Geometric Realizations of Lusztig’s Symmetries of Symmetrizable Quantum Groups

Let U be the quantum group and f be the Lusztig’s algebra associated with a symmetrizable generalized Cartan matrix. The algebra f can be viewed as the positive part of U. Lusztig introduced some symmetries T _i on U for all i ∈ I. Since T _i (f) is not contained in f, Lusztig considered two subalgebras _i f and ⁱ f of f for any i ∈ I, where _i f={x ∈ f | T _i (x) ∈ f} and \({^{i}\mathbf {f}}=\{x\in \mathbf {f}\,\,|\,\,T^{-1}_{i}(x)\in \mathbf {f}\}\). The restriction of T _i on _i f is also denoted by \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\). The geometric realization of f and its canonical basis are introduced by Lusztig via some semisimple complexes on the variety consisting of representations of the corresponding quiver. When the generalized Cartan matrix is symmetric, Xiao and Zhao gave geometric realizations of Lusztig’s symmetries in the sense of Lusztig. In this paper, we shall generalize this result and give geometric realizations of _i f, ⁱ f and \(T_{i}:{_{i}\mathbf {f}}\rightarrow {^{i}\mathbf {f}}\) by using the language ’quiver with automorphism’ introduced by Lusztig.     

Rational frames of minimal twist along space curves under specified boundary conditions

An adapted orthonormal frame (f₁(ξ),f₂(ξ),f₃(ξ)) on a space curve r(ξ), ξ ∈ [ 0, 1 ] comprises the curve tangent \(\mathbf {f}_{1}(\xi ) =\mathbf {r}^{\prime }(\xi )/|\mathbf {r}^{\prime }(\xi )|\) and two unit vectors f₂(ξ),f₃(ξ) that span the normal plane. The variation of this frame is specified by its angular velocity Ω = Ω₁f₁ + Ω₂f₂ + Ω₃f₃, and the twist of the framed curve is the integral of the component Ω₁ with respect to arc length. A minimal twist frame (MTF) has the least possible twist value, subject to prescribed initial and final orientations f₂(0),f₃(0) and f₂(1),f₃(1) of the normal–plane vectors. Employing the Euler–Rodrigues frame (ERF) — a rational adapted frame defined on spatial Pythagorean–hodograph curves — as an intermediary, an exact expression for an MTF with Ω₁ = constant is derived. However, since this involves rather complicated transcendental terms, a construction of rational MTFs is proposed by the imposition of a rational rotation on the ERF normal–plane vectors. For spatial PH quintics, it is shown that rational MTFs compatible with the boundary conditions can be constructed, with only modest deviations of Ω₁ about the mean value, by a rational quartic normal–plane rotation of the ERF. If necessary, subdivision methods can be invoked to ensure that the rational MTF is free of inflections, or to more accurately approximate a constant Ω₁. The procedure is summarized by an algorithm outline, and illustrated by a representative selection of computed examples.     

Some matrix functional equations

We investigate the pair of matrix functional equations G(x)F(y) = G(xy) and G(x)G(y) = F(y/x), featuring the two independent scalar variables x and y and the two N×N matrices F(z) andG(z) (with N an arbitrary positive integer and the elements of these two matrices functions of the scalar variable z). We focus on the simplest class of solutions, i.e., on matrices all of whose elements are analytic functions of the independent variable. While in the scalar (N = 1) case this pair of functional equations only possess altogether trivial constant solutions, in the matrix (N > 1) case there are nontrivial solutions. These solutions satisfy the additional pair of functional equations F(x)G(y) = G(y/x) andF(x)F(y) = F(xy), and an endless hierarchy of other functional equations featuring more than two independent variables.     

Unique solvability of the stationary Fokker-Planck equation in a class of positive functions

We study the stationary Focker-Planck equation Δu ? div(u f) = 0 with a given vector field f of the class C ₀ ^∞ (R ⁿ ) on the basis of a fixed point principle that generalizes the contraction mapping method. Next, we introduce a parameter in the equation and prove the unique solvability of the equation Δu ? div(uγ f) = 0 with the parameter in the class of positive slowly increasing functions. We reveal the analytic dependence of the positive solution u on the parameter γ. Pointwise estimates for positive solutions are proved.     

Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R ⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ? R ⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.     

