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Abstract This paper is concerned with blow-up solutions to the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type { u t = ∇ ⋅ (∇ u m − u q − 1 ∇ v) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R N (N ≥ 3), m ≥ 1 , q ≥ 2. As the basis on this study, it was recently shown that there exist radial initial data such that the corresponding solutions blow up in the case q > m + 2 N ([5]). In the parabolic–elliptic case Sugiyama [27] established behavior of blow-up solutions; however, behavior in the parabolic–parabolic case has not been studied. The purpose of this paper is to give many finite-time blow-up solutions and behavior of blow-up solutions in a neighborhood of blow-up time in the parabolic–parabolic case. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the regularity of the weak solutions to a quasilinear parabolic systems which is a generalization of p -Laplacian of the type u t i − (A α i (∇ u)) x α = f i (x , t , u , ∇ u) , i = 1 , … , N where the main part satisfies some ellipticity and f i satisfies certain growth conditions. We prove boundedness of the solutions and the gradients of solutions to the systems by the means of the energy estimates and a nonlinear iteration procedure of the Moser type in this paper. [ABSTRACT FROM AUTHOR]

When studying non-symmetric nonlocal operators on R d : L f ( x ) = ∫ R d ( f ( x + z ) − f ( x ) − ∇ f ( x ) ⋅ z 1 { | z | ⩽ 1 } ) κ ( x , z ) | z | d + α d z , where 0 < α < 2 , d ⩾ 1 , and κ ( x , z ) is a function on R d × R d that is bounded between two positive constants, it is customary to assume that κ ( x , z ) is symmetric in z . In this paper, we study heat kernel of L and derive its two-sided sharp bounds without the symmetric assumption κ ( x , z ) = κ ( x , − z ) . In fact, we allow the kernel κ to be time-dependent and x → κ ( t , x , z ) to be only locally β -Hölder continuous with Hölder constant possibly growing at a polynomial rate in | z | . We also derive gradient estimate when β ∈ ( 0 ∨ ( 1 − α ) , 1 ) as well as fractional derivative estimate of order θ ∈ ( 0 , ( α + β ) ∧ 2 ) for the heat kernel. Moreover, when α ∈ ( 1 , 2 ) , drift perturbation of the time-dependent non-local operator L t with drift in Kato's class is also studied in this paper. As an application, when κ ( x , z ) = κ ( z ) does not depend on x , we show the boundedness of nonlocal Riesz's transformation: for any p > 2 d / ( d + α ) , ‖ L 1 / 2 f ‖ p ≍ ‖ Γ ( f ) 1 / 2 ‖ p , where Γ ( f ) : = 1 2 L ( f 2 ) − f L f is the carré du champ operator associated with L , and L 1 / 2 is the square root operator of L defined by using Bochner's subordination. Here ≍ means that both sides are comparable up to a constant multiple. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [ABSTRACT FROM AUTHOR]

This paper examines the Euler–Rodrigues formula in dual 3-space D 3 by analyzing its variations such as vectorial form, exponential map, point–line theory and quaternions which have some intrinsic relations. Contrary to the Euclidean case, dual rotation in dual 3-space corresponds to a screw motion in Euclidean 3-space. This paper begins by explaining dual motion in terms of the given dual axis and angle. It will then go on to express dual Euler–Rodrigues formula with algebraic methods. Furthermore, an application of dual Euler–Rodrigues formula to point–line geometry is accomplished and point–line displacement operator is obtained by dual Euler–Rodrigues formula. Finally, dual Euler–Rodrigues formula is presented with the help of dual Euler–Rodrigues parameters that is expressed as a dual quaternion. [ABSTRACT FROM AUTHOR]

In this paper, we describe three systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials, and establish some useful connections between them in terms of three basic operators that are related to them. Furthermore, we investigate boundedness properties of two other operators, both as convolution operators in the translation invariant case where we use Fourier transforms and for the weights related to the relevant orthogonal polynomials. We consider only the most important but also simplest case of L 2 -spaces. However, in subsequent papers, we intend to extend the study to L p -spaces ( 1 < p < ∞ ) . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions to a class of p ( x ) -curl systems arising in electromagnetism. The results obtained in this paper extend several contributions concerning the p -curl operator and we focus on new existence results which are due to the presence of variable exponent. To our best knowledge, our results are new even in the semilinear case. [ABSTRACT FROM AUTHOR]

