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Summary:
The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space.
We derive a certain linear differential equation of the first order, which previously did not appear in the literature and which is
satisfied by any Darboux rotation field of a smooth surface.
We show that, for some surfaces, this additional equation is functionally independent of the three standard equations that
the Darboux rotation field satisfies (and by which it is determined).
As a consequence of this additional equation, we prove the maximum principle for the components of the Darboux rotation
field for disk-homeomorphic surfaces with strictly positive Gaussian curvature.
Summary:
We construct some sphere-homeomorphic flexible self-intersection-free polyhedron in Euclidean
3-space whose all dihedral angles change during some flex. The polyhedron has 26 vertices,
72 edges, and 48 faces. To study the properties of the polyhedron, we use traditional geometric
constructions and reasoning as well as symbolic calculations in the Wolfram Mathematica
software system.
Summary:
We study the problem of existence of two affine-equivalent bar-and-joint frameworks in Euclidean d-space
which have prescribed combinatorial structure and edge lengths.
We prove that theoretically this problem is always solvable, but we cannot propose any practical algorithm for its solution.
Summary:
The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes
have isometric developments then they are congruent.
In other words, we can decide whether two polyhedra are isometric or not by using their developments only.
In this article, we study a similar problem about whether it is possible, using only the developments of two convex
polyhedra of Euclidean 3-space, to understand that these polyhedra are (or are not) affine-equivalent.
Summary:
Given two convex octahedra in Euclidean 3-space, we find conditions on their natural developments
which are necessary and sufficient for these octahedra to be affinely equivalent to each other.
Summary:
We prove that first-order flexes of smooth surfaces in Euclidean 3-space, which are tangent to the set of all nonrigid surfaces,
can be extended to second-order flexes.
Summary:
In 1968, N.V. Efimov proved the following remarkable theorem:
Let f:R2→R2∈C1
det f'(x)<0 for all x∈R2 and let there exist
a function a(x)>0 and constants C1≥0, C2≥0
such that the inequalities
|1/a(x)-1/a(y)|≤C1|x-y|+C2
and
|det f'(x)|≥a(x)|curl f(x)|+a2(x)
hold true for all x, y∈R2.
Then f(R2) is a convex domain and f maps R2 onto f(R2) homeomorhically.
Here curl f(x) stands for the curl of f at x∈R2.
This article is an overview of analogues of this theorem, its generalizations
and applications in the theory of surfaces, theory of global inverse functions,
as well as in the study of the Jacobian Conjecture and the global asymptotic stability of dynamical systems.
Summary:
Being motivated by the theory of flexible polyhedra, we study the Dirichlet and Neumann eigenvalues for the Laplace operator in
special bounded domains of Euclidean d-space. The boundary of such a domain is an embedded simplicial complex which allows
a continuous deformation (a flex), under which each simplex of the complex moves as a solid body and the change in the spatial
shape of the domain is achieved through a change of the dihedral angles only. The main result of this article is that both the
Dirichlet and Neumann spectra of the Laplace operator in such a domain do not necessarily remain unaltered during the flex of its boundary.
Summary:
We derive fundamentally new equations that are satisfied by first-order
flexes of a flexible polyhedron. Moreover, we indicate two sources of such new
equations. These sources are the Dehn invariants and rigidity matrix. The
equations derived provide us with fundamentally new necessary conditions for
the extendibility of a first-order flex of a polyhedron to its flex.
Summary:
We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space,
calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing
its dihedral angles only. We prove that the polyhedron is not flexible if for each of its edges the following holds true:
the length of this edge is not a linear combination with rational coefficients of the lengths of the remaining edges.
We prove also that if a polyhedron is flexible, then some linear combinations of its dihedral angles remain constant during the flex.
In this case, the coefficients of such a linear combination do not alter during the flex, are integers, and do not equal to zero simultaneously.
Summary:
We choose some special unit vectors
n1,...,n5
in R3 and denote by
L ⊂ R5
the set of all points
(L1,...,L5) ∈ R5
with the following property:
there exists a compact convex polytope
P ⊂ R3
such that the vectors
n1,...,n5
(and no other vector) are unit outward normals to the faces of
P and the perimeter of the face with the outward normal
nk
is equal to Lk for all k=1,...,5.
