Some conjectures on modular representations of affine sl<Subscript>2</Subscript> and Virasoro algebra

We conjecture an explicit bound on the prime characteristic of a field, under which theWeyl modules of affine sl₂ and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset construction for affine sl₂ is valid.     

An ansatz for sl<Subscript>2</Subscript>-invariant <Emphasis Type="Italic">R</Emphasis>-matrices

We study spectral decomposition of regular sl ₂-invariant R-matrices R(λ) by the method of reduction of the Yang-Baxter equation to subspaces of a given spin. Restrictions on the possible structure of several leading coefficients in the spectral decomposition are derived. The origin and structure of the exceptional solution for spin s = 3 are explained. A similar analysis is performed for constant R-matrices. In particular, it is shown that the permutation matrix ℙ is a “rigid” solution. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 100–118.     

Approximation in the <Emphasis Type="BoldItalic">M</Emphasis><Subscript>2</Subscript>/<Emphasis Type="BoldItalic">G</Emphasis><Subscript>2</Subscript>/1 Queue with Preemptive Priority

The main purpose of this paper is to use the strong stability method to approximate the characteristics of the M ₂/G ₂/1 queue with preemptive priority by those of the classical M/G/1 queue. The latter is simpler and more exploitable in practice. After perturbing the arrival intensity of the priority requests, we derive the stability conditions and next obtain the stability inequalities with an exact computation of constants. From those theoretical results, we elaborate an algorithm allowing us to verify the approximation conditions and to provide the made numerical error. In order to have an idea about the efficiency of this approach, we consider a concrete example whose results are compared with those obtained by simulation.     

The spectrum of the Dirac operator on SU<Subscript>2</Subscript>/Q<Subscript>8</Subscript>

We compute the fundamental Dirac operator for the three-parameter-family of homogeneous Riemannian metrics and the four different spin structures on SU₂/Q₈, where Q₈ denotes the group of quaternions. We deduce its spectrum for the Berger metrics and show the sharpness of Christian Bär’s upper bound for the smallest Dirac eigenvalue in the particular case where SU₂/Q₈ is a homogeneous minimal hypersurface of S ⁴.     

Integral geometry under G<Subscript>2</Subscript> and Spin(7)

A Hadwiger-type theorem for the exceptional Lie groups G₂ and Spin(7) is proved. The algebras of G₂ or Spin(7) invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formulas for these groups are determined.     

Wheels, wheeling, and the Kontsevich integral of the Unknot

We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne [De]) for the relation between the two natural products on the space of uni-trivalent diagrams. The two formulas use the related notions of “Wheels” and “Wheeing”. We prove these formulas ‘on the level of Lie algebras’ using standard techniques from the theory of Vassiliev invariants and the theory of Lie algebras. In a brief epilogue we report on recent proofs of our full conjectures, by Kontsevich [Ko2] and by DBN, DPT, and T. Q. T. Le, [BLT]. This paper is available electronically     

Approximation of periodic functions by Steklov averages in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

In the space L₂ of periodic functions, we establish exact (in the sense of constants) estimates from below for the deviation of the Steklov functions of the first and second order in terms of the modulus of continuity of the second order. Similar results are also established for even continuous periodic functions with nonnegative Fourier coefficients in the space C. Bibliography: 5 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 79–90     

On constant <Emphasis Type="Italic">Uq</Emphasis>(<Emphasis Type="Italic">sl</Emphasis><Subscript>2</Subscript>)-invariant <Emphasis Type="Italic">R</Emphasis>-matrices

We consider the spectral resolution of a Uq (sl ₂)-invariant solution R of the constant Yang–Baxter equation in the braid group form. It is shown that if the two highest coefficients in this resolution are not equal, then R is either the Drinfeld R-matrix or its inverse. Bibliography: 13 titles.     

A Steady-State Approximation for the E<Superscript>x</Superscript><Subscript>k</Subscript><Emphasis Type="Italic">/M/</Emphasis>1 Queue

A simple approximate formula for the steady-state average number of customers in the E^x_k/M/1 queueing system is derived and evaluated. The formula is very accurate: the errors incurred are small and insensitive to the shape of the batch-size distribution and to the values of the parameters involved; in the special case where the batch size has a geometric distribution, the formula appears to be exact.     

Groups in NTP<Subscript>2</Subscript>

We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP₂ theory.     

A Short Remark on Weierstrass Points at Infinity on <Emphasis Type="Italic">X</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>)

We point out that the formalism of the trace map and reduction modulo p can be used to give a short proof for the fact first proved by Ogg that is not a Weierstrass point on X₀(pM) where p is a prime not dividing M and the genus of X₀(M) is zero.     

A Differential Calculus on the Z<Subscript>3</Subscript>-graded Quantum Group GL<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2)

In this work, we introduce Z₃-graded quantum group GL\({_q(2,{\mathbb C})}\) with the help of Z₃-graded quantum plane and a Z₃-graded bicovariant differential calculus on the Z₃-graded quantum group GL _q (2). The corresponding Z₃-graded quantum Lie superalgebra is obtained.     