In this paper we characterize Birkhoff–James orthogonality of linear operators defined on a finite dimensional real Banach space X . We also explore the left symmetry of Birkhoff–James orthogonality of linear operators defined on X . Using some of the related results proved in this paper, we finally prove that T ∈ L ( l p 2 ) ( p ≥ 2 , p ≠ ∞ ) is left symmetric with respect to Birkhoff–James orthogonality if and only if T is the zero operator. [ABSTRACT FROM AUTHOR]

In this paper we study a particular class of bilinear multipliers which are given by Littlewood-Paley decompositions. In the first part of the paper, we show that if Κ (ξ - η) is a bilinear multiplier for (p, q, r), 1≤ p, q≤ ∞ satisfying the Hölder's condition 1p+1q=1r and have support inside [0, 1), then its periodization Κ ♯ (ξ) =Φ j⊂ ZΚ (ξ - j) is also a bilinear multiplier for the same triplet (p, q, r). Further, we show that for a given triplet (p, q, r) of exponents outside the local L2-range, there exists sequence {Κ j}j⊂ Z of uniformly bounded bilinear multipliers so that the function σ (ξ) =Φ j⊂ ZΚ j (ξ) is not a bilinear multiplier for the triplet (p, q, r). In the second part, we describe several results for bilinear multipliers of the type m (ξ, η) which are similar to the first part in nature. In particular, we point out that the results described by P. Honzik (2014) [13] can be generalized to a more general setting. [ABSTRACT FROM AUTHOR]

We study the 1-d isentropic Euler equations with time-dependent damping { ∂ t ρ + ∂ x ( ρ u ) = 0 , ∂ t ( ρ u ) + ∂ x ( ρ u 2 ) + ∂ x p ( ρ ) = − μ ( 1 + t ) λ ρ u , ρ | t = 0 = 1 + ε ρ 0 ( x ) , u | t = 0 = ε u 0 ( x ) . In a previous paper [8] , we have proven that, when λ = 1 , μ > 2 , the 1-D Euler equations have global existence of small data solutions. However in this paper, we will show that, when the damping, with respect to time, decays faster or equal to 2 1 + t , the C 1 solution of the above system will blow up in finite time. More precisely, when λ = 1 , 0 ≤ μ ≤ 2 or λ > 1 , μ ≥ 0 , we will give a finite upper bound for the lifespan. Combining the results in this paper and [8] , we see that, when the damping decays with time like μ ( 1 + t ) λ , the critical exponents for λ , μ to separate the global existence and finite-time blow up of small data solutions are λ = 1 , μ = 2 . [ABSTRACT FROM AUTHOR]

In this paper we investigate a new class of bounded operators called weighted shifts on directed trees introduced recently in [11] . This class is a natural generalization of the so called weighted bilateral, unilateral and backward shift operators. In the first part of the paper we calculate the asymptotic limit and the isometric asymptote of a contractive weighted shift on a directed tree and that of the adjoint. Then we use the asymptotic behaviour and similarity properties in order to obtain cyclicity results. We also show that a weighted backward shift operator is cyclic if and only if there is at most one zero weight. [ABSTRACT FROM AUTHOR]

In this paper, the consistency of the nearest neighbor estimator of the density function based on widely orthant dependent (WOD, in short) samples is investigated. The convergence rate of strong consistency, the complete consistency, the uniformly complete consistency and uniformly strong consistency of the nearest neighbor estimator of the density function based on WOD samples are established. Our results established in the paper generalize or improve the corresponding ones for independent samples and some negatively dependent samples. [ABSTRACT FROM AUTHOR]

