Darij GrinbergKarlsruhe / Munich
(Germany) A
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update 7 Sep 2009 |
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I am sorry for the inconvenience.)
This website is dedicated to the Geometry of
the Triangle, and more generally to Euclidean Geometry. This area
of mathematics, standing somewhere between Recreational
Mathematics and Algebraic Geometry, today goes through a new
resurrection. The renewed interest in Euclidean Geometry can be
seen in Clark Kimberling's Encyclopedia
of Triangle Centers, in the journal Forum
Geometricorum, on Dick
Klingens' Geometry pages (Dutch), in
the MathLinks
forum, or in the Yahoo newsgroup "Hyacinthos" (in honor of the geometer Emile Michel Hyacinthe
Lemoine). More links can be found in the link list.
Click here for the FAQ.
It can answer some of the questions you wanted to ask me.
This site is currently on two different
servers:
(1) http://de.geocities.com/darij_grinberg/ this will soon expire | (2) http://www.cip.ifi.lmu.de/~grinberg/ .
Please use (2) for linking, because server (1) will soon be
history.
I have also put the
whole website as a ZIP file (well, two ZIP files, to be
honest) on Google Sites, because server (2) is not particularly
reliable and server (1) is closing down.
Publications, papers, notes
German papers on Elementary
Mathematics / Deutschsprachige Aufsätze über
Elementarmathematik
(Arbeiten über Elementargeometrie und Lösungen
von BWM- und QEDMO-Aufgaben)
Solutions to review problems
(Some problems from Mathematical Reflections and,
from 2010(?) on, American Mathematical Monthly, with my
solutions.)
Non-geometric works
A few texts I have written for diverse purposes, arranged according to potential usefulness (i. e., the first in the list are probably the most useful; this means that at least from the middle of the list on, you will only find boring bad-written stuff that noone cares about).
Darij Grinberg, A few facts on
integrality (version 5
September 2009).
PDF file.
Old version (version 20
August 2009) with slightly weaker Theorem 1 but better
proof (it's the same argument, but the generalization
made it harder to write up).
This is a four-part note about commutative algebra. Rings
mean commutative rings with unity.
Part 1 (Integrality over rings) is an
(over-formalized) writeup of proofs to some known and
less known results about integrality over rings. If A is
a subring of a ring B, and n is an integer, then an
element u of B is said to be n-integral over A
if there exists a monic polynomial P of degree n with
coefficients in A such that P(u) = 0. We show that:
- (Theorem 1) An element u of B is n-integral over A if
and only if there exists an n-generated (= generated by n
elements) A-submodule U of B such that uU is a subset of
U and such that v = 0 for every v in B satisfying vU = 0.
- (Theorem 1 as well) An element u of B is n-integral
over A if and only if there exists an n-generated (=
generated by n elements) A-submodule U of B such that 1
lies in U and uU is a subset of U.
- (Theorem 1 in the alternative
version) An element u of B is n-integral over A if
and only if there exists a faithful n-generated (=
generated by n elements) A-submodule U of some B-module
such that uU is a subset of U.
- (Theorem 2) If a_{0}, a_{1}, ..., a_{n}
are elements of A and v is an element of B such that
SUM_{i=0}^{n} a_{i}v^{i}
= 0, then SUM_{i=0}^{n-k} a_{i+k}v^{i}
is n-integral over A for every 0 <= k <= n. (This
result, and its Corollary 3, generalize exercise
2-5 in J.
S. Milne's Algebraic Number Theory.)
- (Theorem 4) If some element v of B is m-integral over
A, and some element u of B is n-integral over A[v], then
u is nm-integral over A. (This is a known fact. I derive
it from Theorem 1, just as most people do. Maybe I will
also write up a different proof using resultants.)
- (Theorem 5) Any element of A is 1-integral over A. If
some element x of B is m-integral over A, and some
element y of B is n-integral over A, then x+y and xy are
nm-integral over A. (This is known again. I use Theorem 4
to prove this.)
- (Corollary 6) Let v be an element of B, and n and m two
positive integers. Let P be a polynomial of degree n-1
with coefficients in A, and let u = P(v). If vu is
m-integral over A, then u is nm-integral over A. (This
follows from Theorems 2 and 5 but may turn out useful,
though I don't expect much.)
