... observations we get the following
51
st
International Mathematical Olympiad
Astana, Kazakhstan 2010
Problems with Solutions
10
We turn to the problem. First, suppose that g(k) = g(ℓ) for some k, ℓ ∈ N. Then ... and N, with ℓ ≤ s and such that a
n
= a
ℓ
+a
n−ℓ
for all n ≥ N.
Solution 1. First, from the problem conditions we have that each a
n
(n > s) can be expressed as
a
n
= a
j
1...
... theory for very weak solutions of the
A
-harmonic equations with w(x) ≡ 1 have been considere d [4], and the regularity the-
ory for very solutions of obstacle problems with w(x) ≡ 1 have been ... gives a Caccioppoli-type estimate for solutions to obstacle problems with
weight, which is closely related to the local regularity theory for very wea k solutions of
the
A
-harmoni...
...
www.MATHVN.com - Anh Quang Le, PhD
www.MathVn.com - Math Vietnam
MEASURE and INTEGRATION
Problems with Solutions
Anh Quang Le, Ph.D.
October 8, 2013
30 CHAPTER 2. LEBESGUE MEASURE ON R
Solution
(a) ... measurable.
(ii) For every ε > 0, there exists an closed set C ⊂ E with µ
∗
L
(E \ C) ≤ ε.
(iii) There exists a F
σ
-set F ⊂ E with µ
∗
L
(E \ F ) = 0.
Problem 18
Let Q be the se...
... Toronto
Problems and Solutions
Dr. David McKeown
Medical Officer of Health
November 2007
Air Pollution Illnesses from Traffic 11
emissions (which include PAHs) may be associated with breast ... associated
with increased cardiac
problems, including
heart attacks
Cancer
There is an increasing body of literature that suggests that vehicle emissions are also
associated wi...
... anode.
At present, anodes with Pt
–
Ru catalyst (with about 50% Ru) are used in the
majority of PEMFCs designed to work with technical hydrogen. It follows from
experiments with pure hydrogen (free ... counterelec-
trode), the front side of the electrode that is in contact with the electrolyte
is covered with a hydrophilic blocking layer having fine pores with a cap-
illary pressu...
... <t
T
<t
T+1
= 1.
Such problems without impulses effects have been solved before, for example, in [1–3].
But as far as we know the publication on the solvability of m-point problems with im-
pulses is ... Corporation
Boundary Value Problems
Volume 2007, Article ID 97067, 8 pages
doi:10.1155/2007/97067
Research Article
Solvability of Second-Order m-Point Boundary Value Problems...
... triangle?
SOLUTIONS
I. Introductory problems -
39
II. The representation
of
integers as sums
and
products - 52
III. Combinatorial problems on
the
chessboard - 76
IV. Geometric problems ... 102
V.
Problems on the binomial coefficients -
125
VI.
Problems on computing probabilities -
141
VII. Experiments with infinitely many possible
outcomes - 194
VIII. Experiments...
... Boundary Value Problems 2011, 2011:18
http://www.boundaryvalueproblems.com/content/2011/1/18
Page 2 of 9
RESEARC H Open Access
Sign-changing solutions for some nonlinear
problems with strong resonance
Aixia ... theory with applications to differential equations. CBMS Reg Cof Ser
Math. 65 (1986)
doi:10.1186/1687-2770-2011-18
Cite this article as: Qian: Sign-changing solutions for s...