Clones with finitely many relative $${\mathcal {R}}$$ -classes

For each clone C{\mathcal {C}} on a set A there is an associated equivalence relation analogous to Green’s R{\mathcal {R}} -relation, which relates two operations on A if and only if each one is a substitution instance of the other using operations from C{\mathcal {C}} . We study the clones for which there are only finitely many relative R{\mathcal {R}} -classes.     

On the intersection of a Hermitian curve with a conic

Let ${\mathcal{H}}${\mathcal{H}} be a Hermitian curve and let Γ be a conic of PG(2, q ²). In this paper we determine the possible intersection configurations between Γ and H{\mathcal{H}} under the hypotheses that Γ and H{\mathcal{H}} either share two points with the same tangent lines or contain a common Baer subconic. Moreover, the intersection configurations between a degenerate Hermitian curve and a conic sharing a Baer subconic are also determined.     

Induced Graph Packing Problems

Let H{\mathcal{H}} be a set of undirected graphs. The induced H{\mathcal{H}} -packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of H{\mathcal{H}} . In this paper we focus on the case when H{\mathcal{H}} consists of factor-critical graphs and a certain family of ‘propellers’. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai–Edmonds type structure theorem and a Berge–Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when H{\mathcal{H}} consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.     

On the Right (Left) Invertible Completions for Operator Matrices

Let ${\mathcal {H}_{1}}Let H₁{\mathcal {H}_{1}} and H₂{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H₁), B ? B(H₂){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H₂, H₁){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H₁, H₂){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained.     

Repeated and Continuous Interactions in Open Quantum Systems

We consider a finite quantum system S{\mathcal {S}} coupled to two environments of different nature. One is a heat reservoir R{\mathcal {R}} (continuous interaction) and the other one is a chain C{\mathcal {C}} of independent quantum systems E{\mathcal {E}} (repeated interaction). The interactions of S{\mathcal {S}} with R{\mathcal {R}} and C{\mathcal {C}} lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S+R{\mathcal {S}+\mathcal {R}} and show that it satisfies a second law of thermodynamics. We analyze a model where both S{\mathcal {S}} and E{\mathcal {E}} are two-level systems and obtain the asymptotic state explicitly (at lowest order in the interaction strength). Even though R{\mathcal {R}} and C{\mathcal {C}} are not directly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W ^*-dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We analyze the full system via rigorous perturbation theory in the coupling strength, and do not resort to any scaling limit, like e.g. weak coupling limits, or any other approximations in order to derive some master equation.     

