The Nonexistence of Pseudoquaternions in {\mathbb{C}^{2\times 2}}

The field of quaternions, denoted by \mathbbH{\mathbb{H}} can be represented as an isomorphic four dimensional subspace of \mathbbR^4×4{\mathbb{R}^{4\times 4}}, the space of real matrices with four rows and columns. In addition to the quaternions there is another four dimensional subspace in \mathbbR^4×4{\mathbb{R}^{4\times 4}} which is also a field and which has – in connection with the quaternions – many pleasant properties. This field is called field of pseudoquaternions. It exists in \mathbbR^4×4{\mathbb{R}^{4\times 4}} but not in \mathbbH{\mathbb{H}}. It allows to write the quaternionic linear term axb in matrix form as Mx where x is the same as the quaternion x only written as a column vector in \mathbbR⁴{\mathbb{R}^4}. And M is the product of the matrix associated with the quaternion a with the matrix associated with the pseudoquaternion b.     

Compact Surfaces with Constant Gaussian Curvature in Product Spaces

We prove that the only compact surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive constant Gaussian curvature in \mathbbH²×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in \mathbbS²×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds are attained, the surface is again a piece of a rotational complete surface.     

A generalisation of the Hopf construction and harmonic morphisms into {\mathbb {S}^2}

In this paper, we construct a new family of harmonic morphisms ${\varphi:V^5\to\mathbb{S}^2}In this paper, we construct a new family of harmonic morphisms j:V⁵?\mathbbS²{\varphi:V^5\to\mathbb{S}^2}, where V ⁵ is a 5-dimensional open manifold contained in an ellipsoidal hypersurface of \mathbbC⁴ = \mathbbR⁸{\mathbb{C}^4\,=\,\mathbb{R}^8}. These harmonic morphisms admit a continuous extension to the completion V^*5{{V^{\ast}}^5}, which turns out to be an explicit real algebraic variety. We work in the context of a generalization of the Hopf construction and equivariant theory.     

$${\mathbb{Z}_2\mathbb{Z}_4}$$-linear codes: rank and kernel

A code C{{\mathcal C}} is \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C{{\mathcal C}} by deleting the coordinates outside X (respectively, Y) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-additive codes under an extended Gray map are called \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes. In this paper, the invariants for \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a \mathbbZ₂\mathbbZ₄{{\mathbb{Z}_2\mathbb{Z}_4}}-linear code for each possible pair (r, k) is given.     

Heat flows for a nonconvex Signorini type problem in $$ {\mathbb{R}^{N}} $$

We prove the existence of a global heat flow u : Ω ×  \mathbbR⁺ ? \mathbbR^N {\mathbb{R}^{+}} \to {\mathbb{R}^{N}}, N > 1, satisfying a Signorini type boundary condition u(∂Ω ×  \mathbbR⁺ {\mathbb{R}^{+}}) ⊂  \mathbbRⁿ {\mathbb{R}^{n}}), n \geqslant 2 n \geqslant 2 , and \mathbbR^N {\mathbb{R}^{N}}) with boundary ∂ [`(W)] \bar{\Omega } such that φ(∂Ω) ⊂ \mathbbR^N {\mathbb{R}^{N}} is given by a smooth noncompact hypersurface S. Bibliography: 30 titles.     

Quasiregularly elliptic link complements

We extend the theorem of B. Daniel about the existence and uniqueness of immersions into \mathbbSⁿ × \mathbbR or \mathbbHⁿ × \mathbbR{\mathbb{S}^{n}\,\times\,\mathbb{R}\, {\rm or}\, \mathbb{H}^{n}\,\times\,\mathbb{R}} to the Riemannian product of two space forms. More precisely, we prove the existence and uniqueness of an isometric immersion of a Riemannian manifold into the Riemannian product of two space forms.     