Interpolation of operators of weak type (ϕ<Subscript>0</Subscript>, ψ<Subscript>0</Subscript>, ϕ<Subscript>1</Subscript>, ψ<Subscript>1</Subscript>) in Lorentz spaces

We prove theorems on interpolation of quasilinear operators of weak type (ϕ₀, ψ₀, ϕ₀, ψ₁) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and convex functions ϕ₀(t), ψ₀(t), ϕ₁(t), and ψ₁(t). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005.     

Power Sums Related to Semigroups S(d<Subscript>1</Subscript>, d<Subscript>2</Subscript>, d<Subscript>3</Subscript>)

The explicit formulas for the sums of positive powers of the integers unrepresentable by the triple of integers , are derived.     

On Compactness of Regular Integral Operators in the Space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

In this paper we obtain a sufficient condition for quite continuity of Fredholm type integral operators in the space L₁(a, b). Uniform approximations by operators with degenerate kernels of horizontally striped structures are constructed. A quantitative error estimate is obtained. We point out the possibility of application of the obtained results to second kind integral equations, including convolution equations on a finite interval, equations with polar kernels, one-dimensional equations with potential type kernels, and some transport equations in non-homogeneous layers.     

Maximal Poisson Commutative subalgebras for truncated parabolic subalgebras of maximal index in sl<Subscript>n</Subscript>

We show that several properties of the semisimple algebras carry over to a certain family of parabolic subalgebras of maximal index in sl_n. More precisely we prove an analogue of Kostant's slice theorem [B. Kostant, Amer. J. Math. 85 (1963), 327-404] for these algebras and construct a maximal Poisson commutative subalgebra in the symmetric algebra, following the theory presented in [A.S. Mishchenko and A.T. Fomenko, Math. USSR-Izv. 12 (1978), 371-389]. These results are quite remarkable since these algebras do not admit appropriate sl₂-triples.     

Distance-Regular Shilla Graphs with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript> = <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript>

A Shilla graph is defined as a distance-regular graph of diameter 3 with second eigen-value θ₁ equal to a₃. For a Shilla graph, let us put a = a₃ and b = k/a. It is proved in this paper that a Shilla graph with b₂ = c₂ and noninteger eigenvalues has the following intersection array:

$$\left\{ {\frac{{{b^2}\left( {b - 1} \right)}}{2},\frac{{\left( {b - 1} \right)\left( {{b^2} - b + 2} \right)}}{2},\frac{{b\left( {b - 1} \right)}}{4};1,\frac{{b\left( {b - 1} \right)}}{4},\frac{{b{{\left( {b - 1} \right)}^2}}}{2}} \right\}$$

If Γ is a Q-polynomial Shilla graph with b₂ = c₂ and b = 2r, then the graph Γ has intersection array

$$\left\{ {2tr\left( {2r + 1} \right),\left( {2r + 1} \right)\left( {2rt + t + 1} \right),r\left( {r + t} \right);1,r\left( {r + t} \right),t\left( {4{r^2} - 1} \right)} \right\}$$

and, for any vertex u in Γ, the subgraph Γ₃(u) is an antipodal distance-regular graph with intersection array

$$\left\{ {t\left( {2r + 1} \right),\left( {2r - 1} \right)\left( {t + 1} \right),1;1,t + 1,t\left( {2r + 1} \right)} \right\}$$

The Shilla graphs with b₂ = c₂ and b = 4 are also classified in the paper.

     

Regularity for weakly (K<Subscript>1</Subscript>, K<Subscript>2</Subscript>)-quasiregular mappings

In this paper, we first give the definition of weakly (K₁,K₂-quasiregular mappings, and then by using the Hodge decomposition and the weakly reverse Hölder inequality, we obtain their regularity property: For anyq ₁ that satisfies\(0< K_1 n^{(n + 4)/2} 2^{n + 1} \times 100^{n^2 } [2^{3n/2} (2^{5n} + 1)](n - q_1 )< 1\), there existsp ₁=p ₁(n,q ₁,K ₁,K ₂)>n, such that any (K₁, K₂)-quasiregular mapping\(f \in W_{loc}^{1,q_1 } (\Omega ,R^n )\) is in fact in\(W_{loc}^{1,p_1 } (\Omega , R^n )\). That is, f is (K₁,K₂)-quasiregular in the usual sense.     

Simply connected symplectic 4-manifolds with <Emphasis Type="Italic">b</Emphasis><Subscript>2</Subscript><Superscript>+</Superscript>=1 and <Emphasis Type="Italic">c</Emphasis><Subscript>1</Subscript><Superscript>2</Superscript>=2

In this article we construct a new simply connected symplectic 4-manifold with b₂⁺=1 and c₁²=2 which is homeomorphic, but not diffeomorphic, to a rational surface by using rational blow-down technique. As a corollary, we conclude that a rational surface admits an exotic smooth structure. Mathematics Subject Classification (2000) 53D05, 14J26, 57R55, 57R57