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In the current article, we extend Voiculescu's version of Berger-Shaw Theorem to properly infinite semifinite von Neumann algebras acting on separable Hilbert spaces.

The hypergeometric product formula describes the product of two hypergeometric functions with arguments xt and yt, respectively. The result is a power series in t whose coefficients are hypergeometric polynomials. Brychkov has extended the product formula such that in one argument the variable t can be squared. We obtain a further extension such that in both arguments the variable t can be raised to positive integer powers.

The semilinear space-time fractional Schrodinger equation is considered. First, we give the explicit form for the fundamental solutions by using the Fox H-functions in order to establish some L s decay estimates. After that, we give some space-time estimates for the mild solutions from which the local well-posedness is derived on some proper Banach space.

We consider an asymmetric lottery Blotto game with two agents and n items, where both agents wish to maximize their probability of winning a majority value of all n items. Duffy and Matros [2] showed that if there exists a Nash equilibrium, then the equilibrium is unique, and it is found in an explicit expression. They also provided sufficient conditions for the existence of a Nash equilibrium in the cases of n = 3 and n = 4 . In this paper, we prove that the lottery Blotto game always has a unique Nash equilibrium for any value of n.

Inverse nodal problems for Dirac operators on a finite interval [ 0 , π ] are studied. We prove that the specification of twin-dense nodal subsets on the interior interval [ ( 1 − e 0 ) π 2 , ( 1 + e 1 ) π 2 ] , 0 e 0 , e 1 ≤ 1 , uniquely determines the operator. We also discuss the case e 1 = 0 .

Let K be a compact Hausdorff space and let H be a real or complex Hilbert space with dim ( H R ) ≥ 2 . We prove that the space C ( K , H ) , of all H-valued continuous functions on K, equipped with the supremum norm, satisfies the Mazur–Ulam property, that is, if Y is any real Banach space, every surjective isometry Δ from the unit sphere of C ( K , H ) onto the unit sphere of Y admits a unique extension to a surjective real linear isometry from C ( K , H ) onto Y. Our strategy relies on the structure of C ( K ) -module of C ( K , H ) and several results in JB⁎-triple theory. For this purpose we determine the facial structure of the closed unit ball of a real JB⁎-triple and its dual space.

In this paper, we generalize the concepts of isotropic vectors and complex symmetric operators from Hilbert spaces to Banach spaces via their dual spaces. With this extension we show the existence of isotropic vectors on Banach spaces whose dimension is at least two and the relation between the simplicity of an eigenvalue and the non-existence of its isotropic eigenvectors.

The paper treats a nonlinear eigenvalue problem that describes propagation of a transverse-magnetic wave in a plane dielectric waveguide having perfectly conducted walls at both sides. The dielectric's permittivity is characterised by an arbitrary monotonically increasing self-action nonlinearity. The full set of guided modes is described by eigenvalues of the corresponding (nonlinear) Maxwell operator with appropriate boundary conditions. We give a comprehensive analysis of this problem and develop an original approach to study its solvability and properties of solutions. Several results about existence of the eigenvalues are proved, their distribution and asymptotic are found; zeros of the eigenfunctions and their location are also determined; criterion of periodicity for the eigenfunctions is found, comparison theorem is derived, etc. It is shown that unbounded nonlinearities lead to appearance of nonlinearised solutions. This results in the existence of a novel guided regime that cannot be described within the framework of perturbation of linear guided wave regimes.

We deal with the following nonlinear Schrodinger equation with magnetic field and critical growth: { ( e i ∇ − A ( x ) ) 2 u + V ( x ) u = f ( | u | 2 ) u + | u | 2 ⁎ − 2 u in R N , u ∈ H 1 ( R N , C ) , where e > 0 is a small parameter, N ≥ 3 , 2 ⁎ = 2 N N − 2 is the critical Sobolev exponent, A ∈ C 1 ( R N , R N ) is a magnetic vector potential, V : R N → R is a continuous positive potential having a local minimum and f : R → R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for e > 0 small enough.