In this paper we employ a C-algebra approach to the study of Fredholm properties of differential and pseudodifferential operators on the half line. The algebra investigated in this paper has compact commutators, so the Gelfand theory applies to the quotient algebra, and we obtain an explicit description of the corresponding maximal ideal space and necessary and sufficient conditions for Fredholmness of operators in the algebra. [ABSTRACT FROM AUTHOR]

Let B denote the unit ball in RN with boundary S. For a non-negative C2 subharmonic function f on B and ∈S, we define the Lusin square area integral Sα(,f) by Sα(f)=[Γα(1-|x|)2-NΔf2(x)dx]12,where for α>1, Γα={x∈B:|x-|<α(1-|x|)} is the non-tangential approach region at ∈S, and Δ is the Laplacian in RN. In the paper we will prove the following: Let f be a non-negative subharmonic function such thatfpois subharmonic for somepo>0. Iffpp=sup0po, then for everyα>1,Sα(•,f)p≤Aα,pfpfor some constantAα,pindependent of f. The above result includes the known results for harmonic or holomorphic functions in the Hardy Hp spaces, as well as for a system F=(u1,...,uN) of conjugate harmonic functions for which it is known that |F|p=(Σuj2)p/2 is subharmonic for p≥(N-2)/(N-1),N≥3. We also consider analogues of the functions g and g\* of Littlewood-Paley, and introduce the function gλ\*, λ>1, defined bygλ\*(,f)=[B(1-|y|)Δf2(y)Kλ(y,)dy]12,whereKλ(y,)=(1-|y|)(λ-1)(N-1)|y-|λ(N-1). In the paper we prove that the inequality gλ\*(•,f)p≤Cpfp holds for all λ≥N/(N-1) when p≥2, and for λ>3-p whenever 1(2N-3)/(N-1). [ABSTRACT FROM AUTHOR]

The concept of a relatively weakly injective pair of operator systems is introduced and studied in this paper, motivated by relative weak injectivity in the C*-algebra category. E. Kirchberg [11] proved that the C\*-algebra C\*(F∞) of the free group F∞ on countably many generators characterises relative weak injectivity for pairs of C\*-algebras by means of the maximal tensor product. One of the main results of this paper shows that C\*(F∞) also characterises relative weak injectivity in the operator system category. A key tool is the theory of operator system tensor products [9,10]. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for null curves in E13 to be W-slant helix in terms of their curvature functions. Mainly, throughout this paper relationships between the null W-slant helices and their pseudo spherical images are obtained. Furthermore, some illustrative examples of the null W-slant helices and their pseudo spherical indicatrices in E13 are demonstrated. [ABSTRACT FROM AUTHOR]

Abstract We prove that all possible values of the deficiency indices r , s of a symmetric (formally self-adjoint) differential expression M with complex coefficients which satisfy the well known inequalities are realized. Subject to these inequalities, in 1974 Kogan and Rofe-Beketov showed that all values of r , s which differ by no more than 1 can be realized, in 1978, 1979 Gilbert showed that their difference can be arbitrarily large provided the order of M is large enough. In this paper we show that all values of r and s are realized. [ABSTRACT FROM AUTHOR]

Abstract We obtain a simple two-sided inequality for the ratio L ν (x) / L ν − 1 (x) in terms of the ratio I ν (x) / I ν − 1 (x) , where L ν (x) is the modified Struve function of the first kind and I ν (x) is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for I ν (x) / I ν − 1 (x) to immediately deduce bounds for L ν (x) / L ν − 1 (x). We note some consequences and obtain further bounds for L ν (x) / L ν − 1 (x) by adapting techniques used to bound the ratio I ν (x) / I ν − 1 (x). We apply these results to obtain new bounds for the condition numbers x L ν ′ (x) / L ν (x) , the ratio L ν (x) / L ν (y) and the modified Struve function L ν (x) itself. Amongst other results, we obtain two-sided inequalities for x L ν ′ (x) / L ν (x) and L ν (x) / L ν (y) that are given in terms of x I ν ′ (x) / I ν (x) and I ν (x) / I ν (y) , respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce BBI spaces (“big balls of itself”), which based on the notion of BPI spaces (“big pieces of itself”) used by David and Semmes to study self-similarity. We prove that the “self-similar” construction described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior. We apply this result to self-conformal sets satisfying the WSC and prove that positive Lebesgue measure implies nonempty interior for such sets. This generalizes Zerner's corresponding result for self-similar sets. [ABSTRACT FROM AUTHOR]

