Discrete-Time Dichotomous Well-Posed Linear Systems and Generalized Schur–Nevanlinna–Pick Interpolation

We introduce a class of matrix-valued functions W called “L²- regular”. In case W is J-inner, this class coincides with the class of “strongly regular J-inner” matrix functions in the sense of Arov–Dym. We show that the class of L²-regular matrix functions is exactly the class of transfer functions for a discrete-time dichotomous (possibly infinite-dimensional) input-state-output linear system having some additional stability properties. When applied to J-inner matrix functions, we obtain a state-space realization formula for the resolvent matrix associated with a generalized Schur–Nevanlinna–Pick interpolation problem. Communicated by Daniel Alpay Submitted: August 20, 2006; Accepted: September 13, 2006     

Optimal Semi-Iterative Methods for Complex SOR with Results from Potential Theory

We consider the application of semi-iterative methods (SIM) to the standard (SOR) method with complex relaxation parameter ω, under the following two assumptions: (1) the associated Jacobi matrix J is consistently ordered and weakly cyclic of index 2, and (2) the spectrum σ(J) of J belongs to a compact subset Σ of the complex plane , which is symmetric with respect to the origin. By using results from potential theory, we determine the region of optimal choice of for the combination SIM–SOR and settle, for a large class of compact sets Σ, the classical problem of characterising completely all the cases for which the use of the SIM-SOR is advantageous over the sole use of SOR, under the hypothesis that . In particular, our results show that, unless the outer boundary of Σ is an ellipse, SIM–SOR is always better and, furthermore, one of the best possible choices is an asymptotically optimal SIM applied to the Gauss–Seidel method. In addition, we derive the optimal complex SOR parameters for all ellipses which are symmetric with respect to the origin. Our work was motivated by recent results of M.Eiermann and R.S. Varga.Dedicated to Professor Richard S. Varga in recognition of his substantial contributions to the subject of the paper.     

J-pluripolar subsets and currents on almost complex manifolds

We prove that every almost complex submanifold of an almost complex manifold is locally J-pluripolar. This generalizes a result of Rosay for J-holomorphic submanifolds. Our second main result is an almost complex version of El Mir’s theorem for the extension of positive currents across locally complete pluripolar sets. As a consequence, we extend some results proved by Dabbek–Elkhadhra–El Mir and Dinh–Sibony in the standard complex case. We also obtain a version of the well-known results of Federer and Bassanelli for flat and \mathbb C{\mathbb {C}}-flat currents in the almost complex setting.     

On the Differential Geometric Characterization of The Lee Models

This paper pertains to the J-Hermitian geometry of model domains introduced by Lee (Mich. Math. J. 54(1), 179–206, 2006; J. Reine Angew. Math. 623, 123–160, 2008). We first construct a Hermitian invariant metric on the Lee model and show that the invariant metric actually coincides with the Kobayashi-Royden metric, thus demonstrating an uncommon phenomenon that the Kobayashi-Royden metric is J-Hermitian in this case. Then we follow Cartan’s differential-form approach and find differential-geometric invariants, including torsion invariants, of the Lee model equipped with this J-Hermitian Kobayashi-Royden metric, and present a theorem that characterizes the Lee model by those invariants, up to J-holomorphic isometric equivalence. We also present an all dimensional analysis of the asymptotic behavior of the Kobayashi metric near the strongly pseudoconvex boundary points of domains in almost complex manifolds.     

The quantum orbifold cohomology of weighted projective spaces

We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S ¹-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities.     

The Patterson–Sullivan Embedding and Minimal Volume Entropy for Outer Space

Motivated by Bonahon’s result for hyperbolic surfaces, we construct an analogue of the Patterson–Sullivan–Bowen–Margulis map from the Culler–Vogtmann outer space CV (F _k ) into the space of projectivized geodesic currents on a free group. We prove that this map is a continuous embedding and thus obtain a new compactification of the outer space. We also prove that for every k ≥ 2 the minimum of the volume entropy of the universal covers of finite connected volume-one metric graphs with fundamental group of rank k and without degree-one vertices is equal to (3k − 3) log 2 and that this minimum is realized by trivalent graphs with all edges of equal lengths, and only by such graphs. Received: December 2005, Accepted: March 2006     

Neighborhood of an Embedded <Emphasis Type="Italic">J</Emphasis>-Holomorphic Disc

We study the deformation space of an embedded J-holomorphic disc D in an almost complex surface (X,J). Every such J is shown to be equivalent to a small deformation of a certain model structure J _β along D, where β : D→ℂ is a complex valued function whose modulus |β| is a biholomorphic invariant. Furthermore, we find a nonlinear invertible operator mapping the space of all small J-holomorphic deformations of the given J-holomorphic disc onto the space of small holomorphic deformations of the standard disc in ℂ².     

Two methods for avoiding hereditary completeness

A family of vectors of a Hubert space H is said to be hereditarily complete if it posses a biorthogonal family {x_n′;n≥1}((x_n,x_k′)=δ_nk) and if any elementx, xε H can be reconstructed in terms of the component of its Fourier series, i.e., x∈V((x,x′_n)x_n:n≥1),∀x∈H. In the paper we indicate two simple methods for constructing nonhereditary complete minimal families having a total biorthogonal family, which just not long ago has caused well-known difficulties (see Ref. Zh. Mat., 1975, 7B802). The first method consists in the fact that a given pair of biorthogonal families Y, Y′ of the space H′,H′⊂H is represented as the projection of the families of the same type but already complete in H.. Clearly, in this case cannot be hereditarily complete. The second method consists in considering linear deformation _n :n⩾1 of the orthogonal bases_n: n⩾1; here A is an unbounded operator of a special type. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 183–188, 1976.     