Part 2 (Integrality over ideal semifiltrations)
gives a common generalization to integrality over rings
(as considered in Part 1) and integrality
over ideals (a less known, but still important
notion).
We define an ideal semifiltration of a ring A as
a sequence (I_{i})_{i>=0.} of ideals of A such
that I_{0} = A and I_{a}I_{b}
is a subset of I_{a+b} for
any a >= 0 and b >= 0. (This notion is weaker than
that of an ideal filtration, since we do not require that
I_{n+1} is a subset of I_{n} for every n >= 0.)
If A is a subring of a ring B, if (I_{i})_{i>=0} is an ideal
semifiltration of A, and if n is an integer, then an
element u of B is said to be n-integral over (A,(I_{i})_{i>=0}) if
there exists a monic polynomial P of degree n with
coefficients in A such that P(u) = 0 and the i-th
coefficient of P lies in I_{deg P - i}
for every i in {0, 1, ..., deg P}.
While this notion is much more general than integrality
over rings (which is its particular case when (I_{i})_{i>=0}
= (A)_{i>=0}) and
integrality over ideals (which is its particular case
when B = A and (I_{i})_{i>=0} = (I^{i})_{i>=0} for some fixed ideal
I), it still can be reduced to basic integrality over
rings by a base change. Namely:
- (Theorem 7) The element u of B is n-integral over (A,(I_{i})_{i>=0})
if and only if the element uY of the polynomial ring B[Y]
is n-integral over the Rees algebra A[(I_{i})_{i>=0}*Y]. (This Rees algebra
A[(I_{i})_{i>=0}*Y]
is defined as the subring I_{0}Y^{0}
+ I_{1}Y^{1} + I_{2}Y^{2} + ... of the
polynomial ring A[Y]. Not that I would particularly like
the notation A[(I_{i})_{i>=0}*Y], but I have not
seen a better one.)
(The idea underlying this theorem is not new, but I
haven't seen it stated in standard texts on integrality.)
Using this reduction, we can generalize Theorems 4 and 5:
- (Theorem 8, generalizing Theorem 5) An element of A is
1-integral over (A,(I_{i})_{i>=0}) if and only if it
lies in I_{1}. If some
element x of B is m-integral over (A,(I_{i})_{i>=0}), and some element y
of B is n-integral over (A,(I_{i})_{i>=0}), then x+y is
nm-integral over (A,(I_{i})_{i>=0}). If some element x of
B is m-integral over (A,(I_{i})_{i>=0}), and some element y
of B is n-integral over A (not necessarily over (A,(I_{i})_{i>=0})
!), then xy is nm-integral over (A,(I_{i})_{i>=0}).
- (Theorem 9, generalizing Theorem 4) If some element v
of B is m-integral over A, and some element u of B is
n-integral over (A[v], (I_{i}A[v])_{i>=0}), then u is
nm-integral over (A,(I_{i})_{i>=0}).
Note that Theorem 9 doesn't seem to yield Theorem 8 as
easily as Theorem 5 could be derived from Theorem 4 !
Part 3 (Generalizing to two ideal
semifiltrations) continues Part 2, generalizing
a part of it even further:
Let A be a subring of a ring B. Let (I_{i})_{i>=0} and (J_{i})_{i>=0} be two ideal
semifiltrations of A. Then, (I_{i}J_{i})_{i>=0}
is an ideal semifiltration of A, as well. Now, we can
give a "relative" version of Theorem 7:
- (Theorem 11) An element u of B is n-integral over (A,(I_{i}J_{i})_{i>=0}) if and only if the
element uY of the polynomial ring B[Y] is n-integral over
the (A_{[I]}, (J_{i}A_{[I]})_{i>=0}), where A_{[I]} is a shorthand for the
Rees algebra A[(I_{i})_{i>=0}*Y].
Using this, we can generalize the xy part of Theorem 8
even further:
- (Theorem 13) If some element x of B is m-integral over
(A,(I_{i})_{i>=0}),
and some element y of B is n-integral over (A,(J_{i})_{i>=0}),
then xy is nm-integral over (A,(I_{i}J_{i})_{i>=0}).
