Wydawnictwa

Streszczenie: On regular periodic solutions to the Navier-Stokes equations. Case C

AbstractWe find a global a priori estimate for solutions to the Navier-Stokes equations with periodic boundary conditions guaranteeing inview of the Serrin type condition the existence of global regular so-lutions. We derive the following estimate‖V(t)‖H1(Ω)≤c,(1)whereVis the velocity of the fluid.The estimate (1) is proved in two steps. First we derive a globalestimate guaranteeing the existence of global regular solutions toweakly compressible Navier-Stokes equations with large second vis-cosity, density close to a constant and gradient part of velocity small.Next we show that solutions to the Navier-Stokes equations remainclose to solutions to the weakly compressible Navier-Stokes equationsif the corresponding initial data and external forces are sufficientlyclose

Streszczenie: On regular periodic solutions to the Navier-Stokes equations. Case B

We find a global a priori estimate for solutions to the Navier-Stokes equations with periodic boundary conditions guaranteeing inview of the Serrin type condition the existence of global regular so-lutions. We derive the following estimate‖V(t)‖H1(Ω)≤c,(1)whereVis the velocity of the fluid.The estimate (1) is proved in two steps. First we derive a globalestimate guaranteeing the existence of global regular solutions toweakly compressible Navier-Stokes equations with large second vis-cosity, density close to a constant and gradient part of velocity small.Next we show that solutions to the Navier-Stokes equations remainclose to solutions to the weakly compressible Navier-Stokes equationsif the corresponding initial data and external forces are sufficientlyclose.

Streszczenie: Global regular periodic solutions to equations of weakly compressible barotropic fluid motions. Case C

We consider barotropic motions described by the compressibleNavier-Stokes equations in a box with periodic boundary conditions.We are looking for density%in the form%=a+η, whereais aconstant andη|t=0is sufficiently small inH2-norm. We assume ex-istence of potentialsφandψsuch thatv=∇φ+ rotψ+ –∫vdx.Next we assume that∇φ|t=0is sufficiently small inH2-norm too.Finally, we assume that the second viscosity coefficientνis suffi-ciently large. Then we prove long time existence of solutions suchthatv∈L∞(0,T;H2(Ω))∩L2(0,T;H3(Ω)),v,t∈L∞(0,T;H1(Ω))∩L2(0,T;H2(Ω)), where the existence timeTis proportional toν.Next forTsufficiently large we obtain thatv(T) is correspondinglysmall so global existence is proved using the methods appropriate forproblems with small data.Mathematical Subject Classification (2010):35A01, 35B10,76N10Key words and phrases:compressible Navier-Stokes equations,periodic solutions, density close to a constant, gradient part of veloc-ity small, large second viscosity, existence of global regular solutions.

Streszczenie: Global regular periodic solutions to equations of weakly compressible barotropic fluid motions. Case B

We consider barotropic motions described by the compressibleNavier-Stokes equations in a box with periodic boundary conditions.We are looking for density%in the form%=a+η, whereais aconstant andη|t=0is sufficiently small inH2-norm. We assume ex-istence of potentialsφandψsuch thatv=∇φ+ rotψ+ –∫vdx.Next we assume that∇φ|t=0is sufficiently small inH2-norm too.Finally, we assume that the second viscosity coefficientνis suffi-ciently large. Then we prove long time existence of solutions suchthatv∈L∞(0,T;H2(Ω))∩L2(0,T;H3(Ω)),v,t∈L∞(0,T;H1(Ω))∩L2(0,T;H2(Ω)), where the existence timeTis proportional toν.Next forTsufficiently large we obtain thatv(T) is correspondinglysmall so global existence is proved using the methods appropriate forproblems with small data.Mathematical Subject Classification (2010):35A01, 35B10,76N10Key words and phrases:compressible Navier-Stokes equations,periodic solutions, density close to a constant, gradient part of veloc-ity small, large second viscosity, existence of global regular solutions

Streszczenie: Tridendriform Algebras and Combinatorial Hopf Algebras

Abstract 710

Abstract. We extend the denition of tridendriform bialgebra by in- troducing a weight q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural struc- ture of brace algebra, related by a distributive law. This data is called q- Gerstenhaber-Voronov algebras. We prove the equivalence between the categories of connected q-tridendriform bialgebras and of q-Gerstenhaber- Voronov algebras. The space spanned by surjective maps, as well as the space spanned by parking functions, have natural structures of q- tridendriform bialgebras, denoted ST(q) and PQSym(q), in such a way that ST(q) is a sub-tridendriform bialgebra of PQSym(q). Finally we show that the bialgebra ofM-permutations dened by T. Lam and P. Pylyavskyy may be endowed with a natural structure of q-tridendriform algebra which is a quotient of ST(q).

