]
Summary:
Using the Green's theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3space to a line integral of a special vector field, which immediately yields the following wellknown theorem: the total mean curvature of a closed smooth surface in the Euclidean 3space is stationary under an infinitesimal flex.
MathSciNet review [MR2603844 (2010m:53006)]:
For any compact oriented smooth surface S in
R^{3} the number
H_{S}=(1/2)∫_{S}
(κ_{1}+κ2)dS,
where κ_{1} and κ_{2} are the
principal curvatures of S, is called the integral
mean curvature of S.
Let ψ_{v}:
S×(ε,ε)→R^{3},
given by (x,t)→x+tv(x),
be an infinitesimal transformation defined by a vector field
v. Let S_{t} ={ψ_{v}
(x,t)  x∈S} be the set of deformed
surfaces, and let n(x,t) be the unit normal
vector to S_{t} at x.
For each t∈(ε,ε),
let n'(x,t)=∂n/∂t(x,t).
The variation of the integral curvature H_{S}
is H'_{S}=H'_{St}(0).
Let m(x) be the vector field on S
defined by m(x)=n(x,0)×
n'(x,0), where × is the crossproduct in
R^{3}. In this paper, the author proves that on
a compact oriented smooth surface S in
R^{3} and for an infinitesimal transformation
ψ_{v}, the variation H'_{S}
of the integral mean curvature H_{S} of S
can be expressed as H'_{S}=(1/2)
∫_{∂S}m(x).dx,
where ∂S is the boundary of S.
[Reviewed by Andrew Bucki]
Summary:
This is an extended version of a lecture given on October 4, 2004
at the research seminar ``Differential geometry and applications''
(headed by Academician A.T. Fomenko) at Moscow State University.
The paper contains an overview of available (but far from wellknown)
results about the Blaschke addition of convex bodies, some new theorems
on the monotonicity of the volume of convex bodies (in particular, convex polyhedra with
parallel faces) as well as description of software for visualization of polyhedra
with prescribed outward normals and face areas.
Summary:
Classical H.Minkowski theorems on existence and uniqueness of convex
polyhedra with prescribed directions and areas of faces as well as the wellknown
generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov
are extended to a class of nonconvex polyhedra which are called polyhedral
herissons and may be described as polyhedra with injective spherical image.
MathSciNet review [MR2110761 (2005k:52034)]:
A polyhedral herisson (hedgehog) P is a generalization
of a convex polyhedron: each polygonal face Q_{j}
of P has a unit normal n_{j};
if Q_{j} and Q_{k} share
an edge, then
n_{j}+n_{k}≠0,
so that there is a shortest geodesic path l_{jk}
on the unit sphere S^{2} joining
n_{j} and n_{k};
two such curves l_{jk} meet at most in a
common endpoint; finally, the curves l_{jk}
partition S^{2} into convex spherical polygons.
In other words, the spherical image of a polyhedral herisson
is injective, though the normal vectors to the faces may point
inside rather than outside the surface. In this paper,
the author generalizes to polyhedral herissons some facts
about convex polyhedra, such as the uniqueness theorem of
A.D. Aleksandrov [Convex polyhedra, Translated from
the 1950 Russian edition by N.S. Dairbekov, S.S. Kutateladze
and A.B. Sossinsky, Springer, Berlin, 2005; MR2127379 (p. 107)],
and the more familiar uniqueness and existence theorems of
Minkowski. [Reviewed by P. McMullen]
Summary:
We prove that (nonimmersed) flexible polyhedra do exist in the Minkowski
3space and each of them preserves the (generalized) volume and the (total) mean
curvature during a flex. To prove the latter result, we introduce the notion of the angle
between two arbitrary nonnull nonzero vectors in the Minkowski plane.
Zentralblatt review [Zbl 1028.52013]:
The author considers flexibility of polyhedra in a Minkowski 3space, i.e., in a linear space
consisting of all ordered triples (x_{1}, x_{2}, x_{3}) endowed with the scalar product
(x,y) = x_{1} y_{1} + x_{2} y_{2}  x_{3} y_{3}.
