A note on isoparametric polynomials

We show that any homogeneous polynomial solution of |\nabla F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm |x|^m (for even m's) or it is a composition of a Chebychev polynomial and a Cartan-M\"unzner polynomial.Comment: 6 page

Protein Kinase CK2α Maintains Extracellular Signal-regulated Kinase (ERK) Activity in a CK2α Kinase-independent Manner to Promote Resistance to Inhibitors of RAF and MEK but Not ERK in BRAF Mutant Melanoma

The protein kinase casein kinase 2 (CK2) is a pleiotropic and constitutively active kinase that plays crucial roles in cellular proliferation and survival. Overexpression of CK2, particularly the α catalytic subunit (CK2α, CSNK2A1), has been implicated in a wide variety of cancers and is associated with poorer survival and resistance to both conventional and targeted anticancer therapies. Here, we found that CK2α protein is elevated in melanoma cell lines compared with normal human melanocytes. We then tested the involvement of CK2α in drug resistance to Food and Drug Administration-approved single agent targeted therapies for melanoma. In BRAF mutant melanoma cells, ectopic CK2α decreased sensitivity to vemurafenib (BRAF inhibitor), dabrafenib (BRAF inhibitor), and trametinib (MEK inhibitor) by a mechanism distinct from that of mutant NRAS. Conversely, knockdown of CK2α sensitized cells to inhibitor treatment. CK2α-mediated RAF-MEK kinase inhibitor resistance was tightly linked to its maintenance of ERK phosphorylation. We found that CK2α post-translationally regulates the ERK-specific phosphatase dual specificity phosphatase 6 (DUSP6) in a kinase dependent-manner, decreasing its abundance. However, we unexpectedly showed, by using a kinase-inactive mutant of CK2α, that RAF-MEK inhibitor resistance did not rely on CK2α kinase catalytic function, and both wild-type and kinase-inactive CK2α maintained ERK phosphorylation upon inhibition of BRAF or MEK. That both wild-type and kinase-inactive CK2α bound equally well to the RAF-MEK-ERK scaffold kinase suppressor of Ras 1 (KSR1) suggested that CK2α increases KSR facilitation of ERK phosphorylation. Accordingly, CK2α did not cause resistance to direct inhibition of ERK by the ERK1/2-selective inhibitor SCH772984. Our findings support a kinase-independent scaffolding function of CK2α that promotes resistance to RAF- and MEK-targeted therapies

The differential-algebraic and bi-Hamiltonian integrability analysis of the Riemann type hierarchy revisited

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic hierarchy is revisited, its new Lax type representation and Poisson structures constructed in exact form. The related bi-Hamiltonian integrability and compatible Poissonian structures of the generalized Riemann type hierarchy are also discussed.Comment: 18 page

Permutable entire functions satisfying algebraic differential equations

It is shown that if two transcendental entire functions permute, and if one of them satisfies an algebraic differential equation, then so does the other one.Comment: 5 page

Uncertainty Principle for Control of Ensembles of Oscillators Driven by Common Noise

We discuss control techniques for noisy self-sustained oscillators with a focus on reliability, stability of the response to noisy driving, and oscillation coherence understood in the sense of constancy of oscillation frequency. For any kind of linear feedback control--single and multiple delay feedback, linear frequency filter, etc.--the phase diffusion constant, quantifying coherence, and the Lyapunov exponent, quantifying reliability, can be efficiently controlled but their ratio remains constant. Thus, an "uncertainty principle" can be formulated: the loss of reliability occurs when coherence is enhanced and, vice versa, coherence is weakened when reliability is enhanced. Treatment of this principle for ensembles of oscillators synchronized by common noise or global coupling reveals a substantial difference between the cases of slightly non-identical oscillators and identical ones with intrinsic noise.Comment: 10 pages, 5 figure

Kochen-Specker Vectors

We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in n-dim Hilbert space, H^n, n>3 to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in R^n, on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all 4-dim KS vector systems containing up to 24 vectors were generated and described, all 3-dim vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.Comment: 19 pages, 6 figures, title changed, introduction thoroughly rewritten, n-dim rotation of KS vectors defined, original Kochen-Specker 192 (117) vector system translated into MMP diagram notation with a new graphical representation, results on Tkadlec's dual diagrams added, several other new results added, journal version: to be published in J. Phys. A, 38 (2005). Web page: http://m3k.grad.hr/pavici

Doppler-free frequency modulation spectroscopy of atomic erbium in a hollow cathode discharge cell

The erbium atomic system is a promising candidate for an atomic Bose-Einstein condensate of atoms with a non-vanishing orbital angular momentum ($L \neq 0$) of the electronic ground state. In this paper we report on the frequency stabilization of a blue external cavity diode laser system on the 400.91 $nm$ laser cooling transition of atomic erbium. Doppler-free saturation spectroscopy is applied within a hollow cathode discharge tube to the corresponding electronic transition of several of the erbium isotopes. Using the technique of frequency modulation spectroscopy, a zero-crossing error signal is produced to lock the diode laser frequency on the atomic erbium resonance. The latter is taken as a reference laser to which a second main laser system, used for laser cooling of atomic erbium, is frequency stabilized

Fuchs versus Painlev\'e

We briefly recall the Fuchs-Painlev\'e elliptic representation of Painlev\'e VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kind, $K$ and $E$, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants, are equivalent to the second order operator $L_E$ which has $E$ as solution (or, for off-diagonal correlations to the direct sum of $L_E$ and $d/dt$). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second order linear differential operator $L_E$ being replaced by an isomonodromic system of two third-order linear partial differential operators associated with $\Pi_1$, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of $L_E$). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlev\'e non-linear ODE's, Fuchsian linear ODE's and elliptic curves. In particular the elliptic representation of Painlev\'e VI has to be generalized to an ``Appellian'' representation of Garnier systems.Comment: Dedicated to the : Special issue on Symmetries and Integrability of Difference Equations, SIDE VII meeting held in Melbourne during July 200

Differential-Algebraic Integrability Analysis of the Generalized Riemann Type and Korteweg-de Vries Hydrodynamical Equations

A differential-algebraic approach to studying the Lax type integrability of the generalized Riemann type hydrodynamic equations at N = 3; 4 is devised. The approach is also applied to studying the Lax type integrability of the well known Korteweg-de Vries dynamical system.Comment: 11 page