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In recent years, the problem of estimating Hankel determinants has attracted the attention of many mathematicians. Their research have been focused mainly on deriving the bounds of H 2 , 2 {H_{2,2}} or H 3 , 1 {H_{3,1}} over different subclasses of 𝒮 {\mathcal{S}} . Only in a few papers third Hankel determinants for non-univalent functions were considered. In this paper, we consider two classes of analytic functions with real coefficients. The first one is the class 𝒯 {\mathcal{T}} of typically real functions. The second object of our interest is 𝒦 ℝ ⁢ ( i ) {\mathcal{K}_{\mathbb{R}}(i)} , the class of functions with real coefficients which are convex in the direction of the imaginary axis. In both classes, we find lower and upper bounds of the third Hankel determinant. The results are sharp.

In this paper, we develop the theory of a class of casual differential equations on a time scale. An existence theorem for casual differential equations on a time scale is given under mixed Lipschitz and compactness conditions by the fixed point theorem in Banach algebra due to Dhage. Some fundamental differential inequalities on a time scale are also presented which are utilized to investigate the existence of extremal solutions. The comparison principle on casual differential equations on a time scale is established. Our results in this paper extend and improve some well-known results.

In this paper, the global exponential stability and periodicity are investigated for delayed neural network models with continuous coefficients and piecewise constant delay of generalized type. The sufficient condition for the existence and uniqueness of periodic solutions of the model is established by applying Banach’s fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with generalized piecewise constant delay, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. This paper ends with a brief conclusion.

This paper gives an explicit formula of the asymptotic expansion of the Kobayashi–Royden metric on the punctured sphere ℂ ⁢ ℙ 1 ∖ { 0 , 1 , ∞ } {\mathbb{CP}^{1}\setminus\{0,1,\infty\}} in terms of the exponential Bell polynomials. We prove a local quantitative version of the Little Picard’s Theorem as an application of the asymptotic expansion. Furthermore, the approach in the paper leads to the interesting consequence that the coefficients in the asymptotic expansion are rational numbers. Furthermore, the explicit formula of the metric and the conclusion regarding the coefficients apply to more general cases of ℂ ⁢ ℙ 1 ∖ { a 1 , … , a n } {\mathbb{CP}^{1}\setminus\{a_{1},\ldots,a_{n}\}} , n ≥ 3 {n\geq 3} , as well, and the metric on ℂ ⁢ ℙ 1 ∖ { 0 , 1 3 , - 1 6 ± 3 6 ⁢ i } {\mathbb{CP}^{1}\setminus\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}} will be given as a concrete example of our results.

This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L ( 1 , φ ; κ ) ⁢ ( G ) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X ) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X} . An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

We define the concepts of commutator (Krylov) transitive and strongly commutator (Krylov) transitive Abelian p-groups. These two innovations are respectively non-trivial generalizations of the notions of commutator fully transitive and strongly commutator fully transitive p-groups from a paper of Chekhlov and Danchev (J. Group Theory, 2015). They are also commutator socle-regular in the sense of Danchev and Goldsmith (J. Group Theory, 2014). Various results from there and from a paper of Goldsmith and Strüngmann (Houston J. Math., 2007) are considerably extended to this new point of view. We also define and explore the concept of a commutator weakly transitive Abelian p-group, comparing its properties with those of the aforementioned two group classes. Some affirmations, sounding quite curiously, are detected in order to illustrate the pathology of the commutators in the endomorphism rings of p-primary Abelian groups.

This paper studies the bounded approximation property (BAP) in quasi-Banach spaces. In the first part of the paper, we show that the kernel of any surjective operator ℓ p → X {\ell_{p}\to X} has the BAP when X has it and 0 < p ≤ 1 {0 , which is an analogue of the corresponding result of Lusky for Banach spaces. We then obtain and study nonlocally convex versions of the Kadec–Pełczyński–Wojtaszczyk complementably universal spaces for Banach spaces with the BAP.

In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.

Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.

In the present paper, we discuss the approximation properties of Jain–Baskakov operators with parameter c. The paper deals with the modified forms of the Baskakov basis functions. Some direct results are established, which include the asymptotic formula, error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct the King modification of these operators, which preserves the test functions e 0 {e_{0}} and e 1 {e_{1}} . It is shown that these King type operators provide a better approximation order than some Baskakov–Durrmeyer operators for continuous functions defined on some closed intervals.

