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.     

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.     

<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.     

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.     

On Ramanujan sums over the Gaussian integers

This note deals with Ramanujan sums c _m (n) over the ring ?[i], in particular with asymptotics for sums of c _m (n) taken over both variables m, n.     

Curvature inequalities for operators of the Cowen–Douglas class

Let T be an operator tuple in the Cowen–Douglas class B _n (Ω) for Ω ? C ^m . The kernels Ker(T ? w) ^l , for w ∈ Ω, l = 1, 2, ···, define Hermitian vector bundles E _T ^l over Ω. We prove certain negativity of the curvature of E _T ^l . We also study the relation between certain curvature inequality and the contractive property of T when Ω is a planar domain.     

Projections of Patterson-Sullivan measures and the dichotomy of Mohammadi-Oh

Let Γ be some discrete subgroup of SO°(n + 1, R) with finite Bowen-Margulis-Sullivan measure. We study the dynamics of the Bowen-Margulis-Sullivan measure with respect to closed connected subspaces of the N component in some Iwasawa decomposition SO°(n+1, R) = KAN. We also study the dimension of projected Patterson-Sullivan measures along some fixed direction.     

Rings of h-deformed differential operators

We describe the center of the ring Diff _h (n) of h-deformed differential operators of type A. We establish an isomorphism between certain localizations of Diff _h (n) and the Weyl algebra W _n , extended by n indeterminates.     

Random process in a homogeneous Gaussian field

We consider a random process in a spatial-temporal homogeneous Gaussian field V (q , t) with the mean E V = 0 and the correlation function W(|q ? q′|, |t ? t′|) ≡ E[V (q, t)V (q′, t′)], where \( \bold{q} \in {\mathbb{R}^d} \), \( t \in {\mathbb{R}^{+} } \), and d is the dimension of the Euclidean space \( {\mathbb{R}^d} \). For a “density” G(r, t) of the familiar model of a physical system averaged over all realizations of the random field V, we establish an integral equation that has the form of the Dyson equation. The invariance of the equation under the continuous renormalization group allows using the renormalization group method to find an asymptotic expression for G(r, t) as r → ∞ and t → ∞.     

Limit theorems for supremum of Gaussian processes over a random interval

Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) x) is considered, as x →∞.     

Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set

For any bounded domain Ω in ?^m, let B₁(Ω) denote the Cowen-Douglas class of commuting m-tuples of bounded linear operators. For an m-tuple T in the Cowen-Douglas class B₁(Ω), let N_T (w) denote the restriction of T to the subspace \(\cap^m_{i,j=1}{\rm{ker}}(T_i-w_iI)(T_j-w_jI)\). This commuting m-tuple N_T (w) of m + 1 dimensional operators induces a homomorphism \({\rho _{{N_T}\left( w \right)}}\) of the polynomial ring P[z₁, · · ·, z_m], namely, \({\rho _{{N_T}\left( w \right)}}\) (p) = p(N_T (w)), p ∈ P[z₁, · · ·, z_m]. We study the contractivity and complete contractivity of the homomorphism \({\rho _{{N_T}\left( w \right)}}\). Starting from the homomorphism \({\rho _{{N_T}\left( w \right)}}\), we construct a natural class of homomorphisms \(\rho_{N^{(\lambda)}(w)}\), λ > 0, and relate the properties of \(\rho_{N^{(\lambda)}(w)}\) to those of \({\rho _{{N_T}\left( w \right)}}\). Explicit examples arising from the multiplication operators on the Bergman space of Ω are investigated in detail. Finally, it is shown that contractive properties of \({\rho _{{N_T}\left( w \right)}}\) are equivalent to an inequality for the curvature of the Cowen-Douglas bundle E_T. However, we construct examples to show that the contractivity of the homomorphism ρ_T does not follow, even if \({\rho _{{N_T}\left( w \right)}}\) is contractive for all w in Ω.     

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.     

A Maximum Principle for the Controlled Sweeping Process

We consider the free endpoint Mayer problem for a controlled Moreau process, the control acting as a perturbation of the dynamics driven by the normal cone, and derive necessary optimality conditions of Pontryagin’s Maximum Principle type. The results are also discussed through an example. We combine techniques from Sene and Thibault (Journal of Nonlinear and Convex Analysis 15, 647–663, 2014) and from Brokate and Krej?í (Discrete and Continuous Dynamical Systems Series B 18, 331–348, 2013), which in particular deals with a different but related control problem. Our assumptions include the smoothness of the boundary of the moving set C(t), but, differently from Brokate and Krej?í, do not require strict convexity and time independence of C(t). Rather, a kind of inward/outward pointing condition is assumed on the reference optimal trajectory at the times where the boundary of C(t) is touched. The state space is finite dimensional.     

