On latently real matrices and block quaternions

Let a complex n × n matrix A be unitarily similar to its entrywise conjugate matrix [`(A)] \bar{A} . If in the relation [`(A)] = P^*AP \bar{A} = {P^*}AP the unitary matrix P can be chosen symmetric (skew-symmetric), then A is called a latently real matrix (respectively, a generalized block quaternion). The differences in the systems of elementary divisors of these two matrix classes are found that explain why latently real matrices can be made real via unitary similarities, whereas, in general, block quaternions cannot. Bibliography: 5 titles.     

On sufficient conditions for the existence of a unitary congruence transformation of a given complex matrix into a real one

A complex n × n matrix A is said to be nonderogatory if the degree of its minimal polynomial is equal to the degree of the characteristic polynomial. The aim of the paper is to prove the following assertion: Let A[`(A)] A\bar{A} be a nonderogatory matrix with real positive spectrum. Then A can be made real by a unitary congruence transformation if and only if A and [`(A)] \bar{A} are unitarily congruent. Bibliography: 5 titles.     

Improved bounds for the recursion width in congruent type methods for solving systems of linear equations

It is shown that n× n solutions A and B of the matrix equation

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems

In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in \mathbbR²{\mathbb{R}^2} of degree d that in complex notation z = x + i y can be written as

On Integral Conditions for the General Beltrami Equations

Under integral restrictions on dilatations, it is proved existence theorems for the degenerate Beltrami equations with two characteristics [`(?)]f = m?f + n[`(?f)]{{\overline {\partial}}f = \mu {{\partial f}} + \nu{\overline {\partial f}}} and, in particular, to the Beltrami equations of the second type [`(?)]f = n[`(?f)]{{\overline {\partial}}f = \nu{\overline {\partial f}}} that play a great role in many problems of mathematical physics and to the so-called reduced Beltrami equations [`(?)]f = lRe ?f{{\overline {\partial}}f = \lambda {\rm Re}\,{{\partial f}}} that also have significant applications.     

Domains with Hyperbolic Orbit Accumulation Boundary Points

Let Ω be a smoothly bounded pseudoconvex domain in ℂ ⁿ satisfying the condition R. Suppose that its Bergman kernel extends to [`(W)]×[`(W)]\overline{\Omega}\times\overline{\Omega} minus the boundary diagonal set as a locally bounded function. In this paper we show that for each hyperbolic orbit accumulation boundary point p, there exists a contraction f∈Aut(Ω) at p. As an application, we show that Ω admits a hyperbolic orbit accumulation boundary point if and only if it is biholomorphically equivalent to a domain defined by a weighted homogeneous polynomial and that Ω is of finite D’Angelo type.     

Purity for overconvergence

Let X \hookrightarrow[`(X)]{X \hookrightarrow \overline{X}} be an open immersion of smooth varieties over a field of characteristic p > 0 such that the complement is a simple normal crossing divisor and [`(Z)] í Z í [`(X)]{\overline{Z}\subseteq Z \subseteq \overline{X}} closed subschemes of codimension at least 2. In this paper, we prove that the canonical restriction functor between the categories of overconvergent F-isocrystals F-Isoc<sup>f</sup>(X,[`(X)]) ? F-Isoc^f(X\Z,[`(X)]\[`(Z)]){F-{\rm Isoc}^\dagger(X,\overline{X}) \longrightarrow F-{\rm Isoc}^\dagger(X{\setminus}Z, \overline{X}{\setminus}\overline{Z})} is an equivalence of categories. We also give an application of our result to the equivalence of certain categories.     

Unitary similarity of entrywise conjugate matrices and unitary decomplexification

Assume that a complex n × n matrix A is unitarily similar to a real matrix. It is proved that in this case, A is unitarily similar to its entrywise conjugate [`(A)] \bar{A} . An example showing that the reverse implication is in general false is constructed. Bibliography: 1 title.     

Extending the Torelli map to toroidal compactifications of Siegel space

It has been known since the 1970s that the Torelli map M  _g →A  _g , associating to a smooth curve its Jacobian, extends to a regular map from the Deligne–Mumford compactification [`(\operatorname M)]<sub>g</sub>\overline {\operatorname {M}}_{g} to the 2nd Voronoi compactification [`(\operatorname A)]_g^vor\overline {\operatorname {A}}_{g}^{\mathrm {vor}}. We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification [`(\operatorname A)]<sub>g</sub><sup>perf</sup>\overline {\operatorname {A}}_{g}^{\mathrm {perf}} is also regular, and moreover [`(\operatorname A)]_g^vor\overline {\operatorname {A}}_{g}^{\mathrm {vor}} and [`(\operatorname A)]<sub>g</sub><sup>perf</sup>\overline {\operatorname {A}}_{g}^{\mathrm {perf}} share a common Zariski open neighborhood of the image of [`(\operatorname M)]_g\overline {\operatorname {M}}_{g}. We also show that the map to the Igusa monoidal transform (central cone compactification) is not regular for g≥9; this disproves a 1973 conjecture of Namikawa.     

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

The singular values of the hadamard product of a positive semidefinite and a skew-symmetric matrix

Let σ₁(X)≤ · ≤ σ_N(X)≤0 denote the ordered singular values ofan n × n matrix X and let α₁ (X) ≤ α₂(X)≤ · ≤ α_n(X) denote its ordered main diagonal entries (assuming that they are real). Let B be any complex n × n skew-symmetric matrix and ||.|| any unitarily invariant norm. It is shown that for any rea positive semidefinite n × n matrix A.     