The aim of this paper is to discuss the existence of mild solutions for a class of semilinear stochastic partial differential equation with nonlocal initial conditions and noncompact semigroups in a real separable Hilbert space. Combined with the theory of stochastic analysis and operator semigroups, a generalized Darbo’s fixed point theorem and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear term and nonlocal function satisfy some appropriate local growth conditions and a noncompactness measure condition. In addition, the condition of uniformly continuity of the nonlinearity is not required and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted in this paper. An example to illustrate the feasibility of the main results is also given.

In this paper, the complete convergence for the weighted sums of arrays of rowwise extended negatively dependent (END, for short) random variables is established under some mild conditions. In addition, the Marcinkiewicz-Zygmund type strong law of large numbers for arrays of rowwise END random variables is also obtained. The result obtained in the paper generalizes and improves some corresponding ones for independent random variables and some dependent random variables in some extent. By using the complete convergence that we established, we further study the complete consistency for the weighted estimator in a nonparametric regression model based on END errors.

The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is very appropriate for images with flow-like structures. Two convolutions are used in the construction of diffusion tensor for such a model, see [Drblíková, O.—Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (2007), 37–60], [Weickert, J.: Coherence-enhancing diffusion filtering, Int. J. Comput. Vis. 31 (1999), 111–127]. In this paper we introduce the third supplemental convolution in order to enhance the diffusion strategy. First, we briefly present the classical coherence enhancing model and explain our modification. Then the discrete scheme is provided. The core of the paper consists in numerical experiments. Benefits of the additional convolution are discussed and illustrated in the figures.

As a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions E / F {E/F} , where F is a finite unramified field extension of ℚ 2 {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice ( L , H ) {(L,H)} , when a base field is unramified over ℚ {\mathbb{Q}} at a prime ( 2 ) {(2)} .

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) Condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank $2$ non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space $S^3\times \mathbb{R}$ which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on $S^2\times S^3$ and $S^6\times S^7$ which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on $S^3\times S^3$, shedding some light on the long standing general Hopf conjecture., 23 pages. The newest version has strengthened the main results in the paper, and provides more examples. We add a short survey on the most recent progress inspired by this paper in the introduction section

In this paper, we first establish a kind of weighted space-time L 2 {L^{2}} estimate, which belongs to Keel–Smith–Sogge-type estimates, for perturbed linear elastic wave equations. This estimate refines the corresponding one established by the second author [D. Zha, Space-time L 2 L^{2} estimates for elastic waves and applications, J. Differential Equations 263 2017, 4, 1947–1965] and is proved by combining the methods in the former paper, the first author, Wang and Yokoyama’s paper [K. Hidano, C. Wang and K. Yokoyama, On almost global existence and local well posedness for some 3-D quasi-linear wave equations, Adv. Differential Equations 17 2012, 3–4, 267–306] and some new ingredients. Then, together with some weighted Sobolev inequalities, this estimate is used to show a refined version of almost global existence of classical solutions for nonlinear elastic waves with small initial data. Compared with former almost global existence results for nonlinear elastic waves due to John [F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math. 41 1988, 5, 615–666] and Klainerman and Sideris [S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, Comm. Pure Appl. Math. 49 1996, 307–321], the main innovation of our result is that it considerably improves the amount of regularity of initial data, i.e., the Sobolev regularity of initial data is assumed to be the smallest among all the admissible Sobolev spaces of integer order in the standard local existence theory. Finally, in the radially symmetric case, we establish the almost global existence of a low regularity solution for every small initial data in H 3 × H 2 {H^{3}\times H^{2}} .

Arone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of π 0 {\pi_{0}} .

This paper is a companion paper to [G4], where sharp estimates are proven for Fourier transforms of compactly supported functions built out of two-dimensional real-analytic functions. The theorems of [G4] are stated in a rather general form. In this paper, we expand on the results of [G4] and show that there is a class of "well-behaved" functions that contains a number of relevant examples for which such estimates can be explicitly described in terms of the Newton polygon of the function. We will further see that for a subclass of these functions, one can prove noticeably more precise estimates, again in an explicitly describable way., 13 pages

In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.