Analysis of algorithms for an online version of the convoy movement problem

In the convoy movement problem (CMP), a set of convoys must be routed from specified origins to destinations in a transportation network, represented by an undirected graph. Two convoys may not cross each other on the same edge while travelling in opposing directions, a restriction referred to as blocking. However, convoys are permitted to follow each other on the same edge, with a specified headway separating them, but no overtaking is permitted. Further, the convoys to be routed are distinguished based on their length. Particle convoys have zero length and are permitted to traverse an edge simultaneously, whereas convoys with non-zero length have to follow each other, observing a headway. The objective is to minimize the total time taken by convoys to travel from their origins to their destinations, given the travel constraints on the edges. We consider an online version of the CMP where convoy demands arise dynamically over time. For the special case of particle convoys, we present an algorithm that has a competitive ratio of 3 in the worst case and (5/2) on average. For the particle convoy problem, we also present an alternate, randomized algorithm that provides a competitive ratio of (√13?1). We then extend the results to the case of convoys with length, which leads to an algorithm with an O(T+CL) competitive ratio, where T={Max _e t(e)}/{Min _e t(e)}, C is the maximum congestion (the number of distinct convoy origin–destination pairs that use any edge e) and L={Max _c L(c)}/{Min _c L(c)}; here L(c)>0 represents the time-headway to be observed by any convoy that follows c and t(e) represents the travel time for a convoy c on edge e.     

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.     

Representations of the Witt Lie Superalgebra with <Emphasis Type="Italic">p</Emphasis>-character of Height One

Let W(n) be the Witt Lie superalgebra over an algebraically closed field k of characteristic p>2. We study the representations of W(n),n≥3 with p-character of height one. We prove that the Kac module always has a unique simple quotient and all simple modules are given in this way. In fact, most Kac modules are simple. We will determine the sufficient and necessary conditions such that the Kac module is simple. Moreover, composition factors of each Kac module can be determined. It is proved that the number of composition factors in each Kac module is at most 2.     

Directed Partial Orders on Complex Numbers and Quaternions over Non-Archimedean Linearly Ordered Fields

Let F be a non-archimedean linearly ordered field, and C and H be the field of complex numbers and the division algebra of quaternions over F, respectively. In this paper, a class of directed partial orders on C are constructed directly and concretely using additive subgroup of F ⁺. This class of directed partial orders includes those given in Rump and Wang (J. Algebra 400, 1–7, 2014), and Yang (J. Algebra 295(2), 452–457, 2006) as special cases and we conjecture that it covers all directed partial orders on C such that 1 > 0. It turns out that this construction also works very well on H. We note that none of these directed partial orders is a lattice order on C or H.     

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).     

Searching for a Best Least Absolute Deviations Solution of an Overdetermined System of Linear Equations Motivated by Searching for a Best Least Absolute Deviations Hyperplane on the Basis of Given Data

We consider the problem of searching for a best LAD-solution of an overdetermined system of linear equations Xa=z, X∈?^m×n, m≥n, \(\mathbf{a}\in \mathbb{R}^{n}, \mathbf {z}\in\mathbb{R}^{m}\). This problem is equivalent to the problem of determining a best LAD-hyperplane x?a ^T x, x∈? ⁿ on the basis of given data \((\mathbf{x}_{i},z_{i}), \mathbf{x}_{i}= (x_{1}^{(i)},\ldots,x_{n}^{(i)})^{T}\in \mathbb{R}^{n}, z_{i}\in\mathbb{R}, i=1,\ldots,m\), whereby the minimizing functional is of the form

$F(\mathbf{a})=\|\mathbf{z}-\mathbf{Xa}\|_1=\sum_{i=1}^m|z_i-\mathbf {a}^T\mathbf{x}_i|.$

An iterative procedure is constructed as a sequence of weighted median problems, which gives the solution in finitely many steps. A criterion of optimality follows from the fact that the minimizing functional F is convex, and therefore the point a ^?∈? ⁿ is the point of a global minimum of the functional F if and only if 0∈?F(a ^?).

Motivation for the construction of the algorithm was found in a geometrically visible algorithm for determining a best LAD-plane (x,y)?αx+βy, passing through the origin of the coordinate system, on the basis of the data (x _i ,y _i ,z _i ),i=1,…,m.