Character formulas for Feigin–Stoyanovsky’s type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules

Exact sequences of Feigin–Stoyanovsky’s type subspaces for affine Lie algebra \mathfraksl(l+1,\mathbbC)^[\tilde]\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}} lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for \mathfraksl(3,\mathbbC)^[\tilde]\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}, we obtain a new family of character formulas for all Feigin–Stoyanovsky’s type subspaces at the general level.     

Poincaré sections for the horocycle flow in covers of $$\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})$$ and applications to Farey fraction statistics

For a given finite index subgroup \(H\subseteq \mathrm {SL}(2,\mathbb {Z})\), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\) found by Athreya and Cheung to the finite cover \(\mathrm {SL}(2,\mathbb {R})/H\) of \(\mathrm {SL}(2,\mathbb {R})/\mathrm {SL}(2,\mathbb {Z})\). We then use the properties of this section to prove the existence of the limiting gap distribution of various subsets of Farey fractions. Additionally, to each of these subsets of fractions, we extend solutions by Xiong and Zaharescu, and independently Boca, to a Diophantine approximation problem of Erd?s, Szüsz, and Turán.     

Approximating Characteristics of the Nikol’skii–Besov Classes S1,θrBℝd$$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$

Ukrainian Mathematical Journal - We establish the exact-order estimates for the approximation of the classes $$ {S}_{1,\theta}^rB\left({\mathrm{\mathbb{R}}}^d\right) $$ by entire functions of...     

A proof of the Anderson–Badawi $$ \mathbf{rad}\varvec{(I)}^{\varvec{n}} \varvec{\subseteq } {\varvec{I}}$$ formula for $$\varvec{n}$$n-absorbing ideals

In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals.     

Tartar’s conjecture and localization of the quasiconvex hull in $ \mathbb{R}^{{2 \times 2}} $

We give a concrete and surprisingly simple characterization of compact sets K ì \mathbbR^{2 ×2} K \subset \mathbb{R}^{{2 \times 2}} for which families of approximate solutions to the inclusion problem Du∈K are compact. In particular our condition is algebraic and can be tested algorithmically. We also prove that the quasiconvex hull of compact sets of 2 × 2 matrices can be localized. This is false for compact sets in higher dimensions in general.     

Hyperplanes of $DW(5,{\mathbb{K}})$ with ${\mathbb{K}}$ a perfect field of characteristic 2

Let \({\mathbb{K}}\) be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space \(DW(5,{\mathbb{K}})\) that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to 5+N, where N is the number of equivalence classes of the following equivalence relation R on the set \(\{\lambda\in {\mathbb{K}}\,|\,X^{2}+\lambda X+1\mbox{ isirreducible}\) \(\mbox{in }{\mathbb{K}}[X]\}\): (λ ₁,λ ₂)∈R whenever there exists an automorphism σ of \({\mathbb{K}}\) and an \(a\in {\mathbb{K}}\) such that (λ ₂ ^σ )^?1=λ ₁ ^?1 +a ²+a.     

Sharp estimates for derivatives of functions in the Sobolev classes $$ \mathop {W_2^r }\limits^ \circ $$(−1,1)

Explicit formulas are obtained for the maximum possible values of the derivatives f ^(k)(x), x ∈ (−1, 1), k ∈ {0, 1, ..., r − 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r − 1 at the points ±1 and are such that $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 $ \left\| {f^{\left( r \right)} } \right\|_{L_2 ( - 1,1)} \leqslant 1 . As a corollary, it is shown that the first eigenvalue λ _1,r of the operator (−D ²) ^r with these boundary conditions is $ \sqrt 2 $ \sqrt 2 (2r)! (1 + O(1/r)), r → ∞.     

Invariants of 2 × 2 matrices, irreducible {\hbox{SL}(2, \mathbb{C})} characters and the Magnus trace map

We obtain an explicit characterization of the stable points of the action of on the cartesian product G  ^{× n} by simultaneous conjugation on each factor in terms of the corresponding invariant functions. From this, a simple criterion for the irreducibility of representations of finitely generated groups into G is derived. We also obtain analogous results for the action of on the vector space of n-tuples of 2 × 2 complex matrices. For a free group F _n of rank n, we show how to generically reconstruct the 2 ^n-2 conjugacy classes of representations F _n → G from their values under the map considered in Magnus [Math. Zeit. 170, 91–103 (1980)], defined by certain 3n − 3 traces of words of length one and two.      