Let F be a totally real number field and let π be a cuspidal automorphic representation of GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} , which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period p ⁢ ( π ) {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods p ⁢ ( π ) {p(\pi)} and all of their twists p ⁢ ( π ⊗ ξ ) {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that ( 𝔤 , K ) {(\mathfrak{g},K)} -cohomological cusp forms of GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for GSp 4 ⁢ ( 𝔸 F ) {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

We are concerned with the study of a class of non-autonomous eigenvalue problems driven by two non-homogeneous differential operators with variable ( p 1 , p 2 ) {(p_{1},p_{2})} -growth. The main result of this paper establishes the existence of a continuous spectrum consisting in an unbounded interval and the nonexistence of eigenvalues in a neighbourhood of the origin. The abstract results of this paper are described by two Rayleigh-type quotients and the proofs rely on variational arguments.

This paper ascertains the exact boundary blow-up rates of the large positive solutions of a class of cooperative logistic systems involving n species in a general domain of ℝ N {\mathbb{R}^{N}} of class 𝒞 2 + ν {\mathcal{C}^{2+\nu}} , 0 < ν < 1 {0 . The problem models a population divided in groups whose individuals compete with those of the same group, while simultaneously they cooperate with the members of the remaining groups. Our main result provides with the exact blow-up rates along the edges of Ω from the values of the blow-up rates of the underlying uncoupled system. Rather astonishingly, these blow-up rates are independent of the strength of the cooperative effects, which play a secondary role in the analysis carried out in this paper. No previous result of this nature is available in the specialized literature for more than n = 2 {n=2} species.

The main aim of this paper is to present some theorems in order to guarantee existence and uniqueness of best proximity points under rational contractivity conditions using very general test functions. To illustrate the variety of possible test functions, we include some examples of pairs of functions which are included in innovative papers published in the last years. As a consequence, we prove that our results unify and extend some recent results in this field.

This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion 𝒯 ¯ {\overline{\mathcal{T}}} of Teichmüller space. The boundary of 𝒯 ¯ {\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space 𝐇 ¯ {\overline{\bf H}} called the model space. The significance of 𝐇 ¯ {\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of 𝒯 ¯ {\overline{\mathcal{T}}} . The main result of the paper is that certain subsets of 𝐇 ¯ {\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on 𝐇 ¯ {\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.

In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists s 0 ⁢ ( d ) < d ${s_{0}(d) such that if E ⊂ 𝔽 q d ${E\subset{\mathbb{F}}_{q}^{d}}$ , d ≥ 2 ${d\geq 2}$ , with | E | ≥ C ⁢ q s 0 ${|E|\geq Cq^{s_{0}}}$ , then | T d d ⁢ ( E ) | ≥ C ′ ⁢ q ( d + 1 2 ) ${|T^{d}_{d}(E)|\geq C^{\prime}q^{d+1\choose 2}}$ , where T k d ⁢ ( E ) ${T^{d}_{k}(E)}$ denotes the set of congruence classes of k-dimensional simplices determined by k + 1 ${k+1}$ -tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for T k d ⁢ ( E ) ${T^{d}_{k}(E)}$ with 2 ≤ k ≤ d - 1 ${2\leq k\leq d-1}$ . A non-trivial result for T 2 2 ⁢ ( E ) ${T^{2}_{2}(E)}$ in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 4 3 ${\frac{4}{3}}$ , previously established in [4] for the q ≡ 3 ⁢ mod ⁢ 4 ${q\equiv 3\textup{ mod }4}$ case to the case q ≡ 1 ⁢ mod ⁢ 4 ${q\equiv 1\textup{ mod }4}$ , and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in 𝔽 q d ${{\mathbb{F}}_{q}^{d}}$ , where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.

The paper concerns interesting problems related to the field of Complex Analysis, in particular Nevanlinna theory of meromorphic functions. The author have studied certain uniqueness problem on differential polynomials of meromorphic functions sharing a value or small function. The results of this paper are generalizations of some problems studied in [BOUSSAF, K.—ESCASSUT, A.—OJEDA, J.: Complex meromorphic functions f′P′(f), g′P′(g) sharing a small function, Indag. Math. (N.S.) 24 (2013), 15–41] and in [DYAVANAL, R. S.: Uniqueness and value-sharing of differential polynomials of meromorphic functions, J. Math. Anal. Appl. 374 (2011), 335–345].

The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the ℤ \mathbb{Z} /3 ℤ \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor 𝒮 {\mathcal{S}} which is a ℤ \mathbb{Z} /4 ℤ \mathbb{Z} × \times ℤ \mathbb{Z} /2 ℤ \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of 𝒮 {\mathcal{S}} . This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers.