He confirms the existence of (nonimmersed) flexible polyhedra in such a space, and that any of
them preserves the (generalized) volume and the (total) mean curvature during a flex.
His arguments are based on the notion of an angle in a Minkowski 2space.
[Horst Martini (Chemnitz)]
Summary:
We study the existence problem for a local implicit function determined by
a system of nonlinear algebraic equations in the particular case when
the determinant of its Jacobian matrix vanishes at the point under consideration.
We present a system of sufficient conditions that implies existence of a local
implicit function as well as another system of sufficient conditions that
guarantees absence of a local implicit function.
The results obtained are applied to proving new and classical results
on flexibility and rigidity of polyhedra and frameworks.
Zentralblatt review [Zbl 0995.58003]:
The classical implicit function theorem (and its generalizations and modifications) for a function
f : R^{p} × R^{q} → R^{n} provide sufficient conditions under which
f(x, φ(x)) ≡ 0 for some function φ : R^{p} → R^{q} in a neighborhood of a zero
of f, the most important condition being the maximal rank of the partial derivative at this zero.
In the present paper, the author gives sufficient conditions, in case of a polynomial function f,
which apply also if the partial derivative is singular. He also gives necessary conditions for the
existence of φ, which therefore may serve as sufficient conditions for nonexistence.
[Juergen Appell (Wuerzburg)]
Summary:
A new approach to describing a high order flex of polyhedra is given. Sufficient
conditions are derived under which an infinitesimal flex of polyhedra has an extension to an authentic
flex. It is shown that one set of these conditions is fulfilled for Bricard octahedra.
Zentralblatt review [Zbl 0916.52005]:
A closed polyhedron Q in the Euclidean 3space is called flexible if there exists an analytic (with
respect to a parameter t∈ [0,1]) family of polyhedra Q_{t}, such that i) Q
coincides with Q_{0}; ii) for each t∈ [0,1], the polyhedron Q_{t} is isometric to
the polyhedron Q_{0} in the intrinsic metrics; iii) the polyhedron Q_{1} cannot be obtained
from Q_{0} by a rigid motion. The family of polyhedra Q_{t} is called an authentic flex. Let
Q be a closed flexible polyhedra and let us suppose all its faces are triangular. Let us suppose
Q has v vertices (x_{i}, y_{i}, z_{i}) 1≤ i≤ v, and e edges. If vertices
(x_{i}, y_{i}, z_{i}) and (x_{j}, y_{j}, z_{j}) are joined by an edge of Q then a number
e(i,j), 1≤ e(i,j)≤ e is assigned to this edge. Let B : R^{3v}× R^{3v}→
R^{e} be a bilinear form, such that B maps the vectors
X=(x_{1}, y_{1}, z_{1}, x_{2}, y_{2}, z_{2}, ... , x_{v}, y_{v}, z_{v}) and
U=(u_{1}, v_{1}, w_{1}, u_{2}, v_{2}, w_{2}, ... , u_{v}, v_{v}, w_{v}) into a vector whose
e(i,j)th component is equal to
(x_{i}x_{j})(u_{i}u_{j})+(y_{i}y_{j})(v_{i}v_{j})+(z_{i}z_{j})(w_{i} w_{j}). If for
n≥ 1 all vertices i and j are joined by an edge we can write
∑_{m=0}^{n}B(X_{m},X_{nm})=0, where
X_{k}=(x_{1,k}, y_{1,k}, z_{1,k}, x_{2,k}, y_{2,k}, z_{2,k}, ... , x_{v,k}, y_{v,k}, z_{v,k}).
The vector X_{0} + tX_{1} + ... + t^{k}X_{k} is called the kth order infinitesimal
flex of the polyhedron X_{0}. Making use of this bilinear map the author reformulates the
problem of finding out if a polyhedron is flexible to an algebraic problem, and proves some linear
algebra statements that will help to prove the main result: give some sufficient conditions under which
an infinitesimal flex of a polyhedron can be extended to an authentic flex. He also gives computer
results that prove that a set of these conditions is fulfilled for the Bricard octahedra. From the
historical point of view flexible polyhedra have become connected to the attempts of solution to the
open problem to prove that a smooth compact surface in the Euclidean 3space is rigid. For further
details see I. IvanovaKaratopraklieva and I.Kh. Sabitov [J. Math. Sci., New York 70,
No. 2, 16851716 (1994); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 23, 131184
(1991; Zbl 0835.53003)] and I.Kh. Sabitov [Encycl. Math. Sci. 48, 179250 (1992);
translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 196270
(1989; Zbl 0781.53008)].