Our main result reads that L is not
a locally-analytic set, i.e., we prove that, for some point
(L1,...,L5) ∈ L,
it is not possible to find a neighborhood
U ⊂ R5
and an analytic set
A ⊂ R5
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 3-space E3 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 many-dimensional analogue of this
theorem: for n ≥ 2, every n-dimensional cross-polytope in
E2n-2 with all diagonals of the same length and all edges
of the same length necessarily lies in En and hence is
a convex regular cross-polytope. We also apply our theorem
to the study of two-distance 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 non-degenerate non-flexible
polyhedron P in Euclidean 3-space, which is the limit
of a sequence of non-degenerate 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
non-degenerate polyhedra polyhedra which have the
same combinatorial structure as P has is not algebraic.
Summary:
We prove that, both in the Lobachevskij and spherical 3-spaces,
there exist nonconvex compact boundary-free 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 3-space
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
3-sphere?
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 sphere-homeomorphic surface in R3
and let k1(x)≤ k2(x)
be its principal curvatures at the point x.
If the inequalities
k1(x)≤ k≤ k2(x)
hold true with some constant k for all x∈ S
then S is a sphere. The
imphases is on a result of Y. Martinez-Maure who first proved that the
above statement is not valid for convex C2-surfaces. For convenience of
the reader, in addendum we give a Russian translation of that paper by
Y. Martinez-Maure originally published in French in C. R. Acad. Sci.,
Paris, Ser. I, Math. 332 (2001), 41-44.
Summary:
We construct a flexible (non immersed) suspension with a hexagonal equator in Euclidean 3-space
and study its properties related to the Strong Bellows Conjecture which reads as follows:
if an immersed polyhedron P in the Euclidean 3-space 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:R2→R2∈C1 be such that det f(x)<0
for all x∈R2 and let there exist
a function a=a(x)>0 and constants C1≥0, C2≥0
such that the inequalities
|1/a(x)-1/a(y)|≤C1 |x-y|+C2
and
|det f'(x)|≥a(x)>|curl f(x)|+a2(x)
hold true for all x, y∈R2.
Then f(R2) is a convex domain and f maps R2 onto f(R2) homeomorhically.
Here curl f(x) stands for the curl of f at x∈R2.
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 3-dimensional polyhedron
P can be defined as follows.
Let f:R→R be a
Q-linear function such that f(π)=0.
Then D(P)=∑l|l|f(αl),
where the sum runs over all edges l; αl
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 ``scissors-equivalent'', i.e., if
P can be sliced into finitely many pieces and reassembled
to form Q. Dehn showed that Df is not
constant over all polyhedra, implying that not all polyhedra
are scissors-equivalent (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
scissors-equivalent. 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 self-intersecting 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
self-intersecting) 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 scissors-equivalent.
[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 well-known 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⊂R3 is orthogonal
to S at least at two points).
MathSciNet review [MR2795785 (2012f:53007)]:
Associated with (infinitesimal) isometric deformations of
surfaces in R3 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 , 1-21, 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 by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space 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
R3 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)Σl ∈ E
|l|(π-α(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 {Pt} for 0≤t≤1
is a flex of P0, then M(Pt)
is independent of t. Second, the oriented volume of
Pt 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
γw(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=0M(P(t))=0;
this is due to J.R. Alexander
[Trans. Amer. Math. Soc. 288 (1985), no. 2, 661-678;
MR0776397 (86c:52004)]. The second result is that, if
PK is the set of all closed triangulated
polyhedra in R3 with prescribed combinatorial
structure K, then there exists a universal polynomial
pK whose coefficients are universal
polynomials in the edge lengths of P∈PK,
such that the oriented volume of any such P is a root of
pK; this was shown by I.K. Sabitov
[Discrete Comput. Geom. 20 (1998), no. 4, 405-425; MR1651896
(2000f:52010)].
The first result is analytic in nature, while the second
is algebraic. Here it is shown that there is no cross-over,
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. 1-2,
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 3-space to a line integral of a special vector field, which immediately yields the following well-known theorem: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.
MathSciNet review [MR2603844 (2010m:53006)]:
For any compact oriented smooth surface S in
R3 the number
HS=(1/2)∫S
(κ1+κ2)dS,
where κ1 and κ2 are the
principal curvatures of S, is called the integral
mean curvature of S.
Let ψv:
S×(-ε,ε)→R3,
given by (x,t)→x+tv(x),
be an infinitesimal transformation defined by a vector field
v. Let St ={ψv
(x,t) | x∈S} be the set of deformed
surfaces, and let n(x,t) be the unit normal
vector to St at x.
For each t∈(-ε,ε),
let n'(x,t)=∂n/∂t(x,t).
