Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasi-periodic forcing term and in the presence of damping. In the limit of large damping, under some generic non-degeneracy condition on the force, there are quasi-periodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension $d$, for the existence of a quasi-periodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasi-periodic solution is proved to describe a local attractor.Comment: 10 page

Explicit expressions for meromorphic solution of autonomous nonlinear ordinary differential equations

Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational, periodic, elliptic) are found for a wide class of autonomous nonlinear ordinary differential equations

On a Watson-like Uniqueness Theorem and Gevrey Expansions

We present a maximal class of analytic functions, elements of which are in one-to-one correspondence with their asymptotic expansions. In recent decades it has been realized (B. Malgrange, J. Ecalle, J.-P. Ramis, Y. Sibuya et al.), that the formal power series solutions of a wide range of systems of ordinary (even non-linear) analytic differential equations are in fact the Gevrey expansions for the regular solutions. Watson's uniqueness theorem belongs to the foundations of this new theory. This paper contains a discussion of an extension of Watson's uniqueness theorem for classes of functions which admit a Gevrey expansion in angular regions of the complex plane with opening less than or equal to (\frac \pi k,) where (k) is the order of the Gevrey expansion. We present conditions which ensure uniqueness and which suggest an extension of Watson's representation theorem. These results may be applied for solutions of certain classes of differential equations to obtain the best accuracy estimate for the deviation of a solution from a finite sum of the corresponding Gevrey expansion.Comment: 18 pages, 4 figure

Fast projectile stopping power of quantal multi-component strongly coupled plasmas

The Bethe-Larkin formula for the fast projectile stopping power is extended to multi-component plasmas. The results are to contribute to the correct interpretation of the experimental data, which could permit to test the existing and future models of thermodynamic, static, and dynamic characteristics of strongly coupled Coulomb systems.Comment: 4 pages, to appear in PR

Haplotype analysis in Icelandic and Finnish BRCA2 999del5 breast cancer families

To access publisher full text version of this article. Please click on the hyperlink in Additional Links fieldThe 999del5 mutation is the single, strong BRCA2 founder mutation in Iceland and the most common BRCA1/2 founder mutation in Finland. To evaluate the origin and time since spreading of the 999del5 mutation in Iceland and in Finland, we constructed haplotypes with polymorphic markers within and flanking the BRCA2 gene in a set of 18 Icelandic and 10 Finnish 999del5 breast cancer families. All Icelandic families analysed shared a common core haplotype of about 1.7 cM. The common ancestors for the Icelandic families studied were estimated to trace back to 340-1000 years, not excluding the possibility that the mutation was brought to Iceland during the settlement of the country. Analysis of the Finnish families revealed two distinct haplotypes. A rare one, found in three families in the old settlement region in southwestern Finland, shared a four-marker (0.5 cM) core haplotype with the Icelandic 999del5 haplotype. A distinct approximately 6 cM haplotype was shared by seven 999del5 Finnish families estimated to have a common ancestry 140-300 years ago. These families cluster in two geographical regions in Finland, in the very same area as those with the rare haplotype and also in the most eastern, late settlement region of Finland. The results may indicate a common ancient origin for the 999del5 mutation in Iceland and in Finland, but distinct mutational events cannot be ruled out. The surprising finding of the same mutation in two completely different haplotypes in a sparsely populated area in Finland may suggest gene conversion

Pseudo-Hermitian Representation of Quantum Mechanics

A diagonalizable non-Hermitian Hamiltonian having a real spectrum may be used to define a unitary quantum system, if one modifies the inner product of the Hilbert space properly. We give a comprehensive and essentially self-contained review of the basic ideas and techniques responsible for the recent developments in this subject. We provide a critical assessment of the role of the geometry of the Hilbert space in conventional quantum mechanics to reveal the basic physical principle motivating our study. We then offer a survey of the necessary mathematical tools and elaborate on a number of relevant issues of fundamental importance. In particular, we discuss the role of the antilinear symmetries such as PT, the true meaning and significance of the charge operators C and the CPT-inner products, the nature of the physical observables, the equivalent description of such models using ordinary Hermitian quantum mechanics, the pertaining duality between local-non-Hermitian versus nonlocal-Hermitian descriptions of their dynamics, the corresponding classical systems, the pseudo-Hermitian canonical quantization scheme, various methods of calculating the (pseudo-) metric operators, subtleties of dealing with time-dependent quasi-Hermitian Hamiltonians and the path-integral formulation of the theory, and the structure of the state space and its ramifications for the quantum Brachistochrone problem. We also explore some concrete physical applications of the abstract concepts and tools that have been developed in the course of this investigation. These include applications in nuclear physics, condensed matter physics, relativistic quantum mechanics and quantum field theory, quantum cosmology, electromagnetic wave propagation, open quantum systems, magnetohydrodynamics, quantum chaos, and biophysics.Comment: 76 pages, 2 figures, 243 references, published as Int. J. Geom. Meth. Mod. Phys. 7, 1191-1306 (2010

Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I

Classical Schur analysis is intimately connected to the theory of orthogonal polynomials on the circle [Simon, 2005]. We investigate here the connection between multipoint Schur analysis and orthogonal rational functions. Specifically, we study the convergence of the Wall rational functions via the development of a rational analogue to the Szeg\H o theory, in the case where the interpolation points may accumulate on the unit circle. This leads us to generalize results from [Khrushchev,2001], [Bultheel et al., 1999], and yields asymptotics of a novel type.Comment: a preliminary version, 39 pages; some changes in the Introduction, Section 5 (Szeg\H o type asymptotics) is extende

The stability for the Cauchy problem for elliptic equations

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.Comment: 57 pages, review articl

Associations of common breast cancer susceptibility alleles with risk of breast cancer subtypes in BRCA1 and BRCA2 mutation carriers

Peer reviewedPublisher PD

Evaluation of a candidate breast cancer associated SNP in ERCC4 as a risk modifier in BRCA1 and BRCA2 mutation carriers. Results from the Consortium of Investigators of Modifiers of BRCA1/BRCA2 (CIMBA)

Background: In this study we aimed to evaluate the role of a SNP in intron 1 of the ERCC4 gene (rs744154), previously reported to be associated with a reduced risk of breast cancer in the general population, as a breast cancer risk modifier in BRCA1 and BRCA2 mutation carriers. Methods: We have genotyped rs744154 in 9408 BRCA1 and 5632 BRCA2 mutation carriers from the Consortium of Investigators of Modifiers of BRCA1/2 (CIMBA) and assessed its association with breast cancer risk using a retrospective weighted cohort approach. Results: We found no evidence of association with breast cancer risk for BRCA1 (per-allele HR: 0.98, 95% CI: 0.93–1.04, P=0.5) or BRCA2 (per-allele HR: 0.97, 95% CI: 0.89–1.06, P=0.5) mutation carriers. Conclusion: This SNP is not a significant modifier of breast cancer risk for mutation carriers, though weak associations cannot be ruled out. A Osorio1, R L Milne2, G Pita3, P Peterlongo4,5, T Heikkinen6, J Simard7, G Chenevix-Trench8, A B Spurdle8, J Beesley8, X Chen8, S Healey8, KConFab9, S L Neuhausen10, Y C Ding10, F J Couch11,12, X Wang11, N Lindor13, S Manoukian4, M Barile14, A Viel15, L Tizzoni5,16, C I Szabo17, L Foretova18, M Zikan19, K Claes20, M H Greene21, P Mai21, G Rennert22, F Lejbkowicz22, O Barnett-Griness22, I L Andrulis23,24, H Ozcelik24, N Weerasooriya23, OCGN23, A-M Gerdes25, M Thomassen25, D G Cruger26, M A Caligo27, E Friedman28,29, B Kaufman28,29, Y Laitman28, S Cohen28, T Kontorovich28, R Gershoni-Baruch30, E Dagan31,32, H Jernström33, M S Askmalm34, B Arver35, B Malmer36, SWE-BRCA37, S M Domchek38, K L Nathanson38, J Brunet39, T Ramón y Cajal40, D Yannoukakos41, U Hamann42, HEBON37, F B L Hogervorst43, S Verhoef43, EB Gómez García44,45, J T Wijnen46,47, A van den Ouweland48, EMBRACE37, D F Easton49, S Peock49, M Cook49, C T Oliver49, D Frost49, C Luccarini50, D G Evans51, F Lalloo51, R Eeles52, G Pichert53, J Cook54, S Hodgson55, P J Morrison56, F Douglas57, A K Godwin58, GEMO59,60,61, O M Sinilnikova59,60, L Barjhoux59,60, D Stoppa-Lyonnet61, V Moncoutier61, S Giraud59, C Cassini62,63, L Olivier-Faivre62,63, F Révillion64, J-P Peyrat64, D Muller65, J-P Fricker65, H T Lynch66, E M John67, S Buys68, M Daly69, J L Hopper70, M B Terry71, A Miron72, Y Yassin72, D Goldgar73, Breast Cancer Family Registry37, C F Singer74, D Gschwantler-Kaulich74, G Pfeiler74, A-C Spiess74, Thomas v O Hansen75, O T Johannsson76, T Kirchhoff77, K Offit77, K Kosarin77, M Piedmonte78, G C Rodriguez79, K Wakeley80, J F Boggess81, J Basil82, P E Schwartz83, S V Blank84, A E Toland85, M Montagna86, C Casella87, E N Imyanitov88, A Allavena89, R K Schmutzler90, B Versmold90, C Engel91, A Meindl92, N Ditsch93, N Arnold94, D Niederacher95, H Deißler96, B Fiebig97, R Varon-Mateeva98, D Schaefer99, U G Froster100, T Caldes101, M de la Hoya101, L McGuffog49, A C Antoniou49, H Nevanlinna6, P Radice4,5 and J Benítez1,3 on behalf of CIMB