[A.Pereira do Vale (Braga)]
Zentralblatt review [Zbl 0881.52007]:
The recently established Bellows Conjecture asserts that each oriented flexible polyhedron in
E^{3} conserves its volume during a flex. It is also known that each flexible polyhedron in
E^{3} conserves its mean curvature during a flex. In this paper the author constructs a
flexible polyhedron in an open halfsphere S^{3}_{+} which conserves neither volume nor mean
curvature during a flex. The polyhedron is a modification of a polyhedron constructed from four
spherical digons in S^{3}.
[Ch.Leytem (Cruchten)]
Zentralblatt review [Zbl 0879.51012]:
The author constructs a closed flexible polyhedron with selfintersections without using Bricard's octahedron. The polyhedron has the same property as the Bricard octahedron: it preserves volume during flexion according to the bellows conjecture. (The conjecture was proved for any triangulated orientable surface mapped into threespace [see R. Connelly, I. Sabitov and A. Walz, Beitr. Alg. Geom. 38, No. 1, 110 (1997)]). [A.H. Temesvari (Sopron)]
MathSciNet review [MR1302424 (95i:90096)]:
For a rectangular domain Ω in R^{n}
the GaleNikaido theorem states that if
F=(f_{1},...,f_{n}):
Ω → R^{n} is differentiable and
if every principal minor of the Jacobian F'(x)=
(∂f_{i}(x)/∂x_{j})
_{i,j=1,...,n} is positive, then F is an
injective map. This result is known not to hold for arbitrary
convex domains (a counterexample was given by
T. Parthasarathy and G. Ravindran
[in Optimization, design of experiments and graph theory
(Bombay, 1986), 417423, Indian Inst. Tech., Bombay, 1988;
MR0998817 (90f:90180)]) and the author proves the following
negative result: For any integer n ≥ 2 there exist
an ellipsoid Δ ⊂ R^{n} and a
C^{∞}map F : Δ →
R^{n} such that every principal minor of
its Jacobian matrix F'(x) is positive but
F is not injective. In contrast to Parthasarathy's
reliance on explicit equations, the author's proof is
"geometric'' and uses the ideas at the basis of a result
of A.M. Fomin, A.R. Fet, A.D. Myshkis and
A.Ya. Bunt (see Bunt and Myshkis
[Uspekhi Mat. Nauk 10 (N.S.) (1955), no. 1(63),
139142; MR0068618 (16,912b)]), which is closely related
to the GaleNikaido theorem. [Reviewed by J.S. Joel]
Zentralblatt review [Zbl 0883.26008]:
The main result of the paper is the following theorem:
For every integer n ≥ 2 there are an ellipsoid
D ⊂ R^{n} and a mapping
F : D → R^{n} of class
C^{∞} such that each principal minor of the Jacobian matrix F'(x) is positive but F is not injective. [W. Wilczynski (Lodz)]
Zentralblatt review [Zbl 0852.54030]:
Let X ⊂ R^{n} be geometrically acceptable in the sense of Borsuk (X ∈ GA), i.e. every two points can be joined by an arc of finite
length and the intrinsic metric ρ^{∗} on X (where ρ^{∗}(x,y) is the infimum of lengths of arcs in X
joining x and y) is topologically equivalent to the Euclidean metric ρ restricted to X. (Unfortunately, the author does not formulate this
precisely.) The set X ∈ GA is rigid in R^{n} provided that every intrinsic isometry f :
X → f(X) ⊂ R^{n} (i.e. isometry with respect to the intrinsic metrics) is an isometry. The paper concerns the problem of existence of a
kdimensional set X rigid in R^{n} for k ≤ n. The author gives a solution for n = 2 and k = 1.