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      8. Finite-dimensional pointed Hopf algebras over $\mathbb{S}_4$      Gastón Andrés García  Agustín García Iglesias 《Israel Journal of Mathematics》2011,183(1):417-444 Let K be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras whose group of group-likes is \mathbbS4\mathbb{S}_4. We also describe all pointed Hopf algebras over \mathbbS5\mathbb{S}_5 whose infinitesimal braiding is associated to the rack of transpositions.      9. Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities      Zhaoxin JIANG  Sining ZHENG 《数学年刊B辑(英文版)》2012,33(1):107-112 This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.      10. Isoperimetric profile comparisons and Yamabe constants      Jimmy Petean  Juan Miguel Ruiz 《Annals of Global Analysis and Geometry》2011,40(2):177-189 We estimate from below the isoperimetric profile of S2 ×\mathbb R2{S^2 \times {\mathbb R}^2} and use this information to obtain lower bounds for the Yamabe constant of S2 ×\mathbb R2{S^2 \times {\mathbb R}^2} . This provides a lower bound for the Yamabe invariants of products S 2 ×  M 2 for any closed Riemann surface M. Explicitly we show that Y (S 2 ×  M 2) >  (2/3)Y(S 4).      11. Singular oscillatory integrals on {\mathbb{R}^n}      M. Papadimitrakis  I. R. Parissis 《Mathematische Zeitschrift》2010,266(1):169-179 Let ${\mathcal{P}_{d,n}}Let Pd,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRn{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? Pd,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      12. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      13. Vector bundles with infinitely many souls      Igor Belegradek 《Proceedings of the American Mathematical Society》2003,131(7):2217-2221 We construct the first examples of manifolds, the simplest one being , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.       14. Biharmonic submanifolds of {\mathbb{C}P^n}      D. Fetcu  E. Loubeau  S. Montaldo  C. Oniciuc 《Mathematische Zeitschrift》2010,266(3):505-531 We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPn{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS2n+1{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPn{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPn{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.      15. Some remarks on the Kähler geometry of the Taub-NUT metrics      Andrea Loi  Michela Zedda  Fabio Zuddas 《Annals of Global Analysis and Geometry》2012,41(4):515-533 In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on \mathbbC2{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on \mathbbC2{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function associated to a Taub-NUT metric always exists, while the coefficient a 3 of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.      16. On finite quasi-isometric sets in $ \mathbb{R}^n $      Gabriela Putinar 《Archiv der Mathematik》2003,80(2):130-138 We compare two concepts from distance geometry of finite sets: quasi-isometry and isometry. We show that for every n 3 5 n\geq5 there exist sets of n points in \mathbbRn-1 \mathbb{R}^{n-1} that are quasi-isometric and not isometric. By contrast, for finite sets in S1 we show that under some additional hypotheses, quasi-isometric sets are isometric.      17. On Einstein Riemannian <Emphasis Type="Italic">g</Emphasis>-natural metrics on unit tangent sphere bundles      Mohamed Tahar Kadaoui Abbassi  Oldřich Kowalski 《Annals of Global Analysis and Geometry》2010,38(1):11-20 We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1–1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V2 \mathbbRm+1 = SO(m+1)/SO(m-1){V_2 \mathbb{R}^{m+1} = {{SO}}(m+1)/{{SO}}(m-1)} and all g-natural metrics on T1 Sm{T_1 S^m} (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V2 \mathbbRm+1, m 1 3{V_2 \mathbb{R}^{m+1}, m \neq 3}, initially constructed by Kobayashi.      18. Approximation in $L_{p} (\mathbb{R}^{d}), 0 < p < 1$, by linear combinations of characteristic functions of balls      A. B. Aleksandrov 《Journal of Mathematical Sciences》2010,165(4):431-434 We prove that translates of the characteristic function of a ball span the space L p ( \mathbbRd\mathbb{R}^{d}) provided that 0 <p<1 and d ≥ 2. Similar approximation problems are considered for some other functions. Bibliography: 5 titles.      19. Counting Ω-ideals      Igor V. Protasov 《Algebra Universalis》2009,62(4):339-343 Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(An) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 22à0{2^{2^{\aleph_0}}} Ω-ideals in P\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.      20. Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $      T. M. Osipchuk  M. V. Tkachuk 《Ukrainian Mathematical Journal》2011,63(2):266-277 We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space \mathbbH2 {\mathbb{H}^2} that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space \mathbbC2 {\mathbb{C}^2} .

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Finite-dimensional pointed Hopf algebras over $\mathbb{S}_4$

Let K be an algebraically closed field of characteristic 0. We conclude the classification of finite-dimensional pointed Hopf algebras whose group of group-likes is \mathbbS₄\mathbb{S}_4. We also describe all pointed Hopf algebras over \mathbbS₅\mathbb{S}_5 whose infinitesimal braiding is associated to the rack of transpositions.     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.     

