---
_id: '8514'
abstract:
- lang: eng
text: We study the extent to which the Hausdorff dimension of a compact subset of
an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional
space. It is possible that the dimension drops under all such mappings, but the
amount by which it typically drops is controlled by the ‘thickness exponent’ of
the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275).
More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness
exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally)
Lipschitz functions from $B$ to $\mathbb{R}^{m}$ that contains the space of bounded
linear functions. We prove that for almost every (in the sense of prevalence)
function $f \in M$, the Hausdorff dimension of $f(X)$ is at least $\min\{ m, d
/ (1 + \tau) \}$. We also prove an analogous result for a certain part of the
dimension spectra of Borel probability measures supported on $X$. The factor $1
/ (1 + \tau)$ can be improved to $1 / (1 + \tau / 2)$ if $B$ is a Hilbert space.
Since dimension cannot increase under a (locally) Lipschitz function, these theorems
become dimension preservation results when $\tau = 0$. We conjecture that many
of the attractors associated with the evolution equations of mathematical physics
have thickness exponent zero. We also discuss the sharpness of our results in
the case $\tau > 0$.
article_processing_charge: No
article_type: original
author:
- first_name: WILLIAM
full_name: OTT, WILLIAM
last_name: OTT
- first_name: BRIAN
full_name: HUNT, BRIAN
last_name: HUNT
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
citation:
ama: OTT W, HUNT B, Kaloshin V. The effect of projections on fractal sets and measures
in Banach spaces. Ergodic Theory and Dynamical Systems. 2006;26(3):869-891.
doi:10.1017/s0143385705000714
apa: OTT, W., HUNT, B., & Kaloshin, V. (2006). The effect of projections on
fractal sets and measures in Banach spaces. Ergodic Theory and Dynamical Systems.
Cambridge University Press. https://doi.org/10.1017/s0143385705000714
chicago: OTT, WILLIAM, BRIAN HUNT, and Vadim Kaloshin. “The Effect of Projections
on Fractal Sets and Measures in Banach Spaces.” Ergodic Theory and Dynamical
Systems. Cambridge University Press, 2006. https://doi.org/10.1017/s0143385705000714.
ieee: W. OTT, B. HUNT, and V. Kaloshin, “The effect of projections on fractal sets
and measures in Banach spaces,” Ergodic Theory and Dynamical Systems, vol.
26, no. 3. Cambridge University Press, pp. 869–891, 2006.
ista: OTT W, HUNT B, Kaloshin V. 2006. The effect of projections on fractal sets
and measures in Banach spaces. Ergodic Theory and Dynamical Systems. 26(3), 869–891.
mla: OTT, WILLIAM, et al. “The Effect of Projections on Fractal Sets and Measures
in Banach Spaces.” Ergodic Theory and Dynamical Systems, vol. 26, no. 3,
Cambridge University Press, 2006, pp. 869–91, doi:10.1017/s0143385705000714.
short: W. OTT, B. HUNT, V. Kaloshin, Ergodic Theory and Dynamical Systems 26 (2006)
869–891.
date_created: 2020-09-18T10:48:52Z
date_published: 2006-06-01T00:00:00Z
date_updated: 2021-01-12T08:19:48Z
day: '01'
doi: 10.1017/s0143385705000714
extern: '1'
intvolume: ' 26'
issue: '3'
language:
- iso: eng
month: '06'
oa_version: None
page: 869-891
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
issn:
- 0143-3857
- 1469-4417
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: The effect of projections on fractal sets and measures in Banach spaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 26
year: '2006'
...