Abstract In this paper we introduce some local versions of Bishop–Phelps–Bollobás type property for operators. That is, the function η which appears in their definitions depends not only on a given ε > 0 , but also on either a fixed norm-one operator T or a fixed norm-one vector x. We investigate those properties and show differences between local and uniform versions. [ABSTRACT FROM AUTHOR]

For a Blaschke product B of degree d and λ on ∂ D , let ℓ λ be the set of lines joining each distinct two preimages in B − 1 ( λ ) . The envelope of the family of lines { ℓ λ } λ ∈ ∂ D is called the interior curve associated with B . In 2002, Daepp, Gorkin, and Mortini proved the interior curve associated with a Blaschke product of degree 3 forms an ellipse. While let L λ be the set of lines tangent to ∂ D at the d preimages B − 1 ( λ ) and the trace of the intersection points of each two elements in L λ as λ ranges over the unit circle is called the exterior curve associated with B . In 2017, the author proved the exterior curve associated with a Blaschke product of degree 3 forms a non-degenerate conic. In this paper, for a Blaschke product of degree d , we give some geometrical properties that lie between the interior curve and the exterior curve. [ABSTRACT FROM AUTHOR]

Some estimates for the norm and essential norm of the difference of two composition operators between different Hardy spaces are given in this paper. [ABSTRACT FROM AUTHOR]

From the beginning of this century more than thirty papers have been published studying the limit cycles of the discontinuous piecewise linear differential systems with two pieces separated by a straight line, but it remains open the following question: what is the maximum number of limit cycles that this class of differential systems can have? Here we prove that when one of the linear differential systems has a center, real or virtual, then the discontinuous piecewise linear differential system has at most two limit cycles. [ABSTRACT FROM AUTHOR]

In this paper we apply the general results in [10] on algebraic traveling wave solutions for n -th order nonlinear evolution equations in one space dimension to prove a full classification of the algebraic traveling wave solutions of a second order generalized viscous Burgers' equation. [ABSTRACT FROM AUTHOR]

This paper deals with the two-species chemotaxis system with logistic source { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ w ) + μ 1 u ( 1 − u − a 1 v ) , x ∈ Ω , t > 0 , v t = Δ v − χ 2 ∇ ⋅ ( v ∇ w ) + μ 2 v ( 1 − a 2 u − v ) , x ∈ Ω , t > 0 , w t = Δ w − λ w + α u + β v , x ∈ Ω , t > 0 under homogeneous Neumann boundary condition in a smooth bounded domain Ω ⊂ R n ( n ≥ 1 ) . It is proved that in convex domains the problem possesses a unique global bounded solution if μ 1 and μ 2 are large enough. Moreover, we establish the existence of global weak solution for any μ 1 > 0 and μ 2 > 0 . [ABSTRACT FROM AUTHOR]

In this paper we study the geometric location of periodic points of power series defined over fields of positive characteristic p . We find a lower bound for the norm of all nonzero periodic points in the open unit disk of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of p -power iterates of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series. [ABSTRACT FROM AUTHOR]

In this paper, we prove a Schwarz lemma at the boundary for holomorphic self-mappings f of finite dimensional irreducible bounded symmetric domains without assuming the boundary regularity of f . Our result generalizes the previous results obtained for holomorphic self-mappings f of the Euclidean unit ball, or of the classical Cartan domains of type I and of type II which are smooth up to the boundary. [ABSTRACT FROM AUTHOR]

