The q-Characters at Roots of Unity

We consider various specializations of the untwisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.     

Sets of rigged paths with Virasoro characters

Let {M _r,s ^(p,p′)}_{1≤r≤p−1,1≤s≤p′−1} be the irreducible Virasoro modules in the (p,p′)-minimal series. In our previous paper, we have constructed a monomial basis of ⊕ _r=1 ^p−1 M _r,s ^(p,p′) in the case 1<p′/p<2. By ‘monomials’ we mean vectors of the form , where φ _−n ^(r′,r):M _r,s ^(p,p′)→M _r′,s ^(p,p′) are the Fourier components of the (2,1)-primary field and |r ₀,s〉 is the highest weight vector of . In this article, we introduce for all p<p′ with p≥3 and s=1 a subset of such monomials as a conjectural basis of ⊕ _r=1 ^p−1 M _r,1^(p,p′). We prove that the character of the combinatorial set labeling these monomials coincides with the character of the corresponding Virasoro module. We also verify the conjecture in the case p=3.      

On Conjectures of A. Eremenko and A. Gabrielov for Quasi-Exactly Solvable Quartic

We study polynomials p(x) satisfying a differential equation of the form p′′?h′p′ + Hp = 0, where h = x ³/3 + ax and H is a polynomial. We prove a conjecture of A. Eremenko and A. Gabrielov.     

Reality Property of Discrete Wronski Map with Imaginary Step

For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coefficients of the discrete Wronskian are real and the imaginary parts of its roots are bounded by |h|, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model.     

Bayesian approach to recovery of the vertical profile of the stratosphere temperature from the ground-based measurements of solar radiation in millimeter absorption lines of molecular oxygen

We propose a method for recovery of the altitude profile of the stratosphere temperature using ground-based radiometric measurements. The method is based on the Bayesian approach to solving inverse problems and involves determining the probability distribution for the temperature in the entire range of the sounded altitudes. In this case, we use some assumptions on noise in the experimental data and available a priori information about the recovered profile. Using the proposed approach, we compare the recovery results for two methods of approximation and regularization of the recovered profile. It is shown that using the ground-based observations of solar radiation in the oxygen absorption line 27_, it is possible to recover the temperature profile in the altitude range 20–55 km.     

On the room temperature multiferroic BiFeO<Subscript>3</Subscript>: magnetic,dielectric and thermal properties

Magnetic dc susceptibility between 1.5 and 800 K, ac susceptibility and magnetization, thermodynamic properties, temperature dependence of radio and audio-wave dielectric constants and conductivity, contact-free dielectric constants at mm-wavelengths, as well as ferroelectric polarization are reported for single crystalline BiFeO₃. A well developed anomaly in the magnetic susceptibility signals the onset of antiferromagnetic order close to 635 K. Beside this anomaly no further indications of phase or glass transitions are indicated in the magnetic dc and ac susceptibilities down to the lowest temperatures. The heat capacity has been measured from 2 K up to room temperature and significant contributions from magnon excitations have been detected. From the low-temperature heat capacity an anisotropy gap of the magnon modes of the order of 6 meV has been determined. The dielectric constants measured in standard two-point configuration are dominated by Maxwell-Wagner like effects for temperatures T > 300 K and frequencies below 1 MHz. At lower temperatures the temperature dependence of the dielectric constant and loss reveals no anomalies outside the experimental errors, indicating neither phase transitions nor strong spin phonon coupling. The temperature dependence of the dielectric constant was measured contact free at microwave frequencies. At room temperature the dielectric constant has an intrinsic value of 53. The loss is substantial and strongly frequency dependent indicating the predominance of hopping conductivity. Finally, in small thin samples we were able to measure the ferroelectric polarization between 10 and 200 K. The saturation polarization is of the order of 40 μC/cm², comparable to reports in literature.