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

It is well known that SOR iterative methods are convergent for linear systems, whose coefficient matrices are strictly or irreducibly diagonally dominant matrices and strong H-matrices (whose comparison matrices are nonsingular M-matrices). However, the same can not be true in case of those iterative methods for linear systems with weak H-matrices (whose comparison matrices are singular M-matrices). This paper proposes some necessary and sufficient conditions such that SOR iterative methods are convergent for linear systems with weak H-matrices. Furthermore, some numerical examples are given to demonstrate the convergence results obtained in this paper.

The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem., 20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)

The de la Vallée Poussin inequality is a handy tool for the investigation of disconjugacy, and hence, for the oscillation/nonoscillation of differential equations. The results in this paper are extensions of former those of Hartman and Wintner [Quart. Appl. Math. 13 (1955), 330–332] to the impulsive differential equations. Although the inequality first appeared in such an early date for ordinary differential equations, its improved version for differential equations under impulse effect never has been occurred in the literature. In the present study, first, we state and prove a de la Vallée Poussin inequality for impulsive differential equations, then we give some corollaries on disconjugacy. We also mention some open problems and finally, present some examples that support our findings.

In this paper, we introduce the sine extended odd Fréchet-G family of distributions, obtained from two well-established families of distributions of completely different nature: the sine-G and the extended odd Fréchet-G families. A particular focus is put on a very flexible member of this family defin ed with the Nadarajah-Haghighi distribution as a baseline, called the sine extended odd Fréchet Nadarajah-Haghighi distribution. For the theoretical part, the interesting mathematical properties of the family are investigated, including asymptotes, quantile function, linear representations and moments, with application to the introduced special member. Then, the inferential aspects of the sine extended odd Fréchet Nadarajah-Haghighi model are examined. In particular, the parameters are estimated by the maximum likelihood method. Two complementary cases are distinguished: the complete data case and the right censored data case, with the development of appropriate statistical tests. A simulation study is carried out to illustrate the convergence of the obtained estimates. Applications are given for three practicaldata sets, including one having the right censored property, illustrating the applicability of the proposed model.

This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system involving fractional Laplace operators: { ( - Δ ) s ⁢ u = λ ⁢ ρ ⁢ ( x ) ⁢ | v | α - 1 ⁢ v in ⁢ Ω , ( - Δ ) t ⁢ v = μ ⁢ τ ⁢ ( x ) ⁢ | u | β - 1 ⁢ u in ⁢ Ω , u = v = 0 in ⁢ ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x% )\lvert v\rvert^{\alpha-1}v&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta)^{t}v&\displaystyle=\mu\tau(x)\lvert u\rvert^{\beta-1}u&% &\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=v=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{n}\setminus\Omega,\end{aligned}\right. where s , t ∈ ( 0 , 1 ) {s,t\in(0,1)} , α , β > 0 {\alpha,\beta>0} satisfy α ⁢ β = 1 {\alpha\beta=1} , Ω is a smooth bounded domain in ℝ n {\mathbb{R}^{n}} , n ≥ 1 {n\geq 1} , and ρ and τ are continuous functions on Ω ¯ {\overline{\Omega}} and positive in Ω. We establish some maximum principles depending on Ω. In particular, we explicitly characterize the measure of Ω for which the maximum principles corresponding to this problem hold in Ω. For this, we derived an explicit lower estimate of principal eigenvalues in terms of the measure of Ω. Aleksandrov–Bakelman–Pucci (ABP) type estimates for the above systems are also proved. We also show the existence of a viscosity solution for a nonlinear perturbation of the nonhomogeneous counterpart of the above problem with polynomial and exponential growths. As an application of the maximum principles, we measure explicitly how small | Ω | {\lvert\Omega\rvert} has to be to ensure the positivity of the obtained solutions.