A sharp eigenvalue theorem for fractional elliptic equations

The invisibility graph I(X) of a set X ? R ^d is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X. We consider the following three parameters of a set X: the clique number ω(I(X)), the chromatic number χ(I(X)) and the convexity number γ(X), which is the minimum number of convex subsets of X that cover X.We settle a conjecture of Matou?ek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3.We also find sets X in R⁵ with χ(X) = 2, but γ(X) arbitrarily large.     

Multipliers of spherical harmonics and energy of measures on the sphere

We consider the operator,f(Δ) for Δ the Laplacian, on spaces of measures on the sphere inR ^d , show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for theL²-norm off(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure onR ^d with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).     

On the existence of strips inside domains convex in one direction

A plane domain Ω is convex in the positive direction if for every ω ∈ Ω, the entire half-line {ω + t: t ≥ 0} is contained in Ω. Suppose that h maps the unit disk onto such a domain Ω with the normalization h(0) = 0 and lim_t→∞h^?1(h(z) + t) = 1. We show that if ∠lim_z→?1 Re h(z) = ?∞ and ∠lim_z→?1(1 + z)h′(z) = ν ∈ (0, +∞), then Ω contains a maximal horizontal strip of width πν. We also prove a converse statement. These results provide a solution to a problem posed by Elin and Shoikhet in connection with semigroups of holomorphic functions.     

Continuous wavelets on compact manifolds

Let M be a smooth compact oriented Riemannian manifold, and let Δ _M be the Laplace–Beltrami operator on M. Say \({0 \neq f \in \mathcal{S}(\mathbb {R}^+)}\) , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ _M ). We show that K _t is well-localized near the diagonal, in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f (t ²Δ) on \({\mathbb {R}^n}\) . We define continuous \({\mathcal {S}}\)-wavelets on M, in such a manner that K _t (x, y) satisfies this definition, because of its localization near the diagonal. Continuous \({\mathcal {S}}\)-wavelets on M are analogous to continuous wavelets on \({\mathbb {R}^n}\) in \({\mathcal {S}}\) (\({\mathbb {R}^n}\)). In particular, we are able to characterize the Hölder continuous functions on M by the size of their continuous \({\mathcal {S}}\)-wavelet transforms, for Hölder exponents strictly between 0 and 1. If M is the torus \({\mathbb T^2}\) or the sphere S ², and f (s)  =  se ^?s (the “Mexican hat” situation), we obtain two explicit approximate formulas for K _t , one to be used when t is large, and one to be used when t is small.     

On the homotopy type of (<Emphasis Type="Italic">n</Emphasis>-1)-connected (3<Emphasis Type="Italic">n</Emphasis>+1)-dimensional free chain Lie algebra

Let R be a subring ring of Q. We reserve the symbol p for the least prime which is not a unit in R; if R ?Q, then p=∞. Denote by DGL _n ^np , n≥1, the category of (n-1)-connected np-dimensional differential graded free Lie algebras over R. In [1] D. Anick has shown that there is a reasonable concept of homotopy in the category DGL _n ^np . In this work we intend to answer the following two questions: Given an object (L(V), ?) in DGL _n ³ⁿ⁺² and denote by S(L(V), ?) the class of objects homotopy equivalent to (L(V), ?). How we can characterize a free dgl to belong to S(L(V), ?)? Fix an object (L(V), ?) in DGL _n ³ⁿ⁺² . How many homotopy equivalence classes of objects (L(W), δ) in DGL _n ³ⁿ⁺² such that H _* (W, d′)?H _* (V, d) are there? Note that DGL _n ³ⁿ⁺² is a subcategory of DGL _n ^np when p>3. Our tool to address this problem is the exact sequence of Whitehead associated with a free dgl.     

Hankel and Berezin Type Operators on Weighted Besov Spaces of Holomorphic Functions on Polydisks

Let S be the space of functions of regular variation and let ω = (ω₁,..., ω_n), ω_j ∈ S. The weighted Besov space of holomorphic functions on polydisks, denoted by B _p (ω) (0 < p < +∞), is defined to be the class of all holomorphic functions f defined on the polydisk U ⁿ such that \(||f||_{{B_{P(\omega )}}}^P = \int_{{U^n}} {|Df(z){|^p}\prod\limits_{j = 1}^n {{\omega _j}{{(1 - |{z_j}{|^2})}^{P - 2}}dm{a_{2n}}(z) < \infty } } \), where dm_2n(z) is the 2ndimensional Lebesgue measure on U ⁿ and D stands for a special fractional derivative of f.We prove some theorems concerning boundedness of the generalized little Hankel and Berezin type operators on the spaces B _p (ω) and L _p (ω) (the weighted L _p -space).     