He proves that the union of all the rational lines in R^{2} is rigid in R^{2} (a line in R^{2} is rational provided it has an equation a_{0} + a_{1} x_{1} + a_{2} x_{2} = 0 for some rational a_{i}, i= 0,1,2). The proof is very interesting. The next step in this direction was made by Irmina Herburt [Geom. Dedicata 49, No. 2, 221230 (1994; Zbl 0803.54029)]. She gave a method (quite different than that of the author) of constructing (n1)dimensional rigid subsets of R^{n} for arbitrary n ≥ 2. Recently she generalized both methods and strengthened the result replacing the topological dimension by the Hausdorff dimension (the paper is prepared for publication).
[M. Moszynska (Warszawa)]
Summary:
We generalize the theorem of M. and S. Radulescu about an
initial value problem for the equation x^{(n)}=f(t,x)
in a Banach space. In contrast to M. Radulescu and S. Radulescu
[Nonlinear Anal. 4 (1980), no. 5, 951965
and
Rev. Roumaine Math. Pures Appl. 33 (1988), no. 8, 695700],
we do not assume any differentiability of f,
and we impose conditions on the growth of f instead of
f_{x}.
Our proof is elementary and in the case n=1 is based
on A. Wintner's theorem about continuation of solutions of
differential equations in finitedimensional Banach spaces.
Zentralblatt review [Zbl 0894.53043]:
A nonschlicht ndimensional domain with smooth boundary is a set D with the following properties: (a) D is contained in an ndimensional Euclidean manifold M^{n}; (b) the metric space (D,d_{g}) is complete; (c) for every point x ∈ ∂ D there are a neighbourhood V in M^{n}, an isometry ψ : V → R^{n} and a smooth ndimensional manifold N with boundary in R^{n} such that ψ (V ∩ int D) ⊂ int N and ψ (V ∩ ∂ D) ⊂ ∂ N. The author considers the following two problems. Problem A. Consider a subset Δ in the set of nonschlicht domains and let D, D_{∗} ∈ Δ be homeomorphic to each other and have isometric boundaries. Are D, D_{∗} isometric to each other? Problem B. Consider a subset Μ in the set of locally
Euclidean metrics on a domain D. Is a metric g reconstructed uniquely from the values of the corresponding distance between the boundary points? Theorem. Let n = 2, 3, 4, 5, 6 and let Δ be the set of nonschlicht domains which are diffeomorphic to the closed ball D^{n}. Let Μ be the set of locally Euclidean metrics on D^{n} defined on a neighbourhood of D^{n}. Then Problem A for Δ is equivalent to Problem B for Μ. [V. Oproiu (MR 93k:53043)]
Zentralblatt review [Zbl 0769.54038]:
A certain sufficient conditions is given for the isometry of domains resulting from the isometry of their boundaries.
[A. Szybiak (Kitchener)]
MathSciNet review [MR1144540 (93e:26009)]:
There are two theorems on global homeomorphism (i.e. theorems
that give conditions for a local homeomorphism to be a
homeomorphism) for mappings in
R^{n} that were proved by
N. V. Efimov in his work on a generalization of the Hilbert
theorem on the impossibility of isometric immersion of the
Lobachevskii plane into the Euclidean space
R^{n}. (Roughly, the
inequality det f ' (x) ≤ a(x)rot
f(x) + a^{2}(x),
rot f = D_{1}f ^{2}  D_{2}f ^{1},
with a>0 satisfying (1/a(x))  (1/a(y)) ≤
C_{1}x  y + C_{2},
is the condition that guarantees global invertibility of f.)
The author observes that the nature of these theorems, which
have complicated proofs, was not studied from a
functionaltheoretical point of view, and therefore we
have no multidimensional analogues of the theorems.
To partially fill in this gap, the author obtains a theorem
in R^{2} with a hypothesis very similar
to the ones in Efimov's theorems, proving it as a corollary
of the wellknown HadamardLevyJohn global inverse function
theorem.
The paper concludes with a remark that the hypotheses
in Efimov's theorems imply a restriction on the spectral
radius of the mapping (f ' (x))^{1} as
x→ + ∞, which suggests the existence of multidimensional
analogues of Efimov's theorems similar to the HadamardLevyJohn
and Zorich theorems (for quasiconformal mappings).