---
_id: '8515'
abstract:
- lang: eng
text: "We consider the evolution of a set carried by a space periodic incompressible
stochastic flow in a Euclidean space. We\r\nreport on three main results obtained
in [8, 9, 10] concerning long time behaviour for a typical realization of the
stochastic flow. First, at time t most of the particles are at a distance of order
√t away from the origin. Moreover, we prove a Central Limit Theorem for the evolution
of a measure carried by the flow, which holds for almost every realization of
the flow. Second, we show the existence of a zero measure full Hausdorff dimension
set of points, which\r\nescape to infinity at a linear rate. Third, in the 2-dimensional
case, we study the set of points visited by the original set by time t. Such a
set, when scaled down by the factor of t, has a limiting non random shape."
article_processing_charge: No
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: D.
full_name: DOLGOPYAT, D.
last_name: DOLGOPYAT
- first_name: L.
full_name: KORALOV, L.
last_name: KORALOV
citation:
ama: 'Kaloshin V, DOLGOPYAT D, KORALOV L. Long time behaviour of periodic stochastic
flows. In: XIVth International Congress on Mathematical Physics. World
Scientific; 2006:290-295. doi:10.1142/9789812704016_0026'
apa: 'Kaloshin, V., DOLGOPYAT, D., & KORALOV, L. (2006). Long time behaviour
of periodic stochastic flows. In XIVth International Congress on Mathematical
Physics (pp. 290–295). Lisbon, Portugal: World Scientific. https://doi.org/10.1142/9789812704016_0026'
chicago: Kaloshin, Vadim, D. DOLGOPYAT, and L. KORALOV. “Long Time Behaviour of
Periodic Stochastic Flows.” In XIVth International Congress on Mathematical
Physics, 290–95. World Scientific, 2006. https://doi.org/10.1142/9789812704016_0026.
ieee: V. Kaloshin, D. DOLGOPYAT, and L. KORALOV, “Long time behaviour of periodic
stochastic flows,” in XIVth International Congress on Mathematical Physics,
Lisbon, Portugal, 2006, pp. 290–295.
ista: Kaloshin V, DOLGOPYAT D, KORALOV L. 2006. Long time behaviour of periodic
stochastic flows. XIVth International Congress on Mathematical Physics. International
Congress on Mathematical Physics, 290–295.
mla: Kaloshin, Vadim, et al. “Long Time Behaviour of Periodic Stochastic Flows.”
XIVth International Congress on Mathematical Physics, World Scientific,
2006, pp. 290–95, doi:10.1142/9789812704016_0026.
short: V. Kaloshin, D. DOLGOPYAT, L. KORALOV, in:, XIVth International Congress
on Mathematical Physics, World Scientific, 2006, pp. 290–295.
conference:
end_date: 2003-08-02
location: Lisbon, Portugal
name: International Congress on Mathematical Physics
start_date: 2003-07-28
date_created: 2020-09-18T10:48:59Z
date_published: 2006-03-01T00:00:00Z
date_updated: 2021-01-12T08:19:49Z
day: '01'
doi: 10.1142/9789812704016_0026
extern: '1'
language:
- iso: eng
month: '03'
oa_version: None
page: 290-295
publication: XIVth International Congress on Mathematical Physics
publication_identifier:
isbn:
- '9789812562012'
- '9789812704016'
publication_status: published
publisher: World Scientific
quality_controlled: '1'
status: public
title: Long time behaviour of periodic stochastic flows
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2006'
...
---
_id: '8513'
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Maria
full_name: Saprykina, Maria
last_name: Saprykina
citation:
ama: Kaloshin V, Saprykina M. Generic 3-dimensional volume-preserving diffeomorphisms
with superexponential growth of number of periodic orbits. Discrete & Continuous
Dynamical Systems - A. 2006;15(2):611-640. doi:10.3934/dcds.2006.15.611
apa: Kaloshin, V., & Saprykina, M. (2006). Generic 3-dimensional volume-preserving
diffeomorphisms with superexponential growth of number of periodic orbits. Discrete
& Continuous Dynamical Systems - A. American Institute of Mathematical
Sciences (AIMS). https://doi.org/10.3934/dcds.2006.15.611
chicago: Kaloshin, Vadim, and Maria Saprykina. “Generic 3-Dimensional Volume-Preserving
Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits.” Discrete
& Continuous Dynamical Systems - A. American Institute of Mathematical
Sciences (AIMS), 2006. https://doi.org/10.3934/dcds.2006.15.611.