In this paper we find sufficient and necessary conditions under which vector lattice C ( X ) and its sublattices C b ( X ) , C 0 ( X ) and C c ( X ) have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space X . We also consider the countable sup property of C ( X × Y ) . Even when both C ( X ) and C ( Y ) have the countable sup property it is possible that C ( X × Y ) fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when C ( ∏ λ ∈ Λ X λ ) has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we study existence, uniqueness and asymptotic behavior of the Laplace equation with dynamical boundary conditions on regular non-cylindrical domains. We write the problem as a non-autonomous Dirichlet-to-Neumann operator and use form methods in a more general framework to accomplish our goal. A class of non-autonomous elliptic problems with dynamical boundary conditions on Lipschitz domains is also considered in this same context. [ABSTRACT FROM AUTHOR]

Our main target in this paper is to present new sharp bounds for inequalities that result when weighted operator means are filtered through positive linear maps and operator monotone functions. As an application, we prove a refined reverse of the celebrated Golden–Thompson inequality. Furthermore, we show how these inequalities can be squared. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of completely linear degeneracy for quasilinear hyperbolic systems in several space variables, and then get an interesting property for multidimensional hyperbolic conservation laws satisfying our new definition. For applications, we give some examples arising from mathematics and physics at last. [ABSTRACT FROM AUTHOR]

Let Q 1 , . . . , Q q be q slowly moving hypersurfaces in P n ( C ) of degree d i which are located in N -subgeneral position. Let f be a meromorphic mapping from C m into P n ( C ) which is algebraically nondegenerate over the field generated by Q i 's. In this paper, we will prove that, for every ϵ > 0 , there exists a positive integer M such that | | ( q − ( N − n + 1 ) ( n + 1 ) − ϵ ) T f ( r ) ≤ ∑ i = 1 q 1 d i N [ M ] ( r , f ⁎ Q i ) + o ( T f ( r ) ) . Moreover, an explicit estimate for M is given. Our result is an extension of the previous second main theorems for meromorphic mappings and moving hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Choquard equation in R 3 that − ϵ 2 Δ u + V ( x ) u = ϵ μ − 3 [ ( ∫ R 3 P ( y ) | u ( y ) | p | x − y | μ ) P ( x ) | u | p − 2 u + ( ∫ R 3 Q ( y ) | u ( y ) | q | x − y | μ ) Q ( x ) | u | q − 2 u ] , where ϵ > 0 is a parameter, 0 < μ < 3 , 6 − μ 3 < q < p < 6 − μ , the functions V and P are positive and Q may be sign-changing. Via variational methods, we establish the existence of ground states for small ϵ , and investigate the concentration behavior of ground states and show that they concentrate at a global minimum point of the least energy function as ϵ → 0 . [ABSTRACT FROM AUTHOR]

A closed subspace M of the Hardy space H 2 ( D 2 ) over the bidisk is called a submodule if it is invariant under multiplication by coordinate functions z 1 and z 2 . Whether every finitely generated submodule is Hilbert–Schmidt is an unsolved problem. This paper proves that every finitely generated submodule M containing z 1 − φ ( z 2 ) is Hilbert–Schmidt, where φ is any finite Blaschke product. Some other related topics such as fringe operator and Fredholm index are also discussed. [ABSTRACT FROM AUTHOR]

In this paper we investigate a relationship between fully k -rotundity properties, uniform K -monotonicity properties, reflexivity and K -order continuity in symmetric spaces E . We also answer a crucial question whether fully k -rotundity properties might be restricted in definition to E d the positive cone of all nonnegative and decreasing elements of E . We present a complete characterization of decreasing uniform K -monotonicity and K -order continuity in E . It is worth mentioning that we also establish several auxiliary results describing reflexivity in Lorentz spaces Γ p , w and K -order continuity in Orlicz spaces L ψ . Finally, we show an application of discussed geometric properties to the approximation theory. [ABSTRACT FROM AUTHOR]

The Schrödinger–Poisson–Newton equations for crystals with a cubic lattice and one ion per cell are considered. The ion charge density is assumed i) to satisfy the Wiener and Jellium conditions introduced in our previous paper [25] , and ii) to be exponentially decaying at infinity. The corresponding examples are given. We study the linearized dynamics at the ground state. The dispersion relations are introduced via spectral resolution for the non-selfadjoint Hamilton generator using the positivity of the energy established in [25] . Our main result is the dispersion decay in the weighted Sobolev norms for solutions with initial states from the space of continuous spectrum of the Hamilton generator. We also prove the absence of singular spectrum and limiting absorption principle. The multiplicity of every eigenvalue is shown to be infinite. The proofs rely on novel exact bounds and compactness for the inversion of the Bloch generators and on uniform asymptotics for the dispersion relations. We derive the bounds by the energy positivity from [25] . We also use the theory of analytic sets. [ABSTRACT FROM AUTHOR]