[Reviewed by Miodrag Perovic.]
Summary:
The possibility of imbedding ndimensional locally Euclidean metrics in the large in R^{n} is studied by means of the global inverse function theorem in the forms suggested by Hadamard, John, Levy and Plastock. The imbeddability of conformally Euclidean metrics is studied by means of a theorem of Zorich on the removability of an isolated singularity of a locally quasiconformal mapping.
Zentralblatt review [Zbl 0721.51014]:
A function f : (M_{1}, q_{1}) → (M_{2}, q_{2}) of metric spaces (M_{i}, q_{i}), i = 1, 2, is called Kquasiisometric if for every x, y ∈ M_{1} we have K^{1}q_{1}(x, y) ≤ q_{2}(f(x), f(y)) ≤ K q_{1} (x,y). Let D_{1}, D_{2} be bounded subsets of Euclidean nspace R^{n} with n ≥ 2, let D_{1} be strictly convex and let q_{i} denote the restriction to ∂D_{i} of the intrinsic metric of D_{i} induced by the metric of R^{n}, i = 1, 2. The author proves that there is a constant C > 0 such that for every sufficiently small ε ≥ 0 the existence of a (1+ε)quasiisometric function f : (∂D_{1}, q_{1}) → (∂D_{2}, q_{2})
implies the existence of an isometry P for which the Hausdorff distance of D_{1} and P(D_{2}) is at most Cε^{1/4}. [M. Lassak (Bydgoszcz)]
Summary:
We obtain a new formulation of Efimov's differential condition which guarantees that a mapping f : R^{2} → R^{2} is a homeomorphism, and use it to obtain, with the aid of the HadamardLevyJohn global inverse function theorem, differential conditions under which f is not only injective but also surjective.
Zentralblatt review [Zbl 0687.53007]:
Connelly stated the following conjecture: Every banded polyhedron does not change its volume in the process of the bending. Sabitov suggested to consider this conjecture in the case when the bending is an infinitesimal one. His conjecture is: The volume of a closed surface does not change in the process of an infinitesimal bending. In this paper the author shows that Sabitov's conjecture is not true for polyhedra, but it is true for the surfaces of revolution with a regular meridian which does not contain segments, orthogonal to the rotation axis. [S. Hineva (Sofia)]
Zentralblatt review [Zbl 0847.57024]:
Let D be a domain of the space R^{n} (n ≥ 2), and let M_{D} be the inner metric of D. Consider the Hausdorff completion of the metric space (D, μ_{D}). By removing the elements corresponding to the points of D from this completion, one obtains the metric space (S(D), ρ_{D}); S(D) is the generalized boundary of the domain D, ρ_{D} is the relative metric of the generalized boundary. The author proves the following theorem: Any bounded domain of the space R^{n} (n ≥ 2) with a polyhedral boundary is determined uniquely by the relative metric of its generalized boundary in the class of all domains with polyhedral boundaries. [L.A. Aksent'ev (MR 90d:31017)]
Summary: We give some conditions under which a domain in the Euclidean plane is uniquely determined by the relative metric of its boundary. The conditions found are completely different for the following two classes of the domains: (1) the domains whose boundaries are real analytic manifolds, and (2) the domains whose boundaries are topological manifolds. The conditions found are shown to be necessary and sufficient for the class (1) and are shown to be be necessary and sufficient for all domains of the class (2) except some neglectable subset.