Streszczenie: Generic Singularities of Symplectic and Quasi-symplectic Immersions

Abstract 709

Abstract. For any k < 2n we construct a complete system of invariants in the problem of classifying singularities of immersed k- dimensional submanifolds of a symplectic 2n-manifold at a generic double point.

Streszczenie: Positivity of crredit spreads

Credit spread is defined as a difference between forward rates corresponding to bond facing the default risk and forward rates corresponding to risk-free bond. We give conditions under which credit spread is a positive process, considering joint model for rates.

Streszczenie: Exponential moments for HJM models with jumps

General HJM models driven by a Lévy process are considered. Necessary moment conditions for the discounted bond prices to be local martingales are derived. It is proved, under the moment conditions, that the discounted bond prices are local martingales if and only if a generalized HJM condition holds.

Streszczenie: Invariant measures for forward rate HJM model with Lévy noise

We give a sufficient condition for the existence of an invariant measure for Heath-Jarrow-Morton model of forward rate function driven by a Levy process. We also prove that there exists an invariant measure for the model, if noise is small enough.

Streszczenie: Invariant measures for a class of stochastic evolution equations

We give a sufficient condition for the existence of an invariant measure for a stochastic evolution equation with noise driven by a Lévy process.

Streszczenie: On Fourier transforms of distribution semigroups

Abstract 666

A one-to-one correspondence is established between Fourier transforms of ultradistribution semigroups in the sense of Beurling and some class of pseudoresolvents characterized by conditions concerning their domains of existence and growth.

Streszczenie: Data driven score tests of fit for a semiparametric homoscedastic linear regression model

Abstract 665

We propose new tests for testing the validity of a semiparametric random-design linear regression model. The construction consists of several steps. First, we follow the classical idea of overfitting and replace the basic problem by a series of auxiliary subproblems. Next, to test whether extra terms are significant we construct a counterpart of classic score statistic. In passing, a handy way of deriving the efficient score is proposed and developed. Finally, we com- bine the solution with smoothing methods providing guidelines to choose the right subproblem. This leads to data driven score tests for the initial testing problem. We show that under the null model our construction is asymptotically distribution free and illustrate this result by a small simulation study. We also compare the finite sample performace of our tests with the the recent solution introduced by Guerre and Lavergne (2005), as well as to Cram?er-von Mises type construction. The simulation experiment indicates the very good performance of the proposed tests.

Streszczenie: The complex symplectic moduli spaces of uni-model parametric plane curve singularities

Abstract 664

Classification of zero-modal singularities of parametric plane curves under diffeomorphism equivalence is extended to uni-modal singularities. Both the simple and uni-modal singularities of parametric plane curves are classified further under symplectomorphic equivalence. In particular the corresponding cyclic symplectic moduli spaces are reconstructed as a canonical ambient spaces for the diffeomorphism moduli spaces which are no longer Hausdorff spaces.

Streszczenie: Wavelet moment method for Cauchy problem fo the Helmhotz equation

Abstract 662

The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from unexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the ill-posed moment problem equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of unexact Cauchy data, allows to prove convergence and stability of the method.

Streszczenie: Gamma-minimax prediction in exponential families with quadratic variance function

Abstract 661

The problem of prediction of a random variable Y , distribution of which belongs to the one parameter exponential family with the quadratic variance function is considered. The knowledge about priors is introduced by a class Γ of all distributions, where the first two moments are within some given convex and compact set. The Γ-minimax predictor under the squared error loss is obtained.

Streszczenie: Approximate solution of a Cauchy problem for the Helmholtz equation

Abstract 660

A problem of reconstruction of the radiation field in a domain in R³ from experimental data on a part of boundary is considered. For the model problem described by a Cauchy problem for the Helmholtz equation, an approximate method based on regularization in the frequency space is analyzed. Convergence and stability are proved under a suitable choice of regularization parameter. Numerical implementation of the method is discussed.