The variation of the integral curvature HS
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 cross-product in
R3. In this paper, the author proves that on
a compact oriented smooth surface S in
R3 and for an infinitesimal transformation
ψv, the variation H'S
of the integral mean curvature HS of S
can be expressed as H'S=(1/2)
∫∂Sm(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 well-known)
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 well-known
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 Qj
of P has a unit normal nj;
if Qj and Qk share
an edge, then
nj+nk≠0,
so that there is a shortest geodesic path ljk
on the unit sphere S2 joining
nj and nk;
two such curves ljk meet at most in a
common end-point; finally, the curves ljk
partition S2 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 (non-immersed) flexible polyhedra do exist in the Minkowski
3-space 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 non-null nonzero vectors in the Minkowski plane.
Zentralblatt review [Zbl 1028.52013]:
The author considers flexibility of polyhedra in a Minkowski 3-space, i.e., in a linear space
consisting of all ordered triples (x1, x2, x3) endowed with the scalar product
(x,y) = x1 y1 + x2 y2 - x3 y3.
He confirms the existence of (non-immersed) 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 2-space.
[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 : Rp × Rq → Rn provide sufficient conditions under which
f(x, φ(x)) ≡ 0 for some function φ : Rp → Rq 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 non-existence.
[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 3-space is called flexible if there exists an analytic (with
respect to a parameter t∈ [0,1]) family of polyhedra Qt, such that i) Q
coincides with Q0; ii) for each t∈ [0,1], the polyhedron Qt is isometric to
the polyhedron Q0 in the intrinsic metrics; iii) the polyhedron Q1 cannot be obtained
from Q0 by a rigid motion. The family of polyhedra Qt 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 (xi, yi, zi) 1≤ i≤ v, and e edges. If vertices
(xi, yi, zi) and (xj, yj, zj) 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 : R3v× R3v→
Re be a bilinear form, such that B maps the vectors
X=(x1, y1, z1, x2, y2, z2, ... , xv, yv, zv) and
U=(u1, v1, w1, u2, v2, w2, ... , uv, vv, wv) into a vector whose
e(i,j)-th component is equal to
(xi-xj)(ui-uj)+(yi-yj)(vi-vj)+(zi-zj)(wi- wj). If for
n≥ 1 all vertices i and j are joined by an edge we can write
∑m=0nB(Xm,Xn-m)=0, where
Xk=(x1,k, y1,k, z1,k, x2,k, y2,k, z2,k, ... , xv,k, yv,k, zv,k).
The vector X0 + tX1 + ... + tkXk is called the k-th order infinitesimal
flex of the polyhedron X0. 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 3-space is rigid. For further
details see I. Ivanova-Karatopraklieva and I.Kh. Sabitov [J. Math. Sci., New York 70,
No. 2, 1685-1716 (1994); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 23, 131-184
(1991; Zbl 0835.53003)] and I.Kh. Sabitov [Encycl. Math. Sci. 48, 179-250 (1992);
translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 196-270
(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
E3 conserves its volume during a flex. It is also known that each flexible polyhedron in
E3 conserves its mean curvature during a flex. In this paper the author constructs a
flexible polyhedron in an open half-sphere S3+ which conserves neither volume nor mean
curvature during a flex. The polyhedron is a modification of a polyhedron constructed from four
spherical digons in S3.
[Ch.Leytem (Cruchten)]
Zentralblatt review [Zbl 0879.51012]:
The author constructs a closed flexible polyhedron with self-intersections 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 three-space [see R. Connelly, I. Sabitov and A. Walz, Beitr. Alg. Geom. 38, No. 1, 1-10 (1997)]). [A.H. Temesvari (Sopron)]
MathSciNet review [MR1302424 (95i:90096)]:
For a rectangular domain Ω in Rn
the Gale-Nikaido theorem states that if
F=(f1,...,fn):
Ω → Rn is differentiable and
if every principal minor of the Jacobian F'(x)=
(∂fi(x)/∂xj)
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), 417-423, 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 ⊂ Rn and a
C∞-map F : Δ →
Rn 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),
139-142; MR0068618 (16,912b)]), which is closely related
to the Gale-Nikaido 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 ⊂ Rn and a mapping
F : D → Rn 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 ⊂ Rn 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 Rn provided that every intrinsic isometry f :
X → f(X) ⊂ Rn (i.e. isometry with respect to the intrinsic metrics) is an isometry. The paper concerns the problem of existence of a
k-dimensional set X rigid in Rn 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 R2 is rigid in R2 (a line in R2 is rational provided it has an equation a0 + a1 x1 + a2 x2 = 0 for some rational ai, 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, 221-230 (1994; Zbl 0803.54029)]. She gave a method (quite different than that of the author) of constructing (n-1)-dimensional rigid subsets of Rn 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, 951-965
and
Rev. Roumaine Math. Pures Appl. 33 (1988), no. 8, 695-700],
we do not assume any differentiability of f,
and we impose conditions on the growth of f instead of
fx.