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      7. Composition Followed by Differentiation Between H ∞ and Zygmund Spaces      Yongmin Liu  Yanyan Yu 《Complex Analysis and Operator Theory》2012,6(1):121-137 Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DCj : H￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.      8. Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules      Patrice P. Ntumba 《Mediterranean Journal of Mathematics》2010,7(4):445-454 Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.      9. Augmented Teichmüller spaces and orbifolds      Vladimir Hinich  Arkady Vaintrob 《Selecta Mathematica, New Series》2010,16(3):533-629 We study complex analytic properties of the augmented Teichmüller spaces [`(T)]g,n{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces Tg,n{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike Tg,n{\mathcal{T}_{g,n}}, the space [`(T)]g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.      10. Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}      John Erik Fornæss  Nessim Sibony 《Geometric And Functional Analysis》2010,19(5):1334-1377 Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP2{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.      11. Weyl’s Theorem for Functions of Operators and Approximation      Chun Guang Li  Sen Zhu  You Ling Feng 《Integral Equations and Operator Theory》2010,67(4):481-497 Let H{\mathcal{H}} be a complex separable infinite dimensional Hilbert space. In this paper, we characterize those operators T on H{\mathcal{H}} satisfying that Weyl’s theorem holds for f(T) for each function f analytic on some neighborhood of σ(T). Also, it is proved that, given an operator T on H{\mathcal{H}} and ε > 0, there exists a compact operator K with ||K|| < e{\|K\| < \varepsilon} such that Weyl’s theorem holds for T + K.      12. 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.      13. Multiplication Operators on the Lipschitz Space of a Tree      Flavia Colonna  Glenn R. Easley 《Integral Equations and Operator Theory》2010,68(3):391-411 In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L{\mathcal{L}} we call the little Lipschitz space.      14. Nilpotent algebras and affinely homogeneous surfaces      Gregor?Fels  Wilhelm?KaupEmail author 《Mathematische Annalen》2012,353(4):1315-1350 To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.      15. Multisorted dualisability: change of base      B. A. Davey  M. J. Gouveia  M. Haviar  H. A. Priestley 《Algebra Universalis》2011,66(4):331-336 We prove that if a quasivariety A{\mathcal{A}} generated by a finite family M{\mathcal{M}} of finite algebras has a multisorted duality based on M{\mathcal{M}}, then A{\mathcal{A}} has a multisorted duality based on any finite family of finite algebras that generates it.      16. Unstable structures definable in o-minimal theories      Assaf Hasson  Alf Onshuus 《Selecta Mathematica, New Series》2010,16(1):121-143 Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in Neq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.      17. Compactified Picard stacks over $${\overline{\mathcal M}_g}$$      Margarida Melo 《Mathematische Zeitschrift》2009,263(4):939-957 We study algebraic (Artin) stacks over [`(M)]g{\overline{\mathcal M}_{g}} giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get Deligne–Mumford stacks strongly representable over[`(M)]g{\overline{\mathcal M}_{g}} .      18. Pan-H-Linked Graphs      Michael Ferrara  Colton Magnant  Jeffrey Powell 《Graphs and Combinatorics》2010,26(2):225-242 Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n 0(H) isolated vertices and n 1(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165–182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H{\mathcal{H}} and d(G) 3 \fracn+m-k+n1(H)+2n0(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}}, then G contains a spanning H-subdivision with the same ground set as H{\mathcal{H}} . As a corollary to this result, the authors were able to obtain Dirac’s famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3 with d(G) 3 \fracn2{\delta(G)\ge\frac{n}{2}} , then G is hamiltonian. Bondy (J. Comb. Theory Ser. B 11:80–84, 1971) extended Dirac’s theorem by showing that if G satisfied the condition d(G) 3 \fracn2{\delta(G) \ge \frac{n}{2}} then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165–182, 2007) in a similar manner. An H-subdivision H{\mathcal{H}} in G is 1-extendible if there exists an H-subdivision H*{\mathcal{H}^{*}} with the same ground set as H{\mathcal{H}} and |H*| = |H| + 1{|\mathcal{H}^{*}| = |\mathcal{H}| + 1} . If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that d(G) 3 \fracn+m-k+n1(H)+2n0(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}} , then G is pan-H-linked. This result is sharp.      19. Generalized Absolute Values and Polar Decompositions of a Bounded Operator      Piotr Niemiec 《Integral Equations and Operator Theory》2011,71(2):151-160 Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?K|2 ￡ á|T|x, x?Há|T*|y, y?K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function f: \mathbbR+ ? \mathbbR+{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.      20. Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips      D. Danielli  N. Garofalo  D. M. Nhieu 《Mathematische Zeitschrift》2010,265(3):617-637 We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H1{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H1{{\mathbb {H}}^1} the quantity \fracsH(S?B(p,r))rQ-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H1{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.

Composition Followed by Differentiation Between H ∞ and Zygmund Spaces

Let j{\varphi} be an analytic self-map of the unit disk \mathbbD{\mathbb{D}}, H(\mathbbD){H(\mathbb{D})} the space of analytic functions on \mathbbD{\mathbb{D}} and g ? H(\mathbbD){g \in H(\mathbb{D})}. The boundedness and compactness of the operator DC_j : H^￥ ? Z{DC_\varphi : H^\infty \rightarrow { \mathcal Z}} are investigated in this paper.     

Cartan-Dieudonné Theorem for {\mathcal {A}}-Modules

Like the classical Cartan-Dieudonné theorem, the sheaf-theoretic version shows that A{\mathcal {A}}-isometries on a convenient A{\mathcal {A}}-module E{\mathcal {E}} of rank n can be decomposed in at most n orthogonal symmetries (reflections) with respect to non-isotropic hyperplanes. However, the coefficient sheaf of \mathbb C{\mathbb {C}}-algebras A{\mathcal {A}} is assumed to be a PID \mathbb C{\mathbb {C}}-algebra sheaf and, if (E,f){(\mathcal {E},\phi)} is a pairing with f{\phi} a non-degenerate A{\mathcal {A}}-bilinear morphism, we assume that E{\mathcal {E}} has nowhere-zero (local) isotropic sections; but, for Riemannian sheaves of A{\mathcal {A}}-modules, this is not necessarily required.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}

Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP²{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.     

Weyl’s Theorem for Functions of Operators and Approximation

Let H{\mathcal{H}} be a complex separable infinite dimensional Hilbert space. In this paper, we characterize those operators T on H{\mathcal{H}} satisfying that Weyl’s theorem holds for f(T) for each function f analytic on some neighborhood of σ(T). Also, it is proved that, given an operator T on H{\mathcal{H}} and ε > 0, there exists a compact operator K with ||K|| < e{\|K\| < \varepsilon} such that Weyl’s theorem holds for T + K.     

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.     

Multiplication Operators on the Lipschitz Space of a Tree

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L{\mathcal{L}} we call the little Lipschitz space.     

