On eigenvalues of the Schr\"odinger operator with a complex-valued polynomial potential

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k>2 boundary conditions, except for the case d is even and k=d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.Comment: 23 page

On eigenvalues of the Schr\"odinger operator with an even complex-valued polynomial potential

In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schr\"odinger equation with even polynomial potentials of arbitrary degree d with complex coefficients, and k<(d+2)/2 boundary conditions. We show that the spectral determinant in this case consists of two components, containing even and odd eigenvalues respectively. In the case with k=(d+2)/2 boundary conditions, we show that the corresponding parameter space consists of infinitely many connected components

Newton's method and Baker domains

We show that there exists an entire function f without zeros for which the associated Newton function N(z)=z-f(z)/f'(z) is a transcendental meromorphic functions without Baker domains. We also show that there exists an entire function f with exactly one zero for which the complement of the immediate attracting basin has at least two components and contains no invariant Baker domains of N. The second result answers a question of J. Rueckert and D. Schleicher while the first one gives a partial answer to a question of X. Buff.Comment: 6 page

Y-System and Deformed Thermodynamic Bethe Ansatz

We introduce a new tool, the Deformed TBA (Deformed Thermodynamic Bethe Ansatz), to analyze the monodromy problem of the cubic oscillator. The Deformed TBA is a system of five coupled nonlinear integral equations, which in a particular case reduces to the Zamolodchikov TBA equation for the 3-state Potts model. Our method generalizes the Dorey-Tateo analysis of the (monomial) cubic oscillator. We introduce a Y-system corresponding to the Deformed TBA and give it an elegant geometric interpretation.Comment: 12 pages. Minor corrections in Section

The size of Wiman-Valiron disks

Wiman-Valiron theory and results of Macintyre about "flat regions" describe the asymptotic behavior of entire functions in certain disks around points of maximum modulus. We estimate the size of these disks for Macintyre's theory from above and below.Comment: 20 page

Characterization of the Cancer Spectrum in Men With Germline BRCA1 and BRCA2 Pathogenic Variants Results From the Consortium of Investigators of Modifiers of BRCA1/2 (CIMBA)

Importance The limited data on cancer phenotypes in men with germline BRCA1 and BRCA2 pathogenic variants (PVs) have hampered the development of evidence-based recommendations for early cancer detection and risk reduction in this population. Objective To compare the cancer spectrum and frequencies between male BRCA1 and BRCA2 PV carriers. Design, Setting, and Participants Retrospective cohort study of 6902 men, including 3651 BRCA1 and 3251 BRCA2 PV carriers, older than 18 years recruited from cancer genetics clinics from 1966 to 2017 by 53 study groups in 33 countries worldwide collaborating through the Consortium of Investigators of Modifiers of BRCA1/2 (CIMBA). Clinical data and pathologic characteristics were collected. Main Outcomes and Measures BRCA1/2 status was the outcome in a logistic regression, and cancer diagnoses were the independent predictors. All odds ratios (ORs) were adjusted for age, country of origin, and calendar year of the first interview. Results Among the 6902 men in the study (median [range] age, 51.6 [18-100] years), 1634 cancers were diagnosed in 1376 men (19.9%), the majority (922 of 1,376 [67%]) being BRCA2 PV carriers. Being affected by any cancer was associated with a higher probability of being a BRCA2, rather than a BRCA1, PV carrier (OR, 3.23; 95% CI, 2.81-3.70; P <.001), as well as developing 2 (OR, 7.97; 95% CI, 5.47-11.60; P <.001) and 3 (OR, 19.60; 95% CI, 4.64-82.89; P <.001) primary tumors. A higher frequency of breast (OR, 5.47; 95% CI, 4.06-7.37; P <.001) and prostate (OR, 1.39; 95% CI, 1.09-1.78; P = .008) cancers was associated with a higher probability of being a BRCA2 PV carrier. Among cancers other than breast and prostate, pancreatic cancer was associated with a higher probability (OR, 3.00; 95% CI, 1.55-5.81; P = .001) and colorectal cancer with a lower probability (OR, 0.47; 95% CI, 0.29-0.78; P = .003) of being a BRCA2 PV carrier. Conclusions and Relevance Significant differences in the cancer spectrum were observed in male BRCA2, compared with BRCA1, PV carriers. These data may inform future recommendations for surveillance of BRCA1/2-associated cancers and guide future prospective studies for estimating cancer risks in men with BRCA1/2 PVs. This cohort study compares the cancer spectrum and frequencies between male BRCA1 and BRCA2 pathogenic variant carriers. Question Are there cancer phenotype differences between male BRCA1 and BRCA2 pathogenic variant carriers? Findings In this cohort study of 6902 men with a BRCA1 or BRCA2 pathogenic variant, being affected by cancer, particularly breast, prostate, and pancreatic cancers and developing multiple primary tumors, was associated with a higher probability for a man of being a BRCA2, rather than a BRCA1, pathogenic variant carrier. Meaning Surveillance programs in men with BRCA1 and BRCA2 pathogenic variants should be tailored in light of these gene-specific cancer phenotype differences. These results may inform the design of prospective studies on cancer risks in male BRCA1 and BRCA2 pathogenic variant carriers.Peer reviewe

The Schwarzian derivative and the Wiman-Valiron property

Consider a transcendental meromorphic function in the plane with finitely many critical values, such that the multiple points have bounded multiplicities and the inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that if the Schwarzian derivative is transcendental then the function has infinitely many multiple points, the inverse function does not have a direct transcendental singularity over infinity, and infinity is not a Borel exceptional value. The first of these conclusions was proved by Nevanlinna and Elfving via a fundamentally different method

Painleve I, Coverings of the Sphere and Belyi Functions

The theory of poles of solutions of Painleve-I is equivalent to the Nevanlinna problem of constructing a meromorphic function ramified over five points - counting multiplicities - and without critical points. We construct such meromorphic functions as limit of rational ones. In the case of the tritronquee solution these rational functions are Belyi functions.Comment: 33 pages, many figures. Version 2: minor corrections and minor changes in the bibliograph

Seasonal and Diurnal Variation of Geomagnetic Activity: Revised \u3cem\u3eDst\u3c/em\u3e Versus External Drivers

Daily and seasonal variability of long time series of magnetometer data from Dst stations is examined. Each station separately shows a local minimum of horizontal magnetic component near 18 local time (LT) and weakest activity near 06 LT. The stations were found to have different baselines such that the average levels of activity differed by about 10 nT. This effect was corrected for by introducing a new “base method” for the elimination of the secular variation. This changed the seasonal variability of the Dst index by about 3 nT. The hemispheric differences between the annual variation (larger activity during local winter and autumn solstice) were demonstrated and eliminated from the Dst index by addition of two Southern Hemisphere stations to a new index termed Dst6. Three external drivers of geomagnetic activity were considered: the heliographic latitude, the equinoctial effect, and the Russell–McPherron effect. Using the newly created Dst6 index, it is demonstrated that these three effects account for only about 50% of the daily and seasonal variability of the index. It is not clear what drives the other 50% of the daily and seasonal variability, but it is suggested that the station distribution may play a role

Meromorphic traveling wave solutions of the complex cubic-quintic Ginzburg-Landau equation

We look for singlevalued solutions of the squared modulus M of the traveling wave reduction of the complex cubic-quintic Ginzburg-Landau equation. Using Clunie's lemma, we first prove that any meromorphic solution M is necessarily elliptic or degenerate elliptic. We then give the two canonical decompositions of the new elliptic solution recently obtained by the subequation method.Comment: 14 pages, no figure, to appear, Acta Applicandae Mathematica