In this paper, we analyse a Gardner equation with dual power law nonlinearity of any order which has been widely investigated due to its applications in quantum field theory, solid state, plasma and fluid physics. A group theoretic approach is used to perform a comprehensive and detailed analysis of the equation. We derive the symmetry generators of the equation in terms of its arbitrary parameters and used them to obtain symmetry reductions and exact solutions. Furthermore, the conservation laws of the equation are derived via the Noether approach after increasing the order and by the use of the multiplier method. The importance of these conservation laws in finding exact solutions is proved via double reduction theory. Most of these solutions are new and contain many known solutions as special cases. These solutions include important soliton solutions and nontrivial solutions in terms of special functions which are meromorphic in the entire complex plane. Since the Gardner equation can model a variety of wave phenomena in plasma, solid state and fluid physics, these solutions possess significant features in the non-linear mechanics aspects of the work. They can also be used as a basis for solving other related model problems and assessing numerical and approximate analytical methods for nonlinear equations describing solitons in wave mechanics.

In this work, we develop a new matrix exponential on time scales via a cylinder transformation with a component-wise, locally μ Δ -integrable square matrix subscript. Our resulting matrix function can be written in terms of the matrix exponential of a Lebesgue integral added to a logarithmic sum in terms of the gaps of a general time scale. Under strict commutativity conditions, we show our dynamic matrix exponential is equivalent to the one in the standard literature. Finally, we demonstrate that our matrix exponential satisfies a nonlinear dynamic integral equation.

This paper is concerned with input-to-state stability of SFDSs. By using stochastic analysis techniques, Razumikhin techniques and vector Lyapunov function method, vector Razumikhin-type theorem has been established on input-to-state stability for SFDSs. Novel sufficient criteria on the pth moment exponential input-to-state stability are obtained by the established vector Razumikhin-type theorem. When input is zero, an improved criterion on exponential stability is obtained. Two examples are provided to demonstrate validity of the obtained results.

We consider the Dirichlet Laplacian in a straight planar strip perturbed by a bounded periodic symmetric operator. We prove the classical Bethe-Sommerfeld conjecture for this operator, namely, that this operator has finitely many gaps in its spectrum provided a certain special function written as a series satisfies some lower bound. We show that this is indeed the case if the ratio of the period and the width of strip is less than a certain explicit number, which is approximately equal to 0.10121. We also find explicitly the point in the spectrum, above which there is no internal gaps. We then study the case of a sufficiently small period and we prove that in such case the considered operator has no internal gaps in the spectrum. The conditions ensuring the absence are written as certain explicit inequalities.

In this paper, we study the asymptotic behavior of the regular solution to a simplified Ericksen–Leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in R 2 as the Mach number tends to zero. The evolution system consists of the compressible Navier–Stokes equations coupled with the transported heat flow for the averaged molecular orientation. We suppose that the Navier–Stokes equations are characterized by a Navier's slip boundary condition, while the transported heat flow is subject to Neumann boundary condition. By deriving a differential inequality with certain decay property, the low Mach limit of the solutions is verified for all time, provided that the initial data are well-prepared.

In this paper, we study the following Schrodinger-Poisson system { − Δ u + V ( x ) u + λ ϕ u = | u | 4 u + μ f ( u ) , x ∈ R 3 , − Δ ϕ = u 2 , x ∈ R 3 , where V ( x ) is a smooth function and μ , λ > 0 . Under suitable conditions on f, by using constraint variational method and the quantitative deformation lemma, if μ is large enough, we obtain a least-energy sign-changing (or nodal) solution u λ to this problem for each λ > 0 , and its energy is strictly larger than twice that of the ground state solutions. Moreover, we study the asymptotic behavior of u λ as the parameter λ ↘ 0 .

Because of individuals' random walk, such as shopping, travel, work, etc., people have different behaviors and thus have different social contact patterns. Therefore, topology of human social contact networks is time-varying. In this paper, we investigate dynamic characteristics of an SIS network epidemic model with Markovian switching. An epidemic threshold is established for the extinction and permanence of the model, which is related to the steady-state distribution of the Markov chain. An interesting result is that when the epidemic is permanent in one network but extinct in another, under network switching mechanisms, it may be either permanent or extinct depending on the steady-state distribution of the Markov chain. This reveals the important role of the Markov chain in epidemic evolution. This work shows that the epidemic propagation in switched networks is quite different from that of static networks. In addition, based on Lyapunov function method, positive recurrence and ergodicity of stochastic spreading processes are also discussed. Finally, numerical simulations are carried out to illustrate our theoretical results.

In this note, we prove that for a 3-dimensional submanifold in an η-Einstein 5-Sasaki manifold with Einstein constant a ≥ 4 , if the Reeb vector field is tangent to the submanifold and it has parallel mean curvature form, then it is a slant submanifold. Furthermore, if a > 4 , then it is either an invariant submanifold or an anti-invariant submanifold.