Zentralblatt review [Zbl 0604.53024]:
A mapping F : A → R^{n} defined on some subset A of the Euclidean space R^{n} is called a Euclidean isometry if for any two points a, b ∈ A the Euclidean distance between the images F(a) and F(b) equals the Euclidean distance between the points a and b. Two domains D and D' in R^{n} are called isometric if there exists a Euclidean isometry F : D → D'. Suppose that the boundary of the domain D ⊂ R^{n} is a connected manifold of class C^{1} and dimension n1 without boundary. With the use of the metric induced on D from the Euclidean metric of R^{n} one can construct in D the intrinsic metric. From the properties of the boundary of D it is clear that this intrinsic metric of D can be extended (by continuity) onto the boundary of the domain D, yielding
a metric on this boundary. The restriction of this extended metric to the boundary of the domain is called the relative metric of the boundary of D which we denote by ρ_{D}. Assume that the boundaries S and S' of the domains D and D' in R^{n} are connected manifolds of class C^{1} and dimension n1 without boundary. A mapping f : S → S' is called isometric in the relative metrics of the boundaries of the domains D and D' if for all points a, b ∈ S, we have ρ_{D'} (f(a), f(b)) = ρ_{D} (a,b). The boundaries S and S' are said to be isometric in the relative metrics of the boundaries of D and D'. Now, the present author has set forth the studies in his paper [ibid. 25, No.3, 313 (1984; Zbl 0576.53040)] to characterize those pairs of domains of finitedimensional Euclidean space for which isometry of the
boundaries in the relative metrics implies isometry of the domains in the Euclidean metrics [see also the paper of A.P. Kopylov, ibid. 25, No.3, 120131 (1984; Zbl 0594.54025)]. The prospects for a complete solution of this problem are unclear at this stage. However, the author's recent investigation indicates two directions towards its solution, and the first one has been dealt with by him in part 1 [loc. cit.], whereas in this paper he presents a way to approach to the second one with the following theorem: assume that D and D' in R^{n}, n ≥ 3, have nonempty bounded complement and that their boundaries are connected manifolds of class C^{1} and dimension n1 without boundary which are isometric in the relative metrics of the boundaries of D and D'. Then the domains D and D' are isometric. [T. Okubo (Victoria)]
Zentralblatt review [Zbl 0576.53040]:
A domain D of R^{n} is by definition an open and connected subset so that the metric of R^{n} may serve D as its intrinsic metric. Herewith the intrinsic metric of a topological space M is the metric such that the distance between any two points is given by the infimum of the lengths of all rectifiable curves between these two points, and one deals with domains in R^{n} whose intrinsic metric is continuously extendable to a metric on the closure of the domain. In particular, the restriction of this extended metric to the boundary of the domain D, denoted by ρ_{D}, will be called the relative metric. For two domains D and D' in R^{n} with boundaries S and S' , respectively, a mapping f : S → S' is said to be isometric in the relative metrics of the boundaries of D and D' if for any two points x and y
of S it holds that ρ_{D} (f(x), f(y)) = ρ_{D'} (x, y). Then the boundaries S and S' may also be called isometric in the relative metrics if there exists a surjective mapping f : S → S' which is isometric in the relative metrics. Evidently, the boundaries of D and D' are isometric if the relative metrics of D and D' are isometric. However, it is an open problem to ask if the converse statement is true, namely, does isometry of the boundaries of domains in the relative metrics imply isometry of domains themselves ? In this paper the author treats this problem for bounded domains with piecewisesmooth boundary. As its preliminary fact, he proves the following theorem: Let D and D' be bounded domains on R^{n}, n ≥ 2, whose boundaries S and S' are (n1)dimensional differentiable manifolds of class C^{1} without boundary.
If S and S' are isometric in the relative metrics, then D and D' are isometric. This theorem is then used to prove the following general theorem which states that, for bounded domains D and D' of R^{n}, n ≥ 2, with piecewisesmooth boundaries, if their boundaries are isometric in the relative metrics, then the domains are themselves isometric. [T. Okubo (Victoria)]
Summary:
The main concepts of the theory of tensors
are presented. The imphasis is on the basic notions of tensor
algebra and practical skills in calculations involving the Kronecker
delta and LeviCivita symbol. Sixty routine exercises are included.
The article is intendent for junior students of physical,
mathematical, and geophysical departments of universities.
Summary:
Basic notions are given related to the generalized functions (distributions)
and theirs applications to solving differential equations.
The booklet covers a part of a basic course on "Fundamentals of Functional Analysis"
delivered by the author for secondyear students of the physical department of the
Novosibirsk State University.
Contains problems recommended for classroom exercises.
Is destined for university students and teachers.
Summary:
Basic notions are given related to the Fourier transform and its
applications to solving differential equations and digital processing.
The booklet covers a part of a basic course on "Fundamentals of Functional Analysis"
delivered by the author for secondyear students of the physical department of the
Novosibirsk State University.