Mean-Field Theory for Heisenberg Zigzag Ladder: Ground State Energy and Spontaneous Symmetry Breaking

The spin-1/2 zig-zag Heisenberg ladder (J₁ − J₂ model) is considered. A new representation for the model is found and a saddle point approximation over the spin-liquid order parameter is performed. Corresponding effective action is derived and analytically analyzed. We observe the presence of phase transitions at values J₂/J₁ = 0.230971 and J₂/J₁ = 1/2. Communicated by Vincent Rivasseau Dedicated to the memory of Daniel Arnaudon Submitted: February 28, 2006; Accepted: May 15, 2006     

Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L ²-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝⁿ, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.     

Magnetic bubble refraction and quasibreathers in inhomogeneous antiferromagnets

We study the dynamics of magnetic bubble solitons in a two-dimensional isotropic antiferromagnetic spin lattice in the case where the exchange integral J(x, y) is position dependent. In the near-continuum regime, this system is described by the relativistic O(3) sigma model on a space-time with a spatially inhomogeneous metric determined by J. We use the geodesic approximation to describe the low-energy soliton dynamics in this system: the n-soliton motion is approximated by geodesic motion in the moduli space M _n of static n-solitons equipped with the L ² metric γ. We obtain explicit formulas for γ for various natural choices of J(x, y). Based on these, we show that single soliton trajectories are refracted with J⁻¹ being analogous to the refractive index and that this refraction effect allows constructing simple bubble lenses and bubble guides. We consider the case where J has a disk inhomogeneity (with the value J ₊ outside a disk and J ₋ < J ₊ inside) in detail. We argue that for sufficiently large J ₊/J ₋, this type of antiferromagnet supports approximate quasibreathers: two or more coincident bubbles confined within the disk spin internally while their shape oscillates with a generically incommensurate period. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 191–208, July, 2007.     

Half-space type theorems in warped product spaces with one-dimensional factor

This work states some half-space type theorems in a warped product space of the form I ×_ρ M, where is an open interval and M is either a compact n-manifold, or a complete simply connected surface with constant curvature c ≤ 0. Such theorems generalize the classical half-space theorem for minimal surfaces in R ³, obtained by Hoffmann and Meeks (Invent Math 101:373–377, 1990), and recent results for surfaces contained in a slab of R ×_ρ M, obtained by Dajczer and Alías (Comment Math Helvetici 81:653–663, 2006).      

L 3,∞-solutions to the MHD equations

We prove that weak solutions to the MHD system are smooth provided that they belong to the so-called “critical” Ladyzhenskaya-Prodi-Serrin class L_3,∞. Besides the independent interest, this result disproves the hypothesis on existence of collapsing self-similar solutions to the MHD equations for which the generating profile belongs to the space L₃. Thus, we extend the results which were known before for the Navier-Stokes system to the case of the MHD equations. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 112–132.     

Gauge-Invariant Characterization of Yang–Mills–Higgs Equations

Let C → M be the bundle of connections of a principal G-bundle P → M over a pseudo-Riemannian manifold (M,g) of signature (n⁺, n⁻) and let E → M be the associated bundle with P under a linear representation of G on a finite-dimensional vector space. For an arbitrary Lie group G, the O(n⁺, n⁻) × G-invariant quadratic Lagrangians on J¹(C × _M E) are characterized. In particular, for a simple Lie group the Yang–Mills and Yang–Mills–Higgs Lagrangians are characterized, up to an scalar factor, to be the only O(n⁺, n⁻) × G-invariant quadratic Lagrangians. These results are also analyzed on several examples of interest in gauge theory. Submitted: May 19, 2005; Accepted: April 25, 2006     

Stationary distribution of a process of random semi-Markov evolution with delaying screens in the case of balance

We determine a stationary measure for a process defined by a differential equation with phase space on the segment [V ₀, V ₁] and constant values of a vector field that depend on a controlling semi-Markov process with finite set of states. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 381–387, March, 2006.     

Positivity of the self-diffusion matrix of interacting Brownian particles with hard core

We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian balls. Received: 22 September 1997 / Revised version: 15 January 1998     

Functions of the first Baire class with values in metrizable spaces

We show that every mapping of the first functional Lebesgue class that acts from a topological space into a separable metrizable space that is linearly connected and locally linearly connected belongs to the first Baire class. We prove that the uniform limit of functions of the first Baire class ƒ _n: X → Y belongs to the first Baire class if X is a topological space and Y is a metric space that is linearly connected and locally linearly connected. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 568–572, April, 2006.     

An <Emphasis Type="Italic">R</Emphasis> = <Emphasis Type="Italic">T</Emphasis> theorem for imaginary quadratic fields

We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the residual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply the criterion, one needs to show that the quotient of the universal deformation ring R by its ideal of reducibility is cyclic Artinian of order no greater than the order of the congruence module T/J, where J is an Eisenstein ideal in the local Hecke algebra T. The inequality is proven by applying the Main conjecture of Iwasawa Theory for Hecke characters and using a result of Berger [Compos Math 145(3):603–632, 2009]. This strengthens our previous result [Berger and Klosin, J Inst Math Jussieu 8(4):669–692, 2009] to include the cases of an arbitrary p-adic valuation of the L-value, in particular, cases when R is not a discrete valuation ring. As a consequence we show that the Eisenstein ideal is principal and that T is a complete intersection.     

Some classes of Chebyshev systems

Let L be a linear real space of solutions of a linear homogeneous ordinary differential equation with constant coefficients. This space is a Chebyshev space on any interval of length at most π/ω, where ω is maximal among the imaginary parts of the roots of characteristic polynomial. It turns out that this estimate for the length of the interval is exact in a certain sense. Bibliography: 2 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 35–38.