ieee: V. Kaloshin and M. Saprykina, “Generic 3-dimensional volume-preserving diffeomorphisms
with superexponential growth of number of periodic orbits,” Discrete &
Continuous Dynamical Systems - A, vol. 15, no. 2. American Institute of Mathematical
Sciences (AIMS), pp. 611–640, 2006.
ista: Kaloshin V, Saprykina M. 2006. Generic 3-dimensional volume-preserving diffeomorphisms
with superexponential growth of number of periodic orbits. Discrete & Continuous
Dynamical Systems - A. 15(2), 611–640.
mla: Kaloshin, Vadim, and Maria Saprykina. “Generic 3-Dimensional Volume-Preserving
Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits.” Discrete
& Continuous Dynamical Systems - A, vol. 15, no. 2, American Institute
of Mathematical Sciences (AIMS), 2006, pp. 611–40, doi:10.3934/dcds.2006.15.611.
short: V. Kaloshin, M. Saprykina, Discrete & Continuous Dynamical Systems -
A 15 (2006) 611–640.
date_created: 2020-09-18T10:48:43Z
date_published: 2006-05-01T00:00:00Z
date_updated: 2021-01-12T08:19:48Z
day: '01'
doi: 10.3934/dcds.2006.15.611
extern: '1'
intvolume: ' 15'
issue: '2'
language:
- iso: eng
month: '05'
oa_version: None
page: 611-640
publication: Discrete & Continuous Dynamical Systems - A
publication_identifier:
issn:
- 1553-5231
publication_status: published
publisher: American Institute of Mathematical Sciences (AIMS)
quality_controlled: '1'
status: public
title: Generic 3-dimensional volume-preserving diffeomorphisms with superexponential
growth of number of periodic orbits
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 15
year: '2006'
...
---
_id: '869'
abstract:
- lang: eng
text: The impact of synonymous nucleotide substitutions on fitness in mammals remains
controversial. Despite some indications of selective constraint, synonymous sites
are often assumed to be neutral, and the rate of their evolution is used as a
proxy for mutation rate. We subdivide all sites into four classes in terms of
the mutable CpG context, nonCpG, postC, preG, and postCpreG, and compare four-fold
synonymous sites and intron sites residing outside transposable elements. The
distribution of the rate of evolution across all synonymous sites is trimodal.
Rate of evolution at nonCpG synonymous sites, not preceded by C and not followed
by G, is ∼10% below that at such intron sites. In contrast, rate of evolution
at postCpreG synonymous sites is ∼30% above that at such intron sites. Finally,
synonymous and intron postC and preG sites evolve at similar rates. The relationship
between the levels of polymorphism at the corresponding synonymous and intron
sites is very similar to that between their rates of evolution. Within every class,
synonymous sites are occupied by G or C much more often than intron sites, whose
nucleotide composition is consistent with neutral mutation-drift equilibrium.
These patterns suggest that synonymous sites are under weak selection in favor
of G and C, with the average coefficient s∼0.25/Ne∼10-5, where Ne is the effective
population size. Such selection decelerates evolution and reduces variability
at sites with symmetric mutation, but has the opposite effects at sites where
the favored nucleotides are more mutable. The amino-acid composition of proteins
dictates that many synonymous sites are CpGprone, which causes them, on average,
to evolve faster and to be more polymorphic than intron sites. An average genotype
carries ∼107 suboptimal nucleotides at synonymous sites, implying synergistic
epistasis in selection against them.
acknowledgement: This research was supported in part by the Intramural Research Program
of the NIH, National Library of Medicine.