Contains problems recommended for classroom exercises.
Is destined for university students and teachers.
Summary:
This booklet covers a part of a basic twosemester course "Fundamentals of Functional Analysis and
Theory of Functions" delivered by the author for second year students at the physical
department of the Novosibirsk State University. 49 exercises and 6 pictures are included.
Contents: Preface (3). § 1. Concept of Fourier series of 2πperiodic functions and
Fourierseries expansion problem for periodic functions (47). § 2. Fourier series for a function with
arbitrary period (79). § 3. Expansions involving only sine or only cosine functions (911). § 4. The
RiemannLebesgue lemma (1114). § 5. The Dirichlet kernel (1417). § 6. A theorem on
pointwise convergence of Fourier series (1719). § 7. Examples of a Fourier series expansion of a
function and of summation of a numerical series by the use of Fourier series (1920). § 8. Complex
Fourier series (2123). § 9. Term by term differentiation and integration of Fourier series (2325).
§ 10. The best approximation problem and the Bessel inequality (2529). § 11. Uniform
convergence of Fourier series (2930). § 12. The Gibbs phenomenon (3033). § 13. Smoothness
of a function and degree of convergence of its Fourier series (3338). § 14. The Lyapunov equality
(3842). § 15. An application of Fourier series to finding of a harmonic function in a circle through
its boundary values (4248). § 16. The Weierstrass theorems on uniform approximationof a
continuous function by trigonometric and algebraic polynomials (4852). Answers (5354). Index (55).
Summary:
This book covers a part of a basic twosemester course
``Fundamentals of functional analysis and theory of functions''
delivered by the author for the second year students at the
Physical Department of the Novosibirsk State University.
30 exercises included.
Contents (page numbers are given in brackets):
Preface (35). § 1. Linear spaces (69).
§ 2. Normed linear spaces (913).
§ 3. Lebesgue functional spaces (1317).
§ 4. Proof of the Holder inequality (1718).
§ 5. Linear spaces with inner products:
Euclidean and unitary ones (1822).
§ 6. GramSchmidt orthogonalization process (2225).
§ 7. Approximation by vectors from a subspace
and orthogonal projection (2629).
§ 8. Projection into a finitedimensional subspace.
The Bessel inequality (2931).
§ 9. Completeness of an orthogonal system.
The Parseval equality. Closed systems (3134).
§ 10. Hilbert basis. An existence theorem for
Hilbert basis (3436).
§ 11. RieszFischer theorem.
Isomorphism of separable Hilbert spaces (3640).
§ 12. Criterion of completeness of an orthonormal
system in a separable Hilbert space (4042).
§ 13. The trigonometric system of functions as
an example of a complete orthonormal system in
L_{2} ([π,π]) (4244).
Index (4546).
Summary:
This book covers a part of a basic twosemester course
``Fundamentals of functional analysis and theory of functions''
delivered by the first author for the second year students at the
Physical Department of the Novosibirsk State University.
59 exercises included.
Contents:
Preface. § 1. Fredholm and Volterra integral equations
and examples of problems that lead to such equations.
§ 2. HilbertSchmidt integral operator.
§ 3. Solving the equations with degenerate kernels.
§ 4. Fredholm alternative.
§ 5. Equations with a small parameter.
Von Neumann series. The method of successive approximations.
§ 6. Integral equations with symmetric kernels.
§ 7. HilbertSchmidt theorem for integral operators.
Expansion of the solutions of an integral equation in
the eigenfunctions of the kernel.
§ 8. Decomposition of the iterated kernel of an integral
operator in its eigenfunctions. The bilinear formula.
§ 9. Application of the theory of integral operators
to the SturmLiouville problem.
§ 10. Volterra integral equations: a theorem on the
existence and uniqueness of solutions.
Zentralblatt review [Zbl 0857.42011]:
This is a textbook for undergraduate students in physics
studying mathematical analysis (calculus).
The book contains exercises, many physical applications
of the theory and does not contain integral representations
and asymptotics of orthogonal polynomials
(since only real analysis methods are used).