Every simply connected and connected solvable Lie group $G$ admits a simply transitive action on a nilpotent Lie group $H$ via affine transformations. Although the existence is guaranteed, not much is known about which Lie groups $G$ can act simply transitive on which Lie groups $H$. So far the focus was mainly on the case where $G$ is also nilpotent, leading to a characterization depending only on the corresponding Lie algebras and related to the notion of post-Lie algebra structures. This paper studies two different aspects of this problem. First, we give a method to check whether a given action $\rho: G \to \operatorname{Aff}(H)$ is simply transitive by looking only at the induced morphism $\varphi: \mathfrak{g} \to \operatorname{aff}(\mathfrak{h})$ between the corresponding Lie algebras. Secondly, we show how to check whether a given solvable Lie group $G$ acts simply transitive on a given nilpotent Lie group $H$, again by studying properties of the corresponding Lie algebras. The main tool for both methods is the semisimple splitting of a solvable Lie algebra and its relation to the algebraic hull, which we also define on the level of Lie algebras. As an application, we give a full description of the possibilities for simply transitive actions up to dimension $4$., Comment: 22 pages, 8 tables. Comments are welcome

This paper is devoted to the existence of solutions for a class of nonlinear boundary value problems with integral boundary conditions and generalized 𝑝-Laplacian on the positive half-line. We establish sufficient conditions to guarantee the existence of solutions in a special function space by using Leray–Schauder-type arguments. Examples are also given to illustrate the main results.

In this paper, we establish the endpoint estimate ( 0 < p ≤ 1 {0 ) for a trilinear pseudo-differential operator, where the symbol involved is given by the product of two standard symbols from the bilinear Hörmander class B ⁢ S 1 , 0 0 {BS^{0}_{1,0}} . The study of this operator is motivated from the L p {L^{p}} ( 1 < p < ∞ {1 ) estimates for the trilinear Fourier multiplier operator with flag singularities considered in [C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam. 23 2007, 2, 705–742] and Hardy space estimates in [A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on L ∞ L^{\infty} and Hardy spaces, Math. Z. 282 2016, 1–2, 577–613], and the L p {L^{p}} ( 1 < p < ∞ {1 ) estimates for the trilinear pseudo-differential operator with flag symbols in [G. Lu and L. Zhang, L p L^{p} -estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J. 66 2017, 3, 877–900]. More precisely, we will show that the trilinear pseudo-differential operator with flag symbols defined in (1.3) maps from the product of local Hardy spaces to the Lebesgue space, i.e., h p 1 × h p 2 × h p 3 → L p {h^{p_{1}}\times h^{p_{2}}\times h^{p_{3}}\rightarrow L^{p}} with 1 p 1 + 1 p 2 + 1 p 3 = 1 p {\frac{1}{p_{1}}+\frac{1}{p_{2}}+\frac{1}{p_{3}}=\frac{1}{p}} with 0 < p < ∞ {0 (see Theorem 1.6 and Theorem 1.7).

This paper introduces a space of nonabelian modular symbols 𝒮 ⁢ ( V ) {{\mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ω ω + 1 {\omega^{\omega}+1} .

Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.

In this paper, we construct a class of new modules for the quantum group U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] . The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) -module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C ⁢ [ K ± 1 ] \mathbb{C}[K^{\pm 1}] -free modules with finite-dimensional simple modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) , and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over U q ⁢ ( s ⁢ l 2 ) U_{q}(\mathfrak{sl}_{2}) .

The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

The main result of this paper is to give that if b ∈ Lip ⁡ ( ℝ n ) {b\in\operatorname{Lip}(\mathbb{R}^{n})} , h j ∈ BMO ⁡ ( ℝ n ) {h_{j}\in\operatorname{BMO}(\mathbb{R}^{n})} , j = 1 , … , k {j=1,\ldots,k} , k ∈ ℤ + {k\in\mathbb{Z}^{+}} and w ∈ A p {w\in A_{p}} , 1 < p < ∞ {1 , then the multilinear Calderón commutators T Ω , b , h → {T_{\Omega,b,\vec{h}}} with variable kernels are bounded on L p ⁢ ( w ) {L^{p}(w)} . In addition, the authors extend the above result to the Morrey space.

The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire’s sense) invariant measure has, for each q > 0 {q>0} , zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system ( X , T ) {(X,T)} (where X = M ℤ {X=M^{\mathbb{Z}}} is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q > 1 {q>1} , infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s ∈ ( 0 , 1 ) {s\in(0,1)} and each q > 1 {q>1} , zero lower s-generalized and infinite upper q-generalized dimensions.

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}} , then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}} , whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3} , we prove that the solution must be one-dimensional. In dimension {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series., 21 pages

Properties of linear regression of order statistics and their functions are usually utilized for the characterization of distributions. In this paper, based on such statistics, the concept of Pearson covariance and the pseudo-covariance measure of dependence is used to characterize the exponential, Pearson and Pareto distributions.

In this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.