Removable Singularities of Solutions of the Minimal Surface Equation

Suppose that G is a bounded domain in ? ⁿ (n ? 2), E ≠ G is a relatively closed set in G, and 0 < α < 1. We prove that E is removable for solutions of the minimal surface equation in the class C ^1,α(G)_loc if and only if the (n ? 1 + α)-dimensional Hausdorff measure of E is zero.     

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V ₀. A holomorphic principal G-bundle E _G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V ₀, the holomorphic vector bundle E _G (W) over X associated to E _G for W is finite. Given a holomorphic principal G-bundle E _G over X, we prove that the following four statements are equivalent: (1) The principal G-bundle E _G admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite étale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*E _G is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle E _G (W) = E ×  ^G W over X, associated to the principal G-bundle E _G for the G-module W, is finite. (4) The principal G-bundle E _G is finite.     

The relative Chebyshev centers of finite sets in geodesic spaces

In the present paper we estimate variation in the relative Chebyshev radius R _W (M), where M and W are nonempty bounded sets of a metric space, as the sets M and W change. We find the closure and the interior of the set of all N-nets each of which contains its unique relative Chebyshev center, in the set of all N-nets of a special geodesic space endowed by the Hausdorff metric. We consider various properties of relative Chebyshev centers of a finite set which lie in this set.     

Low rank approximation in G0W0 calculations

The single particle energies obtained in a Kohn-Sham density functional theory (DFT) calculation are generally known to be poor approximations to electron excitation energies that are measured in transport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green’s function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is approximated. The G₀W₀ approximation is a widely used technique in which the self energy is expressed as the convolution of a noninteracting Green’s function (G₀) and a screened Coulomb interaction (W₀) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating W₀ at multiple frequencies. In this paper, we discuss how the cost of G₀W₀ calculation can be reduced by constructing a low rank approximation to the frequency dependent part of W₀. In particular, we examine the effect of such a low rank approximation on the accuracy of the G₀W₀ approximation. We also discuss how the numerical convolution of G₀ and W₀ can be evaluated efficiently and accurately by using a contour deformation technique with an appropriate choice of the contour.     

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

It is shown that, for any compact set K ? ? ⁿ (n ? 2) of positive Lebesgue measure and any bounded domain G ? K, there exists a function in the Hölder class C^1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.     

The Index of a 1-Form on a Real Quotient Singularity

Let G be a finite Abelian group acting (linearly) on space ?ⁿ and, therefore, on its complexification ?ⁿ, and let W be the real part of the quotient ?ⁿ/G (in the general case, W ≠ ?ⁿ/G). The index of an analytic 1-form on the space W is expressed in terms of the signature of the residue bilinear form on the G-invariant part of the quotient of the space of germs of n-forms on (?ⁿ, 0) by the subspace of forms divisible by the 1-form under consideration.     

Variational Problems with Unilateral Pointwise Functional Constraints in Variable Domains

We consider a sequence of convex integral functionals F_s: W¹,^p(Ω_s) → ? and a sequence of weakly lower semicontinuous and generally nonintegral functionals G_s: W¹,^p(Ω_s) → ?, where {Ω_s} is a sequence of domains in ?ⁿ contained in a bounded domain Ω ? ?ⁿ (n ≥ 2) and p > 1. Along with this, we consider a sequence of closed convex sets V_s = {v ∈ W¹,^p(Ω_s): v ≥ K_s(v) a.e. in Ω_s}, where K_s is a mapping from the space W¹,^p(Ω_s) to the set of all functions defined on Ω_s. We establish conditions under which minimizers and minimum values of the functionals F_s + G_s on the sets V_s converge to a minimizer and the minimum value of a functional on the set V = {v ∈ W¹,^p(Ω): v ≥ K(v) a.e. in Ω}, where K is a mapping from the space W¹,^p(Ω) to the set of all functions defined on Ω. These conditions include, in particular, the strong connectedness of the spaces W¹,^p(Ω_s) with the space W¹,^p(Ω), the condition of exhaustion of the domain Ω by the domains Ω_s, the Γ-convergence of the sequence {F_s} to a functional F: W¹,^p(Ω) → ?, and a certain convergence of the sequence {G_s} to a functional G: W¹,^p(Ω) → ?. We also assume some conditions characterizing both the internal properties of the mappings K_s and their relation to the mapping K. In particular, these conditions admit the study of variational problems with irregular varying unilateral obstacles and with varying constraints combining the pointwise dependence and the functional dependence of the integral form.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