Nilpotent algebras and affinely homogeneous surfaces

To every nilpotent commutative algebra N{\mathcal{N}} of finite dimension over an arbitrary base field of characteristic zero a smooth algebraic subvariety S ì N{S\subset\mathcal{N}} can be associated in a canonical way whose degree is the nil-index and whose codimension is the dimension of the annihilator A{\mathcal{A}} of N{\mathcal{N}}. In case N{\mathcal{N}} admits a grading, the surface S is affinely homogeneous. More can be said if A{\mathcal{A}} has dimension 1, that is, if N{\mathcal{N}} is the maximal ideal of a Gorenstein algebra. In this case two such algebras N{\mathcal{N}}, [(N)\tilde]{\tilde{\mathcal{N}}} are isomorphic if and only if the associated hypersurfaces S, [(S)\tilde]{\tilde S} are affinely equivalent. If one of S, [(S)\tilde]{\tilde S} even is affinely homogeneous, ‘affinely equivalent’ can be replaced by ‘linearly equivalent’. In case the nil-index of N{\mathcal{N}} does not exceed 4 the hypersurface S is always affinely homogeneous. Contrary to the expectation, in case nil-index 5 there exists an example (in dimension 23) where S is not affinely homogeneous.     

Multisorted dualisability: change of base

We prove that if a quasivariety A{\mathcal{A}} generated by a finite family M{\mathcal{M}} of finite algebras has a multisorted duality based on M{\mathcal{M}}, then A{\mathcal{A}} has a multisorted duality based on any finite family of finite algebras that generates it.     

Unstable structures definable in o-minimal theories

Let M{\mathcal {M}} be a dense o-minimal structure, N{\mathcal {N}} an unstable structure interpretable in M{\mathcal {M}}. Then there exists X, definable in N^eq{\mathcal {N}^{eq}}, such that X, with the induced N{\mathcal {N}}-structure, is linearly ordered and o-minimal with respect to that ordering. As a consequence we obtain a classification, along the lines of Zilber’s trichotomy, of unstable t-minimal types in structures interpretable in o-minimal theories.     

Compactified Picard stacks over $${\overline{\mathcal M}_g}$$

We study algebraic (Artin) stacks over [`(M)]<sub>g</sub>{\overline{\mathcal M}_{g}} giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get Deligne–Mumford stacks strongly representable over[`(M)]_g{\overline{\mathcal M}_{g}} .     

Pan-H-Linked Graphs

Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n ₀(H) isolated vertices and n ₁(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165–182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H{\mathcal{H}} and d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}}, then G contains a spanning H-subdivision with the same ground set as H{\mathcal{H}} . As a corollary to this result, the authors were able to obtain Dirac’s famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3 with d(G) 3 \fracn2{\delta(G)\ge\frac{n}{2}} , then G is hamiltonian. Bondy (J. Comb. Theory Ser. B 11:80–84, 1971) extended Dirac’s theorem by showing that if G satisfied the condition d(G) 3 \fracn2{\delta(G) \ge \frac{n}{2}} then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165–182, 2007) in a similar manner. An H-subdivision H{\mathcal{H}} in G is 1-extendible if there exists an H-subdivision H^*{\mathcal{H}^{*}} with the same ground set as H{\mathcal{H}} and |H^*| = |H| + 1{|\mathcal{H}^{*}| = |\mathcal{H}| + 1} . If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}} , then G is pan-H-linked. This result is sharp.     

Generalized Absolute Values and Polar Decompositions of a Bounded Operator

Generalized absolute values as well as corresponding to them generalized polar decompositions of a bounded linear operator T of a Hilbert space H{\mathcal{H}} into a Hilbert space K{\mathcal{K}} are defined, motivated by the inequality |áTx, y?_K|² ￡ á|T|x, x?_Há|T^*|y, y?_K{|\langle{Tx}, {y}\rangle}_{\mathcal{K}}|^2 \leq \langle|T|x, {x}\rangle_{\mathcal{H}}\langle{|T^{*}|y}, {y}\rangle_{\mathcal{K}} . It is shown that there is a natural bijection between generalized absolute values of T and of T* which sends |T| to |T*|. For a bounded nonnegative operator A on H{\mathcal{H}} and a bounded Borel function f: \mathbbR₊ ? \mathbbR₊{f: \mathbb{R}_+ \to \mathbb{R}_+} , equivalent conditions for A and f(|T|) to be generalized absolute values of T are established and corresponding to them generalized absolute values of T* are determined.     

Sub-Riemannian calculus and monotonicity of the perimeter for graphical strips

We consider the class of minimal surfaces given by the graphical strips ${{\mathcal S}}We consider the class of minimal surfaces given by the graphical strips S{{\mathcal S}} in the Heisenberg group \mathbb H¹{{\mathbb {H}}^1} and we prove that for points p along the center of \mathbb H¹{{\mathbb {H}}^1} the quantity \fracs_H(S?B(p,r))r^Q-1{\frac{\sigma_H(\mathcal S\cap B(p,r))}{r^{Q-1}}} is monotone increasing. Here, Q is the homogeneous dimension of \mathbb H¹{{\mathbb {H}}^1} . We also prove that these minimal surfaces have maximum volume growth at infinity.