Isoperimetric profile comparisons and Yamabe constants

We estimate from below the isoperimetric profile of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} and use this information to obtain lower bounds for the Yamabe constant of S² ×\mathbb R²{S^2 \times {\mathbb R}^2} . This provides a lower bound for the Yamabe invariants of products S ² ×  M ² for any closed Riemann surface M. Explicitly we show that Y (S ² ×  M ²) >  (2/3)Y(S ⁴).     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      12. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      13. Vector bundles with infinitely many souls      Igor Belegradek 《Proceedings of the American Mathematical Society》2003,131(7):2217-2221 We construct the first examples of manifolds, the simplest one being , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.       14. Biharmonic submanifolds of {\mathbb{C}P^n}      D. Fetcu  E. Loubeau  S. Montaldo  C. Oniciuc 《Mathematische Zeitschrift》2010,266(3):505-531 We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPn{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS2n+1{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPn{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPn{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.      15. Some remarks on the Kähler geometry of the Taub-NUT metrics      Andrea Loi  Michela Zedda  Fabio Zuddas 《Annals of Global Analysis and Geometry》2012,41(4):515-533 In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on \mathbbC2{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on \mathbbC2{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function associated to a Taub-NUT metric always exists, while the coefficient a 3 of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.      16. On finite quasi-isometric sets in $ \mathbb{R}^n $      Gabriela Putinar 《Archiv der Mathematik》2003,80(2):130-138 We compare two concepts from distance geometry of finite sets: quasi-isometry and isometry. We show that for every n 3 5 n\geq5 there exist sets of n points in \mathbbRn-1 \mathbb{R}^{n-1} that are quasi-isometric and not isometric. By contrast, for finite sets in S1 we show that under some additional hypotheses, quasi-isometric sets are isometric.      17. On Einstein Riemannian <Emphasis Type="Italic">g</Emphasis>-natural metrics on unit tangent sphere bundles      Mohamed Tahar Kadaoui Abbassi  Oldřich Kowalski 《Annals of Global Analysis and Geometry》2010,38(1):11-20 We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1–1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V2 \mathbbRm+1 = SO(m+1)/SO(m-1){V_2 \mathbb{R}^{m+1} = {{SO}}(m+1)/{{SO}}(m-1)} and all g-natural metrics on T1 Sm{T_1 S^m} (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V2 \mathbbRm+1, m 1 3{V_2 \mathbb{R}^{m+1}, m \neq 3}, initially constructed by Kobayashi.      18. Approximation in $L_{p} (\mathbb{R}^{d}), 0 < p < 1$, by linear combinations of characteristic functions of balls      A. B. Aleksandrov 《Journal of Mathematical Sciences》2010,165(4):431-434 We prove that translates of the characteristic function of a ball span the space L p ( \mathbbRd\mathbb{R}^{d}) provided that 0 <p<1 and d ≥ 2. Similar approximation problems are considered for some other functions. Bibliography: 5 titles.      19. Counting Ω-ideals      Igor V. Protasov 《Algebra Universalis》2009,62(4):339-343 Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(An) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 22à0{2^{2^{\aleph_0}}} Ω-ideals in P\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.      20. Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $      T. M. Osipchuk  M. V. Tkachuk 《Ukrainian Mathematical Journal》2011,63(2):266-277 We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space \mathbbH2 {\mathbb{H}^2} that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space \mathbbC2 {\mathbb{C}^2} .

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Vector bundles with infinitely many souls

We construct the first examples of manifolds, the simplest one being , which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.

     

Biharmonic submanifolds of {\mathbb{C}P^n}

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold [`(M)]{\bar{M}} in \mathbbCPⁿ{\mathbb{C}P^n} and the bitension field of the inclusion of the corresponding Hopf-tube in \mathbbS²ⁿ⁺¹{\mathbb{S}^{2n+1}}. Using this relation we produce new families of proper-biharmonic submanifolds of \mathbbCPⁿ{\mathbb{C}P^n}. We study the geometry of biharmonic curves of \mathbbCPⁿ{\mathbb{C}P^n} and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.     

Some remarks on the Kähler geometry of the Taub-NUT metrics

In this article, we investigate the balanced condition and the existence of an Engliš expansion for the Taub-NUT metrics on \mathbbC²{\mathbb{C}^2} . Our first result shows that a Taub-NUT metric on \mathbbC²{\mathbb{C}^2} is never balanced unless it is the flat metric. The second one shows that an Engliš expansion of the Rawnsley’s function associated to a Taub-NUT metric always exists, while the coefficient a ₃ of the expansion vanishes if and only if the Taub-NUT metric is indeed the flat one.     

On finite quasi-isometric sets in $ \mathbb{R}^n $

We compare two concepts from distance geometry of finite sets: quasi-isometry and isometry. We show that for every n 3 5 n\geq5 there exist sets of n points in \mathbbR^n-1 \mathbb{R}^{n-1} that are quasi-isometric and not isometric. By contrast, for finite sets in S¹ we show that under some additional hypotheses, quasi-isometric sets are isometric.     

On Einstein Riemannian <Emphasis Type="Italic">g</Emphasis>-natural metrics on unit tangent sphere bundles

We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1–1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V₂ \mathbbR^m+1 = SO(m+1)/SO(m-1){V_2 \mathbb{R}^{m+1} = {{SO}}(m+1)/{{SO}}(m-1)} and all g-natural metrics on T₁ S^m{T_1 S^m} (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V₂ \mathbbR^m+1, m 1 3{V_2 \mathbb{R}^{m+1}, m \neq 3}, initially constructed by Kobayashi.     

Approximation in $L_{p} (\mathbb{R}^{d}), 0 < p < 1$, by linear combinations of characteristic functions of balls

We prove that translates of the characteristic function of a ball span the space L _p ( \mathbbR^d\mathbb{R}^{d}) provided that 0 <p<1 and d ≥ 2. Similar approximation problems are considered for some other functions. Bibliography: 5 titles.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

Analytic criterion for linear convexity of Hartogs domains with smooth boundary in $ {\mathbb{H}^2} $

We establish a criterion for the local linear convexity of sets in the two-dimensional quaternion space \mathbbH² {\mathbb{H}^2} that are analogs of bounded Hartogs domains with smooth boundary in the two-dimensional complex space \mathbbC² {\mathbb{C}^2} .