Streszczenie: Symplectic packing of a torus

Abstract 659

We show a construction of a symplectic packing of a four-dimensional ball into a torus.

Streszczenie: Lévy modelling of defaultable bonds

Abstract 658

A market with defaultable bonds modelled by equations with L?vy noise is considered. Conditions under which the market is free of arbitrage are derived.

Streszczenie: L-statistics as nonparametric Quantile Estimators

Abstract 657

Key Words: quantiles, Harrell-Davis estimator, Kaigh-Cheng estimator; L-statistics; optimal estimation.

Streszczenie: Estimating quantiles with Linex Loss Function. Approximation to VaR Estimation

Abstract 656

Sometimes, e.g. in the context of estimating V aR (Value at Risk), underestimating a quantile is less desirable than overestimating it which suggest to measure the error of estimation by an asymmetric loss function. As a loss function when estimat- ing a parameter by an estimator T we take the well known Linex Function exp{f(αT - θ )}g (T ) 1. To estimate the quan- tile of order q 2 (0; 1) of a normal distribution N(; ), we con- struct the optimal estimator in the class of all estimators of the form ?x + k; 1 < k < 1, if is known, or of the form ?x + s, if both parameters and are unknown; here ?x and s are standard estima- tors of and , respectively. To estimate a quantile of an unknown distribution F from the family F of all continuous and strictly in- creasing distribution functions we construct the optimal estimator in the class T of all estimators which are equivariant with respect to monotone transformations of data.

Streszczenie: Relative Poincare lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety

Abstract 645

We study the interplay between the properties of the germ of a singular variety N \subset Rⁿ given in the title and the algebra of vector fields tangent to N. The Poincare lemma property means that any closed differential (p+1)-form vanishing at any point of N is a differential of a p-form which also vanishes at any point of N. In particular, we show that the classical quasi-homogeneity is not a necessary condition for the Poincare lemma property, it can be replaced by quasi-homogeneity with respect to a smooth submanifold of R^n or a chain of smooth submanifolds. We prove that N is quasi-homogeneous if and only if there exists a vector field V, V(0) = 0, which is tangent to N and has positive eigenvalues. We also generalize this theorem to quasi-homogeneity with respect to a smooth submanifold of Rⁿ.

Streszczenie: Effective WLLN, SLLN and CLN in statistical models

Abstract 638

Weak laws of large numbers (WLLN), strong laws of large numbers (SLLN), and central limit theorems (CLT) in statistical models differ from those in probability theory in that they should hold uniformly in the family of distributions specified by the model. If a limit law states that for every \eps > 0 there exists N such that for all n > N the inequalities |\xi_n| < \eps are satisfied and N = N(\eps) is explicitly given than we call the law effective. It is trivial to obtain the effective statistical version of WLLN in the Bernoulli scheme, to get SLLN takes a little while, but CLT does not hold uniformly. Other statistical schemes are also considered.

Streszczenie: On new geometric conditions for some weakly lower semicontinous functionals with application to the rank-one conjecture of Morrey (to apperar in Proc. Roy. Soc. Edingburgh)

Abstract 636

We obtain new geometric necessary conditions for a function $f$ to define a lower semicontinuous functional of the form $I_{f}(u)=\int_{\Omega }f(u)dx$, where $u$ satisfies a given conservation law: $Pu=0$, defined by a differential operator $P$ of degree one with constant coefficients. Those conditions imply the so--called $\Lambda $--convexity condition, known as the rank--one condition when we deal with a functional of the calculus of variations. In particular we derive some new geometric properties of quasiconvex functions and state some new questions related to the rank--one conjecture of Morrey. .

Streszczenie: Between the classical theorem of Young and Convergence Teorem in Set-Valued Analysis

Abstract 635

We obtain a version of Young's Theorem, where Young--like measures can control discontinuous functions. It determines the weak limit of $f(u^{\nu })$ in the space of measures where $f$ is (possibly) discontinuous scalar function, while $\{u^{\nu }\}$ is a sequence of vector--valued functions, which is weakly convergent in some $L^{p}$--space. The Representation Theorem we derive unifies the two well-known theorems in nonlinear analysis: the classical Young's theorem, where continuity of $f$ is assumed, and Convergence Theorem known in set--valued analysis, where $f$ may be discontinuous, but the sequence $\{u^{\nu }\}$ converges strongly. Such approach is dictated by the wide range of discontinuous problems arising in nonlinear PDE. The motivations and applications to various convergence problems are discussed.