Eisenstein derived addition formulas for the Weierstrass zeta function from the addition formula for the cotangent function and the fact that the Weierstrass zeta function can be represented as an infinite sum of the cotangent functions. In this paper, we apply this idea of Eisenstein's to the addition type formula for the double cotangent function, established by the author. We show that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. This formula includes the addition formula for the Weierstrass zeta function, evaluation formulas for the double Eisenstein series introduced by Tsumura and the double shuffle relations for the double Eisenstein series, proved by Gangl-Kaneko-Zagier.

In this paper, we investigate the impact of time delay in CTL immune response on a HTLV-I infection model. By defining basic reproduction number for viral infection R 0 and basic reproduction number for CTL response R C T L , we characterize the model dynamics according to whether these two threshold values are greater than one. Especially, we obtain the global dynamics if R 0 ≤ 1 or R C T L ≤ 1 R 0 , as well as infection persistent result when R 0 > 1 . However, the model dynamics become much richer when R C T L 1 . In this case, we use the time delay as a bifurcation parameter to obtain stability switch result on the positive equilibrium and global bifurcation diagrams for the model system. We also conduct higher-order normal form analysis and apply center manifold theory to classify the rich model dynamics near the double Hopf bifurcation points. Our analysis indicates that time delay in CTL immune response can induce not only Hopf bifurcation and double Hopf bifurcation, but also quasi-periodic orbits (torus) and coexistence of multiple stable periodic solutions.

Let c k l ∈ W 2 , ∞ ( R d , C ) for all k , l ∈ { 1 , … , d } . We consider the divergence form operator A = − ∑ k , l = 1 d ∂ l ( c k l ∂ k ) in L 2 ( R d ) when the coefficient matrix satisfies ( C ( x ) ξ , ξ ) ∈ Σ θ for all x ∈ R d and ξ ∈ C d , where Σ θ be the sector with vertex 0 and semi-angle θ in the complex plane. We show that for all p in a suitable interval the contraction semigroup generated by −A extends consistently to a contraction semigroup on L p ( R d ) . For those values of p we present a condition on the coefficients such that the space C c ∞ ( R d ) of test functions is a core for the generator on L p ( R d ) . We also examine the operator A separately in the more special Hilbert space L 2 ( R d ) setting and provide more sufficient conditions such that C c ∞ ( R d ) is a core.

The known family of woven collections of frames have interesting properties and applications. Yet, they cannot be easily constructed. The present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction. The frames { φ i } i ∈ I and { ψ i } i ∈ I for a Hilbert space H are called partition-woven (or briefly P-woven), if there exists a nontrivial subset σ of I such that the family { g i } i ∈ I represented as the triple ( { φ i } , { ψ i } , σ ) and called a P-weaven frame, is a frame defined by g i = φ i if i ∈ σ and g i = ψ i , otherwise. The paper also studies the class of CP-woven collections of frames which is a subset of the P-woven collections of frames and satisfy a stronger condition. The set of all woven frames is a strict subset of the set of all CP-woven frames which is further a strict subset of the set of all P-woven frames. Several examples and counterexamples are studied to show whether or not a pair of frames is P-woven or CP-woven.

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These types of operators are used to model anomalous diffusion and, for a special choice of the integral kernels, reduce to the fractional Laplace operator on a bounded domain. By means of a nonlocal vector calculus we recast the problems in a weak form, leading to corresponding nonlocal variational equality and inequality problems. We prove optimal regularity results for both problems, including a higher regularity of the solution and the Lagrange multiplier. Based on the regularity results, we analyze the convergence of finite element approximations for a linear problem and illustrate the theoretical findings by numerical results.

Let A be an anti-selfadjoint operator on a Hilbert space H with ‖ A ‖ ≤ 1 2 . We give a sufficient and necessary condition for A to be a commutator of a pair of orthogonal projections, and establish the general representation of all pairs ( P , Q ) of orthogonal projections such that A = P Q − Q P . Then we discuss the path components of the set C A = { ( P , Q ) : A = P Q − Q P } . We prove that the action of unitary group U ( { A } ′ ) is transitive in each path component of C A when A is in generic position. Moreover, we characterize the von Neumann algebra generated by all projections in C A . As an application, we obtain that the set of all commutators of pairs of orthogonal projections is connected.

Transmission boundary value problems for the time-harmonic Maxwell system, the Helmholtz operator, and the perturbed Dirac operator are formulated, using Clifford algebras, on bounded Lipschitz domains of R m with m ≥ 3 . It is shown how the Dirac problem decouples into several Maxwell and Helmholtz problems. Necessary and sufficient conditions are provided for well-posedness in each case, and for the Dirac problem to be equivalent to one or several independent Maxwell problems.