author:
- first_name: Fyodor
full_name: Fyodor Kondrashov
id: 44FDEF62-F248-11E8-B48F-1D18A9856A87
last_name: Kondrashov
orcid: 0000-0001-8243-4694
- first_name: Aleksey
full_name: Ogurtsov, Aleksey Yu
last_name: Ogurtsov
- first_name: Alexey
full_name: Kondrashov, Alexey S
last_name: Kondrashov
citation:
ama: Kondrashov F, Ogurtsov A, Kondrashov A. Selection in favor of nucleotides G
and C diversifies evolution rates and levels of polymorphism at mammalian synonymous
sites. Journal of Theoretical Biology. 2006;240(4):616-626. doi:10.1016/j.jtbi.2005.10.020
apa: Kondrashov, F., Ogurtsov, A., & Kondrashov, A. (2006). Selection in favor
of nucleotides G and C diversifies evolution rates and levels of polymorphism
at mammalian synonymous sites. Journal of Theoretical Biology. Elsevier.
https://doi.org/10.1016/j.jtbi.2005.10.020
chicago: Kondrashov, Fyodor, Aleksey Ogurtsov, and Alexey Kondrashov. “Selection
in Favor of Nucleotides G and C Diversifies Evolution Rates and Levels of Polymorphism
at Mammalian Synonymous Sites.” Journal of Theoretical Biology. Elsevier,
2006. https://doi.org/10.1016/j.jtbi.2005.10.020.
ieee: F. Kondrashov, A. Ogurtsov, and A. Kondrashov, “Selection in favor of nucleotides
G and C diversifies evolution rates and levels of polymorphism at mammalian synonymous
sites,” Journal of Theoretical Biology, vol. 240, no. 4. Elsevier, pp.
616–626, 2006.
ista: Kondrashov F, Ogurtsov A, Kondrashov A. 2006. Selection in favor of nucleotides
G and C diversifies evolution rates and levels of polymorphism at mammalian synonymous
sites. Journal of Theoretical Biology. 240(4), 616–626.
mla: Kondrashov, Fyodor, et al. “Selection in Favor of Nucleotides G and C Diversifies
Evolution Rates and Levels of Polymorphism at Mammalian Synonymous Sites.” Journal
of Theoretical Biology, vol. 240, no. 4, Elsevier, 2006, pp. 616–26, doi:10.1016/j.jtbi.2005.10.020.
short: F. Kondrashov, A. Ogurtsov, A. Kondrashov, Journal of Theoretical Biology
240 (2006) 616–626.
date_created: 2018-12-11T11:48:56Z
date_published: 2006-06-21T00:00:00Z
date_updated: 2021-01-12T08:20:33Z
day: '21'
doi: 10.1016/j.jtbi.2005.10.020
extern: 1
intvolume: ' 240'
issue: '4'
month: '06'
page: 616 - 626
publication: Journal of Theoretical Biology
publication_status: published
publisher: Elsevier
publist_id: '6779'
quality_controlled: 0
status: public
title: Selection in favor of nucleotides G and C diversifies evolution rates and levels
of polymorphism at mammalian synonymous sites
type: journal_article
volume: 240
year: '2006'
...
---
_id: '903'
abstract:
- lang: eng
text: 'Background: Carcinogenesis typically involves multiple somatic mutations
in caretaker (DNA repair) and gatekeeper (tumor suppressors and oncogenes) genes.
Analysis of mutation spectra of the tumor suppressor that is most commonly mutated
in human cancers, p53, unexpectedly suggested that somatic evolution of the p53
gene during tumorigenesis is dominated by positive selection for gain of function.