The Best Constant in the Embedding of WN,1(ℝN) into L ∞ ( ℝ N ) $L^{\infty }(\mathbb {R} ^{N})$

Potential Analysis - We compute the best constant in the embedding of $W^{N,1}(\mathbb {R} ^{N})$ into $L^{\infty }(\mathbb {R} ^{N})$ , extending a result of Humbert and Nazaret in dimensions one...     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

Global $\widetilde{SL(2,R)}$ representations of the Schrödinger equation with singular potential

We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V _λ (x) = λx ^?2 as a representation of \(\widetilde{SL(2,\mathbb{R})}\). The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of \(\widetilde{SL(2,\mathbb{R})}\) ? H ₃, where H ₃ is the three-dimensional Heisenberg group.     

Coincidence of the spaces\mathop {J_2^1 }\limits^ \circ (\Omega ) and for plane domains Ω, having exits at infinity

For a large class of plane domains Ω, having exits at infinity, one shows the coincidence of the spaces of solenoidal vector fields \(\mathop {J_2^1 }\limits^ \circ (\Omega )\) and , which play an important role in the investigation of initial-houndary-value problems for the Navier-Stokes equations.     

On the Structure of Maximal Cohen–Macaulay Modules over the Ring $\boldsymbol{K[[x,y]]/(x^n)}$

We study maximal Cohen–Macaulay modules over the hypersurface ring K being a field. Infinite families of non-isomorphic indecomposable maximal Cohen–Macaulay modules of arbitrary number of minimal generators or of arbitrary rank are constructed. The second author was supported by the CEEX Programme of the Romanian Ministry of Education and Research, contract 2-CEX 06-11-20/25.07.06, and the grant CNCSIS 1055/ 2006.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

Products of Functions in $$H^1_{\rho }({{\mathcal {X}}})$$ and $${\mathrm {BMO}}_{\rho }({{\mathcal {X}}})$$BMOρ(X) over RD-Spaces and Applications to Schrödinger Operators

Let \(({{\mathcal {X}}},d,\mu )\) be an RD-space, \(H^1_{\rho }({{\mathcal {X}}})\), and \({\mathrm {BMO}}_{\rho }({{\mathcal {X}}})\) be, respectively, the local Hardy space and the local BMO space associated with an admissible function \(\rho \). Under an additional assumption that there exists a specific generalized approximation of the identity, the authors prove that the product \(f\times g\) of \(f\in H^1_{\rho }({{\mathcal {X}}})\) and \(g\in {\mathrm {BMO}}_{\rho }({{\mathcal {X}}})\), viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from \(H^1_{\rho }({{\mathcal {X}}})\times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})\) into \(L^1({{\mathcal {X}}})\) and from \(H^1_{\rho }({{\mathcal {X}}}) \times {\mathrm {BMO}}_{\rho } ({{\mathcal {X}}})\) into \(H^{\log }({{\mathcal {X}}})\), which is of wide generality. The authors also give out four applications of this result to Schrödinger operators, respectively, over different underlying spaces, where three of these applications are new.     

Global existence of solutions for a nonlinearly perturbed Keller–Segel system in {\mathbb{R}}^2

We show the global existence of small solution to the perturbed Keller–Segel system of simplified version. Our system has a perturbed nonlinear term of worse sign, therefore the existence and uniqueness of solution is not really obvious. The local existence theorem is obtained by a variational observation for the elliptic part.      

Classical Limit of the Scaled Elliptic Algebra \mathcal{A} ħ,η (\widetilde{\mathfrak{s}\mathfrak{l}}_2)

The classical limit of the scaled elliptic algebra $\mathcal{A}$ ?,η ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic sl2 valued functions on a strip and as an extended algebra of decreasing automorphic sl2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra $\mathcal{A}$ q,p ( $\widetilde{\mathfrak{s}\mathfrak{l}}_2$ ) is also briefly presented.