Let λ ( s ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) s , β ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) s , and η ( s ) = ∑ n = 1 ∞ ( − 1 ) n − 1 n s be the Dirichlet lambda function, its alternating form, and the Dirichlet eta function, respectively. According to a recent historical book by Varadarajan ( [25, p. 70] ), these three functions were investigated by Euler under the notations N ( s ) , L ( s ) , and M ( s ) , respectively. In this paper, we shall present some additional properties for them. That is, we obtain a number of infinite families of linear recurrence relations for λ ( s ) at positive even integer arguments λ ( 2 m ) , convolution identities for special values of λ ( s ) at even arguments and special values of β ( s ) at odd arguments, and a power series expansion for the alternating Hurwitz zeta function J ( s , a ) , which involves a known one for η ( s ) .

Sufficient conditions are given for a vector to be cyclic, or more generally, to generate a spectral subspace for a densely defined unbounded multiplication operator acting on a Hilbert space having an orthonormal basis of eigenvectors with unbounded associated set of eigenvalues. Such operators are shown to have non-spectral invariant subspaces which are closed with respect to a topology defined on the core of the operator. Conditions are also given for such operators to have non-spectral norm-closed invariant subspaces.

The Smith's approximation formula for the gamma function is given by Γ ( x + 1 2 ) = 2 π ( x e ) x ( 2 x tanh ⁡ 1 2 x ) x / 2 ( 1 + O ( 1 x 5 ) ) , as x → ∞ . In this paper, by means of a little known power series, we develop this formula to an asymptotic expansions: Γ ( x + 1 / 2 ) 2 π ( x / e ) x ∼ ( 2 x tanh ⁡ 1 2 x ) x / 2 exp ⁡ ( ∑ n = 3 ∞ [ ( 2 n ) ! − ( 2 n − 1 ) 2 2 n − 1 ] ( 1 − 2 1 − 2 n ) 2 n ( 2 n − 1 ) ( 2 n ) ! B 2 n x 2 n − 1 ) , as x → ∞ , and give an estimate of remainder in the above asymptotic series, where B 2 n is the Bernoulli number. Moreover, as relevant results, Ramanujan type asymptotic expansions for the gamma function Γ ( x + 1 / 2 ) are obtained; an asymptotic expansion for Wallis ratio and an estimate of the remainder are established.

A two-dimensional doubly-periodic, three-phase hexagonal structure is considered. The flow in the structure is generated by three sets of vortexes/sinks/sources, which are the same in each phase and are located in the centers of the hexagons. Complex analysis methods are utilized to reduce the doubly periodic R-linear conjugation problem to the simpler one, Riemann-Hilbert (RH) problem, on a three-sheeted Riemann surface. In turn, the latter problem is reduced to a RH problem involving three joined sectors on the plane, which was previously investigated in [3] . The limiting cases with one non-conducting phase and two phases of the same conductivities are investigated. All solutions derived are verified both numerically and analytically. Examples of relevant flow networks, streamlines and equipotentials, are plotted in the whole structure and separately in each phase.

In this paper we study the Strichartz estimates for the Schrodinger propagator in the context of Wiener amalgam spaces which, unlike the Lebesgue spaces, control the local regularity of a function and its decay at infinity separately. This separability makes it possible to perform a finer analysis of the local and global behavior of the propagator. Our results improve some of the classical ones in the case of large time.

In this paper we establish some results concerning existence and multiplicity of solution for a elliptic problem under integral Dirichlet boundary condition. We mainly use a lower and upper solution method and a result due to Rabinowitz [11] related to the existence of a continuum of solutions of an abstract nonlinear eigenvalue problem.

We investigate systems of the form { A t g : g ∈ G , t ∈ [ 0 , L ] } where A ∈ B ( H ) is a normal operator in a separable Hilbert space H , G ⊂ H is a countable set, and L is a positive real number. Although the main goal of this work is to study the frame properties of { A t g : g ∈ G , t ∈ [ 0 , L ] } , as intermediate steps, we explore the completeness and Bessel properties of such systems from a theoretical perspective, which are of interest by themselves. Beside the theoretical appeal of investigating such systems, their connections to dynamical and mobile sampling make them fundamental for understanding and solving several major problems in engineering and science.