A comprehensive characterization of the set of polynomial curves with rational rotation-minimizing frames

A rotation–minimizing frame (f ₁,f ₂,f ₃) on a space curve r(ξ) defines an orthonormal basis for \(\mathbb {R}^{3}\) in which \(\mathbf {f}_{1}=\mathbf {r}^{\prime }/|\mathbf {r}^{\prime }|\) is the curve tangent, and the normal–plane vectors f ₂, f ₃ exhibit no instantaneous rotation about f ₁. Polynomial curves that admit rational rotation–minimizing frames (or RRMF curves) form a subset of the Pythagorean–hodograph (PH) curves, specified by integrating the form \(\mathbf {r}^{\prime }(\xi )=\mathcal {A}(\xi )\,\mathbf{i} \,\mathcal {A}^{*}(\xi )\) for some quaternion polynomial \(\mathcal {A}(\xi )\). By introducing the notion of the rotation indicatrix and the core of the quaternion polynomial \(\mathcal {A}(\xi )\), a comprehensive characterization of the complete space of RRMF curves is developed, that subsumes all previously known special cases. This novel characterization helps clarify the structure of the complete space of RRMF curves, distinguishes the spatial RRMF curves from trivial (planar) cases, and paves the way toward new construction algorithms.     

<Emphasis Type="Bold">K</Emphasis><Subscript>0</Subscript> of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity

If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group K _O(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of K _O(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding K _O(R) order isomorphic to G(P r i m _R ), where P r i m _R is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as K _O(R) for a suitable semiartinian and unit-regular ring R.     

Trigonometrical fitting conditions for two derivative Runge-Kutta methods

Two derivative Runge-Kutta methods are Runge-Kutta methods for problems of the form y^′ = f(y) that include the second derivative y^″ = g(y) = f^′(y)f(y) and were developed in the work of Chan and Tsai (Numer. Alg. 53, 171–194 2010). Explicit methods were considered and attention was given to the construction of methods that involve one evaluation of f and many evaluations of g per step. In this work, we consider trigonometrically fitted two derivative explicit Runge-Kutta methods of the general case that use several evaluations of f and g per step; trigonometrically fitting conditions for this general case are given. Attention is given to the construction of methods that involve several evaluations of f and one evaluation of g per step. We modify methods with stages up to four, with three f and one g evaluation and with four f and one g, evaluation based on the fourth and fifth order methods presented in Chan and Tsai (Numer. Alg. 53, 171–194 2010). We provide numerical results to demonstrate the efficiency of the new methods using four test problems.     

Congruence by Dissection of Topological Discs—An Elementary Approach to Tarski's Circle Squaring Problem

Let G be a group of affine transformations of the plane R ² and let the family F consist of all topological discs in R ² whose boundary is subject to some smoothness condition (general, rectifiable, piecewise C ¹ , piecewise C ² ). Are any two members D,E ∈ F congruent by dissection with respect to G such that all the pieces in the corresponding dissections of D and E belong to F as well? We give an affirmative answer if G contains all affine transformations and F consists of the discs whose boundary is piecewise C ¹ . An example shows that C ¹ cannot be replaced by C ² . Moreover, if G is either the group of equiaffine transformations or the group of similarities, then congruence by dissection of two convex discs D and E turns out to be essentially equivalent to congruence by dissection of the boundaries bd(D ) and bd(E ).     

Order of magnitude of multiple Fourier coefficients of functions of bounded <Emphasis Type="Italic">p</Emphasis>-variation

For a Lebesgue integrable complex-valued function f defined over the n-dimensional torus \(\mathbb{T}^n \):= [0, 2π) ⁿ , let \(\hat f\)(k) denote the Fourier coefficient of f, where k = (k ₁, … k _n ) ∈ ? ⁿ . In this paper, defining the notion of bounded p-variation (p ≧ 1) for a function from [0, 2π] ⁿ to ? in two diffierent ways, the order of magnitude of Fourier coefficients of such functions is studied. As far as the order of magnitude is concerned, our results with p = 1 give the results of Móricz [5] and Fülöp and Móricz [3].     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).     

Growth rates of algebras,III: finite solvable algebras

We investigate how the behavior of the function d_A(n), which gives the size of a least size generating set for Aⁿ, influences the structure of a finite solvable algebra A.