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 finite-dimensional Banach spaces.
Zentralblatt review [Zbl 0894.53043]:
A non-schlicht n-dimensional domain with smooth boundary is a set D with the following properties: (a) D is contained in an n-dimensional Euclidean manifold Mn; (b) the metric space (D,dg) is complete; (c) for every point x ∈ ∂ D there are a neighbourhood V in Mn, an isometry ψ : V → Rn and a smooth n-dimensional manifold N with boundary in Rn 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 non-schlicht 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 non-schlicht domains which are diffeomorphic to the closed ball Dn. Let Μ be the set of locally Euclidean metrics on Dn defined on a neighbourhood of Dn. 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
Rn 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
Rn. (Roughly, the
inequality |det f ' (x)| ≤ a(x)|rot
f(x)| + a2(x),
rot f = D1f 2 - D2f 1,
with a>0 satisfying |(1/a(x)) - (1/a(y))| ≤
C1|x - y| + C2,
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
functional-theoretical 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 R2 with a hypothesis very similar
to the ones in Efimov's theorems, proving it as a corollary
of the well-known Hadamard-Levy-John 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 Hadamard-Levy-John
and Zorich theorems (for quasiconformal mappings).
[Reviewed by Miodrag Perovic.]
Summary:
The possibility of imbedding n-dimensional locally Euclidean metrics in the large in Rn 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 : (M1, q1) → (M2, q2) of metric spaces (Mi, qi), i = 1, 2, is called K-quasiisometric if for every x, y ∈ M1 we have K-1q1(x, y) ≤ q2(f(x), f(y)) ≤ K q1 (x,y). Let D1, D2 be bounded subsets of Euclidean n-space Rn with n ≥ 2, let D1 be strictly convex and let qi denote the restriction to ∂Di of the intrinsic metric of Di induced by the metric of Rn, 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 : (∂D1, q1) → (∂D2, q2)
implies the existence of an isometry P for which the Hausdorff distance of D1 and P(D2) is at most Cε1/4. [M. Lassak (Bydgoszcz)]
Summary:
We obtain a new formulation of Efimov's differential condition which guarantees that a mapping f : R2 → R2 is a homeomorphism, and use it to obtain, with the aid of the Hadamard-Levy-John 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 Rn (n ≥ 2), and let MD be the inner metric of D. Consider the Hausdorff completion of the metric space (D, μD). By removing the elements corresponding to the points of D from this completion, one obtains the metric space (S(D), ρD); S(D) is the generalized boundary of the domain D, ρD is the relative metric of the generalized boundary. The author proves the following theorem: Any bounded domain of the space Rn (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 → Rn defined on some subset A of the Euclidean space Rn 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 Rn are called isometric if there exists a Euclidean isometry F : D → D'. Suppose that the boundary of the domain D ⊂ Rn is a connected manifold of class C1 and dimension n-1 without boundary. With the use of the metric induced on D from the Euclidean metric of Rn 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 ρD. Assume that the boundaries S and S' of the domains D and D' in Rn are connected manifolds of class C1 and dimension n-1 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 ρD' (f(a), f(b)) = ρD (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, 3-13 (1984; Zbl 0576.53040)] to characterize those pairs of domains of finite-dimensional 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, 120-131 (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 Rn, n ≥ 3, have nonempty bounded complement and that their boundaries are connected manifolds of class C1 and dimension n-1 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 Rn is by definition an open and connected subset so that the metric of Rn 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 Rn 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 ρD, will be called the relative metric. For two domains D and D' in Rn 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 ρD (f(x), f(y)) = ρD' (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 piecewise-smooth boundary. As its preliminary fact, he proves the following theorem: Let D and D' be bounded domains on Rn, n ≥ 2, whose boundaries S and S' are (n-1)-dimensional differentiable manifolds of class C1 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 Rn, n ≥ 2, with piecewise-smooth 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 Levi-Civita 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 second-year 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 second-year students of the physical department of the
Novosibirsk State University.
Contains problems recommended for classroom exercises.
Is destined for university students and teachers.
Summary:
The book is devoted to an exposition of foundations of calculus of variations.
The book was written on the basis of a course of lectures given by the first author at the Physics Department of the Novosibirsk
State University and contains the basic results of classical calculus of variations.
Contains problems recommended for classroom exercises.
Is destined for university students and teachers.