Streszczenie: Regularizatin of discrete ill-posed problems

Abstract 630

The paper concerns conditioning aspects of finite dimensional problems arisen when the Tikhonov regularization is applied to discrete ill-posed problems. A relation between a regularization parameter and sensitivity of regularized solution is investigated. Moreover, it is shown that choice of regularization parameter optimal with respect to the condition number of finite dimensional operator approximating a compact operator gives regularized discrete solutions convergent to the exact generalized solution when dimension of discrete equation and data error are related in a proper way and the data error tends to 0. As an example the method of truncated singular value decomposition with regularization is considered.

Streszczenie: Frequency moments inverse problem and maximum (Shannon vs. Renyi-Tsallis) entropy

Abstract 628

An ill-posed inverse problem which lays in making a choice of probability distribution from set H of distributions is considered, in a non-standard setting where H is given by constraints -- known as frequency moment constraints -- which are non-linear (in probabilities). From perspective of their probabilistic justification(s) two methods for making the choice are studied: a) maximization of Shannon's entropy (MaxEnt), b) maximization of Renyi-Tsallis entropy (maxTent). Concerning MaxEnt, its large deviations asymptotic justification, known as conditioned weak law of large numbers, seems to be yet not developed for this case (of non-convex H and non-unique maximum). Hovewer, MaxProb justification of MaxEnt is valid also in this setting, implying that distribution chosen by maxTent is not the most probable one.

The case of Tsallis' X-frequency constraints is studied, as well. Two sub-classes of the feasible set are constructed, where it can be directly seen that maxTent choice is not only different than the most probable distribution, it is also asymptotically improbable. The findings, taken together, brings from probability viewpoint maxTent under suspect.

Streszczenie: Constructing median-unbiased estimators in one-parameter families of distributions via optimal nonparametric estimation and stochastic ordering,

Abstract 611

If \theta \in \Theta is an unknown real parameter of a distribution under consid- eration, we are interested in constructing an exactly median-unbiased estimator \hat\theta of \theta, i.e. an estimator \hat\theta such that a median Med(\hat\theta) of the estimator equals \theta, uniformly over theta \in \Theta. We shall consider the problem in the case of a fixed sample size n (non-asymptotic approach).

Streszczenie: Generalized Sturm separation theorem,

Abstract 625

A generalization of Sturm separation theorem was given in Advantages of one-dimensional kernels. (Math Nachrichten, 149(1990), 122-147) in the case when a right invertible operator under consideration had the one-dimensional dernel. The purpose of the present paper is to shoe the Sturm theorem without this restrictive assumption

Streszczenie: Locally trivial quantum Hopf fibration,

Abstract 625

The irreducible *-representations of the polynomial algebra 0(S³_pq) of quantum 3-sphere introduced by Calow and Mathes are classified. The K-groups of its universal C*-algebra are shown to coincide with their classical counterparts. The U(1)-actions on 0(S³_pq) corresponding to the classical Hopf fibration is proven to be a Galois action. The thus obrained Hopf-Galois extension is shown to be relatively projective (admitting a strong connection) for any p,q in (0,1), and non-cleft at least in the symmetric case p=q. The latter is obtained by computing an appropriate Chern-Connes pairing.

Streszczenie: Joint replenishment multiproduct inventory problem with continous production and discrete demands

Abstract 623

The basic, deterministic, multi-item inventory problem is considered. The production rates (decision variables) are represented by nonnegative functions defined on the interval $[0,T],$ $T

Streszczenie: The Wider spaces, representations of the convolution algebra L1(R+) and one parameter semigroups of operators,

Abstract - kis

A short proof is presented of the isometric isomorphosm between teh Widder space W(K⁺;E) and the space L(L¹(R⁺);E). The Hille-Yosida generation theorem from the theory of operator semigroups is reproved by an argument involving representations of the convolution algebra L¹(R⁺).