Exotic smooth structures on small 4-manifolds with odd signatures

Let M be (2n-1)\mathbbCP²#2n[`(\mathbbCP)]<sup>2</sup>(2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2} for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \mathbbCP<sup>2</sup>#4[`(\mathbbCP)]²\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2} or 3\mathbbCP²#k[`(\mathbbCP)]²3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2} for k=6,8,10.     

On the sets of branch points of mappings more general than quasiregular

It is shown that if a point x ₀ ∊ ℝ ⁿ , n ≥ 3, is an essential isolated singularity of an open discrete Q-mapping f : D → [`(\mathbb R<sup>n</sup>)] \overline {\mathbb {R}^n} , B <sub>f</sub> is the set of branch points of f in D; and a point z <sub>0</sub> ∊ [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} is an asymptotic limit of f at the point x ₀; then, for any neighborhood U containing the point x ₀; the point z ₀ ∊ [`(f( B<sub>f</sub> ?U ))] \overline {f\left( {B_f \cap U} \right)} provided that the function Q has either a finite mean oscillation at the point x <sub>0</sub> or a logarithmic singularity whose order does not exceed n − 1: Moreover, for n ≥ 2; under the indicated conditions imposed on the function Q; every point of the set [`(\mathbb Rⁿ)] \overline {\mathbb {R}^n} \ f(D) is an asymptotic limit of f at the point x ₀. For n ≥ 3, the following relation is true: [`(\mathbbR<sup>n</sup> )] \f( D ) ì [`(f B_f )] \overline {\mathbb{R}^n } \backslash f\left( D \right) \subset \overline {f\,B_f } . In addition, if ￥ ? f( D ) \infty \notin f\left( D \right) , then the set f B _f is infinite and x₀ ? [`(B_f )] x_0 \in \overline {B_f } .     

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.     

Characterization of field homomorphisms and derivations by functional equations

Summary. Let K and [`(K)] \overline K be fields containing \Bbb Q {\Bbb Q} . We characterize pairs of additive functions f,g: K ?[`(K)] f,g: K \to \overline K satisfying a functional equation¶¶ g(x^ln) = f(x^l)ⁿ     \textrespectively        g(x^ln) = Ax^ln + x^ln-lf(x^l) g(x^{ln}) = f(x^l)^n \quad \text{respectively} \qquad g(x^{ln}) = Ax^{ln} + x^{ln-l}f(x^l) ,¶where n ? \Bbb Z \{0,1} n \in {\Bbb Z} \setminus \{0,1\} , l ? \Bbb N l\in {\Bbb N} and A ? K A \in K .     

Partitions involving D.H. Lehmer numbers

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by [`(a)]<sub>c</sub>{\overline{a}_{c}} the unique integer satisfying 1 ￡ [`(a)]_c ￡ q{1\leq\overline{a}_{c} \leq{q}} and a[`(a)]_c o c(mod q){a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}. Put

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let [`(k)] \bar{k} be a fixed algebraic closure of k. Let X be a smooth geometrically integral k-variety; we set [`(X)] = X ×_k[`(k)] \bar{X} = X{ \times_k}\bar{k} and denote by [`(X)] \bar{X} . In [BvH2] we defined the extended Picard complex of X as the complex of Gal( [`(k)]

Pontrjagin–Thom maps and the homology of the moduli stack of stable curves

We study the singular homology (with field coefficients) of the moduli stack [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} of stable n-pointed complex curves of genus g. Each irreducible boundary component of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}} determines via the Pontrjagin–Thom construction a map from [`(\mathfrakM)]<sub>g, n</sub>{\overline{\mathfrak{M}}_{g, n}} to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This detects many new torsion classes in the homology of [`(\mathfrakM)]_{g, n}{\overline{\mathfrak{M}}_{g, n}}.     

Compact embeddings of besov spaces involving only slowly varying smoothness

We characterize compact embeddings of Besov spaces B _p,r ^0,b (ℝ ⁿ ) involving the zero classical smoothness and a slowly varying smoothness b into Lorentz-Karamata spaces L<sub>p,q;[`(b)]</sub> {L_{p,q;\overline b }}(Ω), where is a bounded domain in ℝ <sup>n</sup> and [`(b)]\overline b is another slowly varying function.     

When is the Uvarov transformation positive definite?

Let L\cal{L} be a positive definite bilinear functional, then the Uvarov transformation of L\cal{L} is given by  U(p,q) = L(p,q) + m p(a)[`(q)](a<sup>-1</sup>) +[`(m)] p([`(a)]<sup>-1</sup>)\,\mathcal{U}(p,q) = \mathcal{L}(p,q) + m\,p(\alpha)\overline{q}(\alpha^{-1}) + \overline{m}\,p(\overline{\alpha}^{-1}) [`(q)]([`(a)])\overline{q}(\overline{\alpha}) where $|\alpha| > 1, m \in \mathbb{C}$|\alpha| > 1, m \in \mathbb{C}. In this paper we analyze conditions on m for U\cal{U} to be positive definite in the linear space of polynomials of degree less than or equal to n. In particular, we show that m has to lie inside a circle in the complex plane defined by α, n and the moments associated with L\cal{L}. We also give an upper bound for the radius of this circle that depends only on α and n. This and other conditions on m are visualized for some examples.