We study the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. The periodicity cell is a rectangle of sides of length l and 1 / l , where l is a positive parameter, and the shape of the cross section of the cylinders is determined by the image of a fixed domain through a diffeomorphism ϕ. We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair ( l , ϕ ) , one defines the average of the longitudinal component of the flow velocity Σ [ l , ϕ ] . Here, we prove that the quantity Σ [ l , ϕ ] depends analytically on the pair ( l , ϕ ) , which we consider as a point in a suitable Banach space.

In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space R n m + n . Firstly, we classify m-dimensional complete spacelike translating solitons in R n m + n , and prove the only complete spacelike translating solitons in R n m + n are the spacelike m-planes. Secondly, we generalize the rigidity theorem of entire spacelike Lagrangian translating solitons to spacelike translating solitons with general codimensions.

In this paper we use variational methods to establish a Berestycki-Lions type result for two classes of quasilinear Schrodinger equations in R N in the zero-mass situation.

This paper investigates the effects of different control measures on the dynamics of A(H7N9) virus transmission from poultry to human. An SI-SIR model with nonlinear incidence and recovery rates is formulated to evaluate the combined impacts of government intervention strategies and available hospital resources. The global stability of the disease-free and endemic equilibria is obtained by constructing Lyapunov and Dulac functions. Furthermore, based on the threshold policy, the SI-SIR model is extended to a novel three-dimensional Filippov one to represent the control measures being triggered once the total number of infected poultry and human reaches the tolerant level I c . Model solutions are able to approach either one real equilibrium or the pseudo-equilibrium, depending on the tolerant threshold. Our results suggest that in order to diminish the outbreak of A(H7N9) virus or lead the number of infections to an expected level, it requires not only adequate hospital resources and certain government interventions, but also a good threshold policy.

In this work we give a Fredholm criteria for the operators in the C ⁎ -algebra generated by certain Toeplitz operators and Fourier multipliers acting on the Hardy space of the bidisc. With help of the obtained results we also completely characterize the essential spectra of quasi-parabolic composition operators on the Hardy spaces of the bi-disc.

A novel infection-age structured eco-epidemiological model with saturation incidence is constructed. The model is regarded as an abstract non-densely defined Cauchy problem, and the condition of the existence of the interior equilibrium is derived. By employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we obtain that the model undergoes a Hopf bifurcation around the interior equilibrium which manifests that this model has a non-trivial periodic orbit that bifurcates from the interior equilibrium when bifurcation parameter τ crosses the bifurcation critical value τ 0 . In other words, the sustained periodic oscillation phenomenon appears. To support and extend our theoretically analytic results, numerical simulations are performed.

We use reflections involving analytic Dirichlet and Neumann data on a real-analytic curve in order to find a representation of solutions to Cauchy problems for harmonic functions in the plane. We apply this representation for finding solutions to Hele-Shaw problems. We also generalize the results by deriving the corresponding formulae for the Helmholtz equation and applying them to a uniform elliptic growth.

The aim of this paper is to study the regularity of the solution of some linear parabolic-elliptic problems in which parabolicity region depends on time. More specifically, this region is the position occupied by a body undergoing a motion (a deformation smoothly evolving in time). The main tool we introduce is a suitable extension of the motion to the entire spatial domain of the PDE. This enables us to reduce the original problem to a parabolic-elliptic problem with variable coefficients and with a parabolicity region independent of time. This problem can be seen as a Lagrangian formulation of our original problem. Next, we obtain regularity results for a class of parabolic-elliptic problems with variable coefficients and fixed parabolicity region. We apply these results to the Lagrangian formulation and, finally, we obtain a regularity result for our original problem.

The purpose of this paper is to study an inhomogeneous Dirichlet problem of a superquadratic nonlinear elliptic system with VMO coefficients in Reifenberg flat domains. We derive pointwise potential estimates for weak solution to the Dirichlet problem. As a consequence, we derive global integrability estimates for solutions with respect to Lorentz and Morrey norms.

In this paper, we investigate the existence of sign-changing solution for a class nonlocal Kirchhoff-Schrodinger-type equation − ( a + b ∫ R 3 | ∇ u | 2 d x ) △ u + V ( x ) u = f ( x , u ) , x ∈ R 3 , where a and b are positive constants. With the help of the constraint variational method and via a direct approach, we show the existence of sign-changing solution and prove that the sign-changing solution has precisely two nodal domains. This work can be regarded as the complement for some results of the literature.