[A.Lukashov (Saratov)]
Summary:
This booklet covers a part of a basic twosemester course "Fundamentals of Functional Analysis and
Theory of Functions" delivered by the author for second year students of the physics department of the Novosibirsk State University.
132 exercises and 16 pictures are included.
Contents: Preface (3). § 1. Originals and their Laplace transforms (56).
§ 2. Linearity of the Laplace transform. Scaling (67).
§ 3. Frequency shifting (78).
§ 4. Differentiation and integration in time domain (812).
§ 5. Frequency differentiation and frequency integration (1214).
§ 6. Using the Laplace transform for solving bondary value problems for ordinary differential equations (1418).
§ 7. Time shifting (1920).
§ 8. Convolution. The Borel theorem (2021).
§ 9. The Duhamel formula (2123).
§ 10. Analyticity of the Laplace transform. The Bromwich integral (2324).
§ 11. Using the Laplace transform in the theory of electrical circuits (2429).
Table of selected Laplace transforms (30).
Summary:
The paper presents
the history of the Sobolev Institute of Mathematics in Novosibirsk, Russia.
Summary:
The paper contains a brief biography of Janos Bolyai,
one of the creators of the noneuclidean geometry.
Summary:
In this survey article we show how inverse and implicit function
theorems do work in the theory of polyhedra.
Namely, we show how they are used for proving classical and
new theorems about polyhedra, such as existence, uniqueness and
rigidity of a convex polyhedron with prescribed envelope,
construction of flexible polyhedra, existence and uniqueness of
a convex polyhedron with prescribed outward unit normal vectors
and surface areas of faces, generalisation of the latter results
to some class of nonconvex polyhedra, extension of infinitesimal
bending to ``continuous'' flexes, tiling a space with polyhedra, etc.
Summary:
In geometry `in the large' many theorems may be reformulated
in such a way that some particular mapping is injective or surjective.
We show that this point of view is very fruitful to prove some wellknown or new
results in geometry `in the large' and, first of all, in the theory of flexible polyhedra.
Summary:
In geometry `in the large' many theorems may be reformulated
in such a way that some particular mapping is injective or surjective.
We show that this point of view is very fruitful to prove some wellknown or new
results in geometry `in the large' and, first of all, in the theory of flexible polyhedra.
The paper presents recollections about my stay in Paris in spring 2001. It contains a story about an incident at avenue des ChampsElysees and historical remarks related to Paulownia tree.
Zentralblatt review:
[Zbl 0966.00048]
Zentralblatt review:
[Zbl 0947.52012]
To crush a closed polyhedral surface in the Euclidean 3space one has to mark new edges on it and
break its faces along those new edges. \par The paper is devoted to an elementary presentation of
recent results by {\it D.~Bleecker} [J. Differ. Geom. 43, No. 3, 505526 (1996; Zbl 0864.52003)]
who showed, in particular, that the surface of a right tetrahedron can be crushed in such a way as to
enlarge its volume. \par The article is based on a lecture given by the author for secondary school
teachers in the framework of the ``International Soros Science Educational Program'' and is aimed at
presenting contemporary developments in mathematics on an elementary level understandable for
secondary school teachers and pupils.
[ N.I.Alexandrova (Novosibirsk) ]
Zentralblatt review [Zbl 0880.52017]:
A part of the theory of closed flexible polyhedral surfaces is considered which does not require
additional knowledge and skills with respect to the secondary school course in stereometry. Bricard
octahedra and Steffen flexible polyhedron are considered in detail. For the first time a selfcontained
elementary proof is published of the fact that the both polyhedra are flexible. The bellows conjecture
is formulated (now an affirmative solution is published by I.Kh. Sabitov [Fundam. Prikl. Mat.
2, 12351246 (1996)] and R. Connelly, I. Sabitov, and A. Walz [Beitr. Algebra Geom.
38, 110 (1997)]). Applications of flexible octahedra to stereochemistry and unsolved problems are
discussed.
The paper is based on a lecture given by the author for secondary school teachers in
the framework of the ``International Soros Science Educational Program'' and is aimed at presenting
contemporary developments in mathematics on an elementary level understandable for secondary
school teachers and pupils. [ N.I.Alexandrova (Novosibirsk)]