Part 4 (Accelerating ideal semifiltrations)
extends Theorem 7:
- (Theorem 16) We will work under the same conditions at
Let s >= 0 be an integer. The element u of B is
n-integral over (A,(I_{si})_{i>=0}) if and only if the
element uY^{s} of the polynomial ring B[Y] is
n-integral over the Rees algebra A[(I_{i})_{i>=0}*Y].
Actually, this can be further generalized in the vein of
Theorem 11 (to Theorem 15).
As a consequence, Theorem 2 is generalized.
Darij Grinberg, The Vornicu-Schur
inequality and its variations
(version 13 August 2007).
PDF file. A copy can be found in
a MathLinks article.
The so-called Vornicu-Schur inequality states that
x(a-b)(a-c) + y(b-c)(b-a) + z(c-a)(c-b) >= 0, where a,
b, c are reals and x, y, z are nonnegative reals. Of
course, this inequality only holds when certain
conditions are imposed on a, b, c, x, y, z, and the
purpose of this note is to collect some of the possible
conditions that make the inequality valid. For instance,
(a >= b >= c and x + z >= y) is one such
sufficient condition (covering the most frequently used
condition (a >= b >= c and (x >= y >= z or x
<= y <= z))). Another sufficient condition is that
x, y, z are the sidelengths of a triangle. An even
weaker, but still sufficient one is that x, y, z are the
squares of the sidelengths of a triangle. A yet different
sufficient condition is that ax, by, cz are the
sidelengths of a triangle - or, again, their squares.
These, and more, conditions are discussed, and some
variations and equivalent versions of the Vornicu-Schur
inequality are shown. The note is not primarily focused
on applications, but a few inequalities that can be
proven using the Vornicu-Schur inequality are given as
exercises.
Darij Grinberg, An inequality
involving 2n numbers (version
22 August 2007).
PDF file.
The main result of this note is the following inequality:
Theorem 1.1. Let a_{1},
a_{2}, ..., a_{n},
b_{1}, b_{2},
..., b_{n }be 2n reals.
Assume that sum_{1<=i<j<=n} a_{i}a_{j }>= 0 or
sum_{1<=i<j<=n} b_{i}b_{j }>= 0. Then,
(sum_{1<=i<=n, 1<=j<=n, i \neq j} a_{i}b_{j})^{2}
>= 4 sum_{1<=i<j<=n} a_{i}a_{j }sum_{1<=i<j<=n} b_{i}b_{j}.
This result can either be deduced from the Aczel
inequality (one of the many variations on
Cauchy-Schwarz), or verified more directly by algebraic
manipulation. It appeared in the 39th Yugoslav Federal
Mathematical Competition 1998 as problem 1 for the 3rd
and 4th grades, but in a weaker form (the reals a_{1}, a_{2},
..., a_{n}, b_{1},
b_{2}, ..., b_{n
}were required to be nonnegative, while we
only require sum_{1<=i<j<=n} a_{i}a_{j }>= 0 or
sum_{1<=i<j<=n} b_{i}b_{j }>= 0).
After proving Theorem 1.1, we apply it to establish some
inequalities, including an n-numbers generalization of
Walther Janous'
a/(b+c) * (v+w) + b/(c+a) * (w+u) + c/(a+b) * (u+v) >=
sqrt(3(vw+wu+uv)) >= 3(vw+wu+uv) / (u+v+w).
Darij Grinberg, Math Time problem
proposal #1 (version 24 August
2007).
PDF file (with solution).
Let x_{1}, x_{2},
..., x_{n} be real numbers
such that x_{1} + x_{2} + ... + x_{n}
= 1 and such that x_{i}
< 1 for every i in {1,2,...,n}. Prove that
sum_{1<=i<j<=n} x_{i}x_{j} / ((1-x_{i})(1-x_{j})) >= n / (2(n-1)).
[Note that we do not require x_{1},
x_{2}, ..., x_{n}
to be nonnegative - otherwise, the problem would be much
easier.]
Darij Grinberg, Generalizations of
Popoviciu's inequality (version
4 March 2009).
PDF version. This note was published in arXiv under arXiv:0803.2958,
but the version there is older (20 March 2008), though
the changes are non-substantial.
Additionally, here you can find a "formal version"
(PDF) of the note (i. e. a version where proofs are
elaborated with more detail; you won't need the formal
version unless you have troubles with understanding the
standard one).