We study the following Cauchy problems for semi-linear structurally damped σ-evolution models: u t t + ( − Δ ) σ u + μ ( − Δ ) δ u t = f ( u , u t ) , u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) with σ ≥ 1 , μ > 0 and δ ∈ ( 0 , σ 2 ) . Here the function f ( u , u t ) stands for the power nonlinearities | u | p and | u t | p with a given number p > 1 . We are interested in investigating L 1 estimates for oscillating integrals in the presentation of the solutions to the corresponding linear models with vanishing right-hand sides by applying the theory of modified Bessel functions and Faa di Bruno's formula. By assuming additional L m regularity on the initial data, we use ( L m ∩ L q ) − L q and L q − L q estimates with q ∈ ( 1 , ∞ ) and m ∈ [ 1 , q ) , to prove the global (in time) existence of small data Sobolev solutions to the above semi-linear models from suitable function spaces basing on L q spaces.

We show that each member of a doubly infinite sequence of highly nonlinear expressions of Bernoulli polynomials, which can be seen as linear combinations of certain higher-order convolutions, is a multiple of a specific product of linear factors. The special case of Bernoulli numbers has important applications in the study of multiple Tornheim zeta functions. The proof of the main result relies on properties of Eulerian polynomials and higher-order Bernoulli polynomials., Comment: 17 pages

In this paper we solve a problem of finding a risk minimizing hedging strategy on a general Markovian market with ratings on which prices are influenced by additional factors and rating. The market is described by a system of SDEs driven by a Wiener process and a compensated Poisson random measure. Rating-sensitive claims are considered. We relate the problem of pricing and hedging the contracts described by a general cash-flow process to solving Cauchy/Dirichlet problems and subsequently to solving some linear system of equations. We illustrate our theory on two examples of different nature. The first is a general exponential Levy model with stochastic volatility, and the second is a generalization of an exponential Levy model with regime switching.

We study the linear stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere S 2 . The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

It is well-known that the Rayleigh–Taylor (RT) problem of an inhomogeneous viscoelastic fluid defined on a bounded domain is unstable, if the elasticity coefficient κ is less than some threshold κ C . In this paper, we rigorously prove the existence of a unique unstable strong solution in the sense of L 1 -norm for the RT problem in Lagrangian coordinates based on a bootstrap instability method, when κ κ C . Applying an inverse transformation of Lagrangian coordinates to the obtained unstable solution, we can further get a unique unstable solution for the RT problem of inhomogeneous viscoelastic fluids in Eulerian coordinates.

We prove two maximal regularity results in spaces of continuous and Holder continuous functions, for a linear Cauchy-Dirichlet problem with a fractional time derivative D t α . This derivative is intended in the sense of Caputo and α is taken in ( 0 , 2 ) . In case α = 1 , we obtain maximal regularity results for parabolic problems already known in mathematical literature.

We study the topological entropy h ( f ) of continuous endomorphisms f of compact-like groups. More specifically, we consider the e-spectrum E t o p ( K ) for a compact-like group K (namely, the set of all values h ( f ) , when f runs over the set E n d ( K ) of all continuous endomorphisms of K). We pay particular attention to the class E ∞ of topological groups without continuous endomorphisms of infinite entropy (i.e., ∞ ∉ E t o p ( K ) ) as well as the subclass E 0 of E ∞ consisting of those groups K with E t o p ( K ) = { 0 } . It turns out that the properties of the e-spectrum and these two classes are very closely related to the topological dimension. We show, among others, that a compact connected group K with finite-dimensional commutator subgroup belongs to E ∞ if and only if dim ⁡ K ∞ and we obtain a simple formula (involving the entropy function) for the dimension of an abelian topological group which is either locally compact or ω-bounded (in particular, compact). Examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results (e.g., compact connected semi-simple groups K with dim ⁡ K = ∞ and K ∈ E 0 , or countably compact connected abelian groups with the same property). Since the class E ∞ is not stable under taking closed subgroups or quotients, we study also the largest subclasses S ( E ∞ ) and Q ( E ∞ ) , respectively, of E ∞ , having these stability properties. We provide a complete description of these two classes in the case of compact groups, that are either abelian or connected. The counterpart for S ( E 0 ) and Q ( E 0 ) is done as well.

Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. Colombeau stochastic processes with independent values and stationary Colombeau stochastic processes are studied with special attention paid to the property of translational invariance of generalized functions. Processes with stationary increments are characterized via stationarity of their gradient. Gaussian stationary solutions are analyzed for linear stochastic partial differential equations with generalized constant coefficients in the framework of Colombeau stochastic processes.