The degree of biholomorphic mappings between special domains in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> preserving 0

Let G _i be a closed Lie subgroup of U(n), Ω _i be a bounded G _i -invariant domain in C ⁿ which contains 0, and \(O{\left( {{\mathbb{C}^n}} \right)^{{G_i}}} = \mathbb{C}\), for i = 1; 2. If f: Ω₁ → Ω₂ is a biholomorphism, and f(0) = 0, then f is a polynomial mapping (see Ning et al. (2017)). In this paper, we provide an upper bound for the degree of such polynomial mappings. It is a natural generalization of the well-known Cartan’s theorem.     

Vector spaces of non-extendable holomorphic functions

In this paper, we analyze the linear structure of the family H _e (G) of holomorphic functions on a domain G of the complex plane that are not analytically continuable beyond the boundary of G. We prove that H _e (G) contains, except for zero, a dense algebra; and, under appropriate conditions, the subfamily of H _e (G) consisting of boundary-regular functions contains dense vector spaces with maximal dimension as well as infinite dimensional closed vector spaces and large algebras. We also consider the case in which G is a domain of existence in a complex Banach space. The results obtained complete or extend a number of previous results by several authors.     

Packing coloring of Sierpiński-type graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in \{1,\ldots ,k\}\), where each \(V_i\) is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs \(ST^n_3\) is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs \(S^n_G\) with respect to all connected graphs G of order 4.     

Spectral asymptotics of a nonself-adjoint elliptic differential operator with an indefinite weight function

In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R ⁿ , where A is an elliptic differential operator of order 2m with domain D(A) ? W _m ^2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure.     

Removable singularities of solutions of linear uniformly elliptic second order equations

Let L be a uniformly elliptic linear second order differential operator in divergence form with bounded measurable real coefficients in a bounded domain G ? ?ⁿ (n ? 2). We define classes of continuous functions in G that contain generalized solutions of the equation L? = 0 and have the property that the compact sets removable for such solutions in these classes can be completely described in terms of Hausdorff measures.     

Linear models for the approximate solution of the problem of packing equal circles into a given domain

The linear models for the approximate solution of the problem of packing the maximum number of equal circles of the given radius into a given closed bounded domain G are proposed. We construct a grid in G; the nodes of this grid form a finite set of points T, and it is assumed that the centers of circles to be packed can be placed only at the points of T. The packing problems of equal circles with the centers at the points of T are reduced to 0–1 linear programming problems. A heuristic algorithm for solving the packing problems based on linear models is proposed. This algorithm makes it possible to solve packing problems for arbitrary connected closed bounded domains independently of their shape in a unified manner. Numerical results demonstrating the effectiveness of this approach are presented.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

Matrix semigroups whose ring commutators have real spectra are realizable

We introduce the numerical spectrum \(\sigma _n(A)\subseteq {\mathbb {C}}\) of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, \(\sigma _n(A)\) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of \(C_0\)-semigroups.     

Bounded Engel elements in groups satisfying an identity

We prove that a residually finite group G satisfying an identity \(w\equiv 1\) and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent.     

Groups equal to a product of three conjugate subgroups

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies \(G = {\left( {B{n_0}B} \right)^2} = B{B^{{n_0}}}B\) where n₀ is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.     

Monotonicity of strata in the stratification of the cone of totally positive matrices

According to a theorem of Bjorner [5], there exists a stratified space whose strata are labeled by the elements of [u, v] for every interval [u, v] in the Bruhat order of a Coxeter group W, and each closed stratum (respectively the boundary of each stratum) has the homology of a ball (respectively of a sphere). In [6], Fomin and Shapiro suggest a natural geometric realization of these stratified spaces for a Weyl group W of a semi-simple Lie group G, and then prove its validity in the case of the symmetric group. The stratified spaces arise as links in the Bruhat decomposition of the totally non-negative part of the unipotent radical of G. In this article, we verify the topological regularity property of the strata formed as a result of Bruhat partial ordering on the elements of theWeyl group (of rank 4) of a semi-simple simply connected algebraic group G which is SL(4,?) in our case here. The Weyl group here is the Coxeter group S ₄.     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).