Summary:
This booklet covers a part of a basic two-semester 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 2π-periodic functions and
Fourier-series expansion problem for periodic functions (4-7). § 2. Fourier series for a function with
arbitrary period (7-9). § 3. Expansions involving only sine or only cosine functions (9-11). § 4. The
Riemann-Lebesgue lemma (11-14). § 5. The Dirichlet kernel (14-17). § 6. A theorem on
pointwise convergence of Fourier series (17-19). § 7. Examples of a Fourier series expansion of a
function and of summation of a numerical series by the use of Fourier series (19-20). § 8. Complex
Fourier series (21-23). § 9. Term by term differentiation and integration of Fourier series (23-25).
§ 10. The best approximation problem and the Bessel inequality (25-29). § 11. Uniform
convergence of Fourier series (29-30). § 12. The Gibbs phenomenon (30-33). § 13. Smoothness
of a function and degree of convergence of its Fourier series (33-38). § 14. The Lyapunov equality
(38-42). § 15. An application of Fourier series to finding of a harmonic function in a circle through
its boundary values (42-48). § 16. The Weierstrass theorems on uniform approximationof a
continuous function by trigonometric and algebraic polynomials (48-52). Answers (53-54). Index (55).
Summary:
This book covers a part of a basic two-semester 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 (3-5). § 1. Linear spaces (6-9).
§ 2. Normed linear spaces (9-13).
§ 3. Lebesgue functional spaces (13-17).
§ 4. Proof of the Holder inequality (17-18).
§ 5. Linear spaces with inner products:
Euclidean and unitary ones (18-22).
§ 6. Gram-Schmidt orthogonalization process (22-25).
§ 7. Approximation by vectors from a subspace
and orthogonal projection (26-29).
§ 8. Projection into a finite-dimensional subspace.
The Bessel inequality (29-31).
§ 9. Completeness of an orthogonal system.
The Parseval equality. Closed systems (31-34).
§ 10. Hilbert basis. An existence theorem for
Hilbert basis (34-36).
§ 11. Riesz-Fischer theorem.
Isomorphism of separable Hilbert spaces (36-40).
§ 12. Criterion of completeness of an orthonormal
system in a separable Hilbert space (40-42).
§ 13. The trigonometric system of functions as
an example of a complete orthonormal system in
L2 ([-π,π]) (42-44).
Index (45-46).
Summary:
This book covers a part of a basic two-semester 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. Hilbert-Schmidt 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. Hilbert-Schmidt 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 Sturm-Liouville 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 two-semester 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 (5-6).
§ 2. Linearity of the Laplace transform. Scaling (6-7).
§ 3. Frequency shifting (7-8).
§ 4. Differentiation and integration in time domain (8-12).
§ 5. Frequency differentiation and frequency integration (12-14).
§ 6. Using the Laplace transform for solving bondary value problems for ordinary differential equations (14-18).
§ 7. Time shifting (19-20).
§ 8. Convolution. The Borel theorem (20-21).
§ 9. The Duhamel formula (21-23).
§ 10. Analyticity of the Laplace transform. The Bromwich integral (23-24).
§ 11. Using the Laplace transform in the theory of electrical circuits (24-29).
Table of selected Laplace transforms (30).
Summary:
This is a problem about a tetrahedron in Euclidean 3-space, which generalizes the Conway circle for a triangle on the plane.
It is published in the "Kvant's Problem Section" and is intended for high school students.
Its solution is published in Kvant, 2023, No. 8, 17-18.
Summary:
The article presents the higher mathematics chair of the physics department at Novosibirsk State University,
describing the history of its creation, its traditions, and methods used by its lecturers. The reader can also find here some
general information about the staff and the courses taught.
Summary:
It is known that the property of a bar-and-joint framework `to be
infinitesimally rigid' is preserved under projective transformations of ambient space.
In this article, we prove that the property of a bar-and-joint framework `to be
globally rigid' is not preserved even under affine transformations of ambient space.
Summary:
The paper presents
the history of the Sobolev Institute of Mathematics in Novosibirsk, Russia.
Summary:
The article presents the higher mathematics chair of the physics department at Novosibirsk State University,
describing the history of its creation, its traditions, and methods used by its lecturers. The reader can also find here some
general information about the staff and the courses taught.
Summary:
The paper contains a brief biography of Janos Bolyai,
one of the creators of the non-euclidean 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 non-convex 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 well-known 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 well-known 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 Champs-Elysees 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 3-space 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, 505-526 (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 self-contained
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, 1235-1246 (1996)] and R. Connelly, I. Sabitov, and A. Walz [Beitr. Algebra Geom.
38, 1-10 (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)]