This conclusion is supported by accumulating experimental evidence of evolution
of new functions of p53 in tumors. These findings prompted a genome-wide analysis
of possible positive selection during tumor evolution. Methods: A comprehensive
analysis of probable somatic mutations in the sequences of Expressed Sequence
Tags (ESTs) from malignant tumors and normal tissues was performed in order to
access the prevalence of positive selection in cancer evolution. For each EST,
the numbers of synonymous and non-synonymous substitutions were calculated. In
order to identify genes with a signature of positive selection in cancers, these
numbers were compared to: i) expected numbers and ii) the numbers for the respective
genes in the ESTs from normal tissues. Results: We identified 112 genes with a
signature of positive selection in cancers, i.e., a significantly elevated ratio
of non-synonymous to synonymous substitutions, in tumors as compared to 37 such
genes in an approximately equal-sized EST collection from normal tissues. A substantial
fraction of the tumor-specific positive-selection candidates have experimentally
demonstrated or strongly predicted links to cancer. Conclusion: The results of
EST analysis should be interpreted with extreme caution given the noise introduced
by sequencing errors and undetected polymorphisms. Furthermore, an inherent limitation
of EST analysis is that multiple mutations amenable to statistical analysis can
be detected only in relatively highly expressed genes. Nevertheless, the present
results suggest that positive selection might affect a substantial number of genes
during tumorigenic somatic evolution.'
acknowledgement: This work was supported by the Intramural Research Program of the
National Library of Medicine at the National Institutes of Health/DHHS. FAK is an
NSF Graduate Fellow. We thank Yuri Pavlov for helpful discussions.
author:
- first_name: Vladimir
full_name: Babenko, Vladimir N
last_name: Babenko
- first_name: Malay
full_name: Basu, Malay K
last_name: Basu
- first_name: Fyodor
full_name: Fyodor Kondrashov
id: 44FDEF62-F248-11E8-B48F-1D18A9856A87
last_name: Kondrashov
orcid: 0000-0001-8243-4694
- first_name: Igor
full_name: Rogozin, Igor B
last_name: Rogozin
- first_name: Eugene
full_name: Koonin, Eugene V
last_name: Koonin
citation:
ama: Babenko V, Basu M, Kondrashov F, Rogozin I, Koonin E. Signs of positive selection
of somatic mutations in human cancers detected by EST sequence analysis. BMC
Cancer. 2006;6. doi:10.1186/1471-2407-6-36
apa: Babenko, V., Basu, M., Kondrashov, F., Rogozin, I., & Koonin, E. (2006).
Signs of positive selection of somatic mutations in human cancers detected by
EST sequence analysis. BMC Cancer. BioMed Central. https://doi.org/10.1186/1471-2407-6-36
chicago: Babenko, Vladimir, Malay Basu, Fyodor Kondrashov, Igor Rogozin, and Eugene
Koonin. “Signs of Positive Selection of Somatic Mutations in Human Cancers Detected
by EST Sequence Analysis.” BMC Cancer. BioMed Central, 2006. https://doi.org/10.1186/1471-2407-6-36.
ieee: V. Babenko, M. Basu, F. Kondrashov, I. Rogozin, and E. Koonin, “Signs of positive
selection of somatic mutations in human cancers detected by EST sequence analysis,”
BMC Cancer, vol. 6. BioMed Central, 2006.
ista: Babenko V, Basu M, Kondrashov F, Rogozin I, Koonin E. 2006. Signs of positive
selection of somatic mutations in human cancers detected by EST sequence analysis.
BMC Cancer. 6.
mla: Babenko, Vladimir, et al. “Signs of Positive Selection of Somatic Mutations
in Human Cancers Detected by EST Sequence Analysis.” BMC Cancer, vol. 6,
BioMed Central, 2006, doi:10.1186/1471-2407-6-36.
short: V. Babenko, M. Basu, F. Kondrashov, I. Rogozin, E. Koonin, BMC Cancer 6 (2006).
date_created: 2018-12-11T11:49:07Z
date_published: 2006-02-09T00:00:00Z
date_updated: 2021-01-12T08:21:47Z
day: '09'
doi: 10.1186/1471-2407-6-36
extern: 1
intvolume: ' 6'
month: '02'
publication: BMC Cancer
publication_status: published
publisher: BioMed Central
publist_id: '6744'
quality_controlled: 0
status: public
title: Signs of positive selection of somatic mutations in human cancers detected
by EST sequence analysis
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
volume: 6
year: '2006'
...