This paper concerns three kinds of seemingly paradoxical real valued functions of one variable. The first two, defined on R , are the celebrated continuous nowhere differentiable functions, known as Weierstrass's monsters, and everywhere differentiable nowhere monotone functions—simultaneously smooth and very rugged—to which we will refer as differentiable monsters. The third kind was discovered only recently and consists of differentiable functions f defined on a compact perfect subset X of R which has derivative equal zero on its entire domain, making it everywhere pointwise contractive, while, counterintuitively, f maps X onto itself. The goal of this note is to show that this pointwise shrinking globally stable map f can be extended to functions f , g : R → R which are differentiable and Weierstrass's monsters, respectively. Thus, we pack three paradoxical examples into two functions. The construction of f is based on the following variant of Jarník's Extension Theorem: For every differentiable function f from a closed P ⊆ R into R there exists its differentiable extension f ˆ : R → R such that f ˆ is nowhere monotone on R ∖ P . [ABSTRACT FROM AUTHOR]

In this paper, we present a general comparison theorem for two divided differences of a three times differentiable function. This gives a unified treatment for (logarithmically) complete monotonicity, monotonicity and inequalities involving some special functions including gamma, psi and polygamma functions. As their consequences, we not only refine and generalize some important results, but also present simple and interesting alternative proofs of certain earlier results. [ABSTRACT FROM AUTHOR]

This paper concerns the global regularity to the Cauchy problem of the 3D heat conducting incompressible Navier–Stokes equations with density-temperature-dependent viscosity and vacuum. Through the t -weighted a priori estimates, we establish the global existence and decay of strong solutions provided the initial energy is suitably small. It should be noted that the absolute temperature can be large initially. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following nonlinear coupled elliptic system ( A ε ) { − ε 2 Δ u + P ( x ) u = μ 1 u 3 + β u v 2 in R N , − ε 2 Δ v + Q ( x ) v = μ 2 v 3 + β u 2 v in R N , u > 0 , v > 0 in R N , u → 0 , v → 0 as | x | → + ∞ , where ε > 0 is a parameter, μ 1 , μ 2 > 0 and β > 0 are constants, and P ( x ) and Q ( x ) are two nonnegative, smooth functions with different nondegenerate critical points and separated zero sets. Due to the Lyapunov–Schmidt reduction method and the Maximum Principle, we show that when β is less than a small positive number, there exists an ε 0 > 0 such that for any 0 < ε < ε 0 , ( A ε ) has a segregated vector solution where ( u ε , v ε ) and u ε is trapped in a neighborhood of the nondegenerate critical points of P ( x ) as well as the zero sets of P ( x ) , and v ε is trapped in a neighborhood of the nondegenerate critical points of Q ( x ) as well as the zero sets of Q ( x ) . Moreover, the amplitudes of u ε (res v ε ) around the nondegenerate critical points and the zero sets of P ( x ) (res Q ( x ) ) are of a different order compared with ε . To the best of our knowledge, these multi-scale solutions to the system have not been obtained previously. [ABSTRACT FROM AUTHOR]

Let R be a finite von Neumann algebra with a faithful tracial state τ and let Δ denote the associated Fuglede–Kadison determinant. In this paper, we characterize all unital bijective maps ϕ on the set of invertible positive elements in R which satisfy Δ ( ϕ ( A ) + ϕ ( B ) ) = Δ ( A + B ) . We show that any such map originates from a τ -preserving Jordan ⁎-automorphism of R (either ⁎-automorphism or ⁎-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equation Δ ( A + B ) = Δ ( A ) + Δ ( B ) in the set of invertible positive operators in R . To this end, we give a new proof of the inequality Δ ( A + B ) ≥ Δ ( A ) + Δ ( B ) , using a generalized version of the Hadamard determinant inequality and conclude that equality holds for invertible B if and only if A is a nonnegative scalar multiple of B . [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem by Joseph Plemelj [16] and Ivan Privalov [17]. [ABSTRACT FROM AUTHOR]