Streszczenie: On ramification values of a polynomial mapping

Abstract 621a

Let C₁,...,C_n be affine algebraic varietes of dimension n-1. There is a proper polynomial mapping F:Cⁿ -> Cⁿ, such that the set of its ramification values containes hypersurfaces C'₁,...,C'_k, which are birationally equivalent to C₁,...,C_k

Streszczenie: On generalized critical values of a polynomial mapping

Abstract 621a

Let f=(f₁,...,f_n):Cⁿ->Cⁿ be a polynomial mapping and let deg f_i

Streszczenie: On an inequality for a free boundary problem for equations of a viscous compressible heat conducting capilary fluid

Abstract 620

We derive an inequality for a local solution of a free boundary problem for a viscous compressible heat-conducting capillary fluid. This inequality is crucial in proving the global existence of solutions belonging to certain anisotropic Sobolev-Slobodetskii space and close to an equilibrium state.

Streszczenie: The most stable estimator of location under integrable contaminants

Abstract 619 if a symmetric distribution is eps-contaminated and contaminants have finite first moments, then median may cease to bi the most robust estimator of location.

Streszczenie: On Tableau consistency in weak theories

Abstract 618

We prove that the G\"odel second incompleteness theorem holds for\hfil\break Tableau consistency and weak arithmetic. Exactly, we prove that $$\D+\Omega_m\not\vdash Tabcons(\D+\Omega_m),$$ for $m\ge 1$.

Also, we show that it is consistent with $\D+\Omega_m$ that a Tableau inconsistency proof of $\D+\Omega_m$ can be found in a cut which is ``slightly" larger than the cut determined by the $m-1$th iteration of $\log$ (theorem 1.1 and 1.2).

Streszczenie: Pseudoresolvents and distributions semigroups

Abstract 617

For pre-distribution semigroups in Kunstmann's sense [K] the generators are defined and their characterization is given in terms of the corresponding pseudoresolvents.

Streszczenie: On global special solutions for Navier-Stokes equations with boundary slip conditions in a cylindrical domain. Existence

Abstract 616

Global existence and uniquness of solutions to Navier-Stokes equations in a cylinder and with boundary slip conditions is proved under the assumptions that the initial motion is sufficiently close to the axially symmetric motion. Then the solution remains close to the axially symmetric solution for all time. Moreover the solution has some properties of the axially symmetric solution. Therfore the existence is proved in the Sobolev weighed spaces where the weight is a power function of the distance to the axis of symmetry. The existence is proved in such classes that velocity belongs to H³ locally.

Streszczenie: On some presentation of graphs

Abstract 615

The paper contains a survey of some mathematical results inspired by investigations in system theory. In part, these results are of purely mathematical character (e.g. a characterization theorem for the category of graph algebras), and partly they are new presentations of known mathematical notins (e.g. the power object catagory). These new presentations give rise to some applications of these mathematical notions in system theory, which will be discussed in the second part of the paper.

Streszczenie: Holomorphic Characterizations of Certains Classes of Distributions

Abstract 613

We give two holomorphic characterizations of the space D'_L^p of distributions for 1<p<∞. Using the almost analytic continuation method we have found sufficient conditions for a function f holomorphic in C\R or even in a smaller region, a stripe along the real line, to have its boundary values in the space D'_L^p(R) for 1 < p <∞. We will also give another characterization of holomorphic functions whose boundary values are in D'_L^p for 1 < p <∞.

We are also concerned with the holomorphic representation of integrable distributions.

We present some applications of the holomorphic representation of distributions.

Streszczenie: Almost sure and moment stability of stochastic partial differential equations

Abstract 612

We study the almost sure and moment stability of a class of stochastic partial differential equations and we present an infinite-dimensional version of a theorem proved for stochastic ordinary differential equations by Arnold, Oeljeklaus and Pardoux. We also investigate how adding a term with white noise influences the stability of a deterministic system. The outcome is quite surprising. It turns out that regardless whether the deterministic system was stable or unstable, after adding a term with sufficiently large noise, it becomes pathwise exponentially stable and unstable in the p-th mean, for p > 1.

Keywords: stochastic partial differential equation, almost sure stability, moment stability, deterministic partial differential equation, stabilization by noise, destabilization by noise.

AMS 2000 Classifications: Primary: 60H15, 35K10; Secondary: 65N25.