We establish a general criterion for inequalities of the
kind
convex combination of f(x_{1}),
f(x_{2}), ..., f(x_{n}) and f(some weighted mean of
x_{1}, x_{2},
..., x_{n})
>= convex combination of f(some other weighted means
of x_{1}, x_{2},
..., x_{n}),
where f is a convex function on an interval I of the real
axis containing the reals x_{1},
x_{2}, ..., x_{n},
to hold. Here, the left hand side contains only one
weighted mean, while the right hand side may contain as
many as possible, as long as there are finitely many. The
weighted mean on the left hand side must have positive
weights, while those on the right hand side must have
nonnegative weights.
This criterion entails Vasile Cîrtoaje's generalization
of the Popoviciu inequality (in its standard and in its
weighted forms) as well as a cyclic inequality that
sharpens another result by Vasile Cîrtoaje. This cyclic
inequality (in its non-weighted form) states that
2 SUM_{i=1}^{n} f(x_{i}) +
n(n-2) f(x) >= n SUM_{s=1}^{n} f(x + (x_{s}
- x_{s+r})/n),
where indices are cyclic modulo n, and x = (x_{1} + x_{2}
+ ... + x_{n})/n.
Darij Grinberg, St. Petersburg 2003:
An alternating sum of zero-sum subset numbers
(version 14 March 2008).
PDF file.
Using a lemma about finite differences (which is proven
in detail), the following two problems are solved:
Problem
1 (Saint Petersburg Mathematical Olympiad 2003).
For any prime p and for any n integers a_{1},
a_{2}, ..., a_{n}
with n >= p, show that the number
SUM_{k=0}^{n} (-1)^{k} *
(number of subsets T of {1, 2, ..., n} with k elements
such that the sum of these k elements is divisible by p)
is divisible by p.
Problem
2 (user named "lzw75" on MathLinks).
Let p be a prime, let m be an integer, and let n >
(p-1)m be an integer. Let a_{1},
a_{2}, ..., a_{n}
be n elements of the vector space F_{p}^{m}. Prove that
there exists a non-empty subset T of {1, 2, ..., n} such
that SUM_{t in T} a_{t} =
0.
I am working on an update of this note focussing
more on the finite differences lemma and less on Problems
1 and 2 (not like I want to remove these
problems, but I want to add more about finite differences
and a few other applications).
Darij Grinberg, Proof of a CWMO problem
generalized (version 7
September 2009).
PDF file.
The point of this note is to prove a result by Dan
Schwarz (appearing as problem
4 (c) on the Romanian MO 2004 for the 9th grade and
as a generalization
of CWMO 2006 problem 8 provided by a MathLinks user named
tanlsth):
Let X be a set. Let n and m >= 1 be two nonnegative
integers such that |X| >= m (n-1) + 1. Let B_{1}, B_{2},
..., B_{n} be n subsets of
X such that |B_{i}| = m for
every i. Then, there exists a subset Y of X such that |Y|
= n and Y has at most one element in common with B_{i} for every i.
Darij Grinberg, An algebraic approach to
Hall's matching theorem
(version 6 October 2007).
PDF file.
This note is quite a pain to read, mostly due to its
length. If you are really interested in the proof, try the abridged version first.
Hall's matching theorem (also called marriage theorem)
has received a number of different proofs in
combinatorial literature. Here is a proof which appears
to be new. However, due to its length, it is far from
being of any particular interest, except for one idea
applied in it, namely the construction of the matrix S.
See the
corresponding MathLinks topic for details.
It turned out that the idea is not new, having been
discovered by Tutte long ago, rendering the above note
completely useless.
Darij Grinberg, Two problems on complex
cosines (version 3 February
2007).
PDF file; a copy (possibly outdated) is also downloadable
from the MathLinks forum as an attachment in the topic
"I cant get it".
This note discusses five properties of sequences of
complex numbers x_{1}, x_{2}, ..., x_{n}
satisfying either the equation
x_{1} = 1/x_{1}
+ x_{2} = 1/x_{2}
+ x_{3} = ... = 1/x_{n-1} + x_{n}
or the equation
x_{1} = 1/x_{1}
+ x_{2} = 1/x_{2}
+ x_{3} = ... = 1/x_{n-1} + x_{n}
= 1/x_{n}.
Two of these properties have been posted on MathLinks and
seem to be olympiad folklore.