Abstract In this paper, suppose that the initial data is sufficiently small, we study the global well-posedness and analyticity of mild solutions to the three-dimensional incompressible Hall-MHD equations. [ABSTRACT FROM AUTHOR]

Let N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) denote the number of representations of n as a x 2 + b y 2 + c z 2 + d w 2 and the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , respectively, where a , b , c , d are positive integers, n is a nonnegative integer and x , y , z , w are integers. Sun established many relations between N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) and posed 23 conjectures. Yao proved five of them by using ( p , k ) -parametrization of theta functions. In this paper, we confirm four conjectures of Sun by employing theta function identities. [ABSTRACT FROM AUTHOR]

It is known that few characterization results of the logistic distribution were available before, although it is similar in shape to the normal one whose characteristic properties have been well investigated. Fortunately, in the last decade, several authors have made great progress in this topic. Some interesting characterization results of the logistic distribution have been developed recently. In this paper, we further provide some new results by the distributional equalities in terms of order statistics of the underlying distribution and the random exponential shifts. The characterization of the closely related Pareto type II distribution is also investigated. [ABSTRACT FROM AUTHOR]

Let S = U S | S | and T = U T | T | be the polar decompositions of S and T in L ( H ) , respectively. We say that S and T have similar positive parts if | S | and | T | are similar. In this paper, we investigate properties of operators that are preserved under this similarity condition. We describe the form for the polar decomposition of an operator when the operator and its adjoint have similar positive parts. For this case, we also study the existence of invariant subspaces under this assumption. [ABSTRACT FROM AUTHOR]

Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the stability with respect to the composition of a homogeneous polynomial with linear operators. In an attempt to explore the underlying nonlinearity in a more consistent way and to bring the subject closer to its roots in Complex Analysis – and, also, taking into account recent results in the field – in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials. [ABSTRACT FROM AUTHOR]

This paper is concerned with a generalized nonlinear hematopoiesis model with variable delays. By utilizing mixed monotone operator fixed point theorem in a cone, we establish some criteria to ensure that the existence of positive almost periodic solution. Two examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This paper studies the nonisotropic chaotic oscillations of the initial-boundary value problem of one-dimensional wave equation with a mixing transport term. It separately considers that the boundary condition at the right-end of the wave equation is a superlinear type and linear perturbation of such type, each causing the total energy of the underlying system to rise and fall due to the interaction with a mixing transport term. For each type of boundary condition, the occurrence of nonisotropic chaotic oscillations is rigorously proved. Numerical examples verify the effectiveness of theoretical prediction. [ABSTRACT FROM AUTHOR]

We consider positive invertible Lamperti operators T f ( x ) = h ( x ) Φ f ( x ) such that Φ has no periodic part. Let A n , T be the sequence of averages of T and M T the ergodic maximal operator. It is obvious that if M T is bounded on some L p , 1 < p < ∞ , then sup ⁡ ‖ A n , T ‖ L p ( ν ) ≤ ‖ M T ‖ L p ( ν ) < ∞ . It is known that the converse is true. In this paper we search the sharp dependence of the norm ‖ M T ‖ L p ( ν ) with respect to sup n ⁡ ‖ A n , T ‖ L p ( ν ) < ∞ . We prove that ‖ M T ‖ L p ( ν ) ≤ C ( p ) ( sup n ∈ N ⁡ ‖ A n , T ‖ L p ( d ν ) ) p ′ , where p ′ = p / ( p − 1 ) is the conjugate exponent and C ( p ) depends only on p . Furthermore, the exponent p ′ is sharp. Our results are closely related to Buckley's theorem about sharp bounds for the Hardy–Littlewood maximal function. [ABSTRACT FROM AUTHOR]