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In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

The aim of this paper is to characterize the solutions Φ : G → M 2(ℂ) of the following matrix functional equations Φ ( x y ) + Φ ( σ ( y ) x ) 2 = Φ ( x ) Φ ( y ) , x , y , ∈ G , {{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, and Φ ( x y ) − Φ ( σ ( y ) x ) 2 = Φ ( x ) Φ ( y ) , x , y , ∈ G , {{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰 n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn , 𝒰 n )}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

In this paper we establish a new critical point theorem for a class of perturbed differentiable functionals without satisfying the Palais–Smale condition. We prove the existence of at least one critical point to such functionals, provided that the perturbation is sufficiently small. The main abstract result of this paper is applied both to perturbed nonhomogeneous equations in Orlicz–Sobolev spaces and to nonlocal problems in fractional Sobolev spaces.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p-Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

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.

In this paper, we develop the elementary theory of inverse semigroups to the cases of type B semigroups. The main aim of this paper is to study proper type B semigroups. We introduce first the concept of a left admissible triple. After obtaining some basic properties of left admissible triple, we give the definition of a Q-semigroup and get a structure theorem of Q-semigroup. In particular, we introduce the notion of an admissible triple and give some characterization of proper type B semigroups. It is proved that an arbitrary Q-semigroup with an admissible triple is an E-unitary type B semigroup.

The presented paper is strictly connected, among others, with the paper On the sum of some alternating series, Comp. Math. Appl. (2011), written by Wituła and Słota. A problem concerning the form of determinants formulated in the cited paper is solved here. Next, the obtained result is adapted to solve some system of linear equations and the description of the sum of alternating series.

In the present paper, we introduce a certain subclass 𝒦q(λ, γ, h) of analytic functions by means of a quasi-subordination. Sharp bounds of the Fekete-Szegő functional for functions belonging to the class 𝒦q(λ, γ, h) are obtained. The results presented in the paper give improved versions for the certain subclasses involving the quasi-subordination and majorization.

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.

The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.

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.

[EN] This paper presents a methodology to quantify computationally the uncertainty in a class of differential equations often met in Mathematical Physics, namely random non-autonomous second-order linear differential equations, via adaptive generalized Polynomial Chaos (gPC) and the stochastic Galerkin projection technique. Unlike the random Frobenius method, which can only deal with particular random linear differential equations and needs the random inputs (coefficients and forcing term) to be analytic, adaptive gPC allows approximating the expectation and covariance of the solution stochastic process to general random second-order linear differential equations. The random inputs are allowed to functionally depend on random variables that may be independent or dependent, both absolutely continuous or discrete with infinitely many point masses. These hypotheses include a wide variety of particular differential equations, which might not be solvable via the random Frobenius method, in which the random input coefficients may be expressed via a Karhunen-Loeve expansion., This work has been supported by the Spanish Ministerio de Economia y Competitividad grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigacion y Desarrollo (PAID), Universitat Politecnica de Valencia. The authors are grateful for the valuable comments raised by the reviewer, which have improved the final version of the paper.

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.

In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

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

The main aim of this paper is to obtain Maia type fixed point results for Ćirić-Prešić contraction condition, following Ćirić L. B. and Prešić S. B. result proved in [Ćirić L. B.; Prešić S. B. On Prešić type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenian. (N.S.), 2007, v 76, no. 2, 143–147] and Luong N. V. and Thuan N. X. result in [Luong, N. V., Thuan, N. X., Some fixed point theorems of Prešić-Ćirić type, Acta Univ. Apulensis Math. Inform., No. 30, (2012), 237–249]. We unified these theorems with Maia’s fixed point theorem proved in [Maia, Maria Grazia. Un’osservazione sulle contrazioni metriche. (Italian) Rend. Sem. Mat. Univ. Padova 40 1968 139–143] and the obtained results are proved is the present paper. An example is also provided.

This paper is devoted to the study of a nonlinear Euler-Bernoulli Beam type equation involving both left and right Caputo fractional derivatives. Differently from the approaches of the other papers where they established the existence of solution for the linear Euler-Bernoulli Beam type equation numerically, we use the lower and upper solutions method with some new results on the monotonicity of the right Caputo derivative. Furthermore, we give the explicit expression of the upper and lower solutions. A numerical example is given to illustrate the obtained results.

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}} .

In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schrödinger problem: ε 2 ⁢ s ⁢ ( - Δ ) s ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u ) in ⁢ ℝ N , \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=f(u)\quad\text{in }\mathbb{R}^{N}, where N > 2 ⁢ s {N>2s} and the nonlinearity f has critical growth. By using the variational approach, we construct a localized bound-state solution concentrating around an isolated component of the positive minimum point of V as ε → 0 {\varepsilon\rightarrow 0} . Our result improves the study made in [X. He and W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 91], in the sense that, in the present paper, the Ambrosetti–Rabinowitz condition and the monotonicity condition on f ⁢ ( t ) / t {f(t)/t} are not required.

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}} .

In this paper, we continue our study of the zero-divisor graphs of lower dismantlable lattices that was started in [PATIL, A.—WAPHARE, B. N.—JOSHI, V.—POURALI, H. Y.: Zero-divisor graphs of lower dismantlable lattices I, Math. Slovaca 67 (2017), 285–296]. The present paper mainly deals with an Isomorphism Problem for the zero-divisor graphs of lattices. In fact, we prove that the zero-divisor graphs of lower dismantlable lattices with the greatest element 1 as join-reducible are isomorphic if and only if the lattices are isomorphic.

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

Let 𝓜(Ω, μ) denote the algebra of all scalar-valued measurable functions on a measure space (Ω, μ). Let B ⊂ 𝓜(Ω, μ) be a set of finitely supported measurable functions such that the essential range of each f ∈ B is a subset of {0,1}. The main result of this paper shows that for any p ∈ (0, ∞), B has strict p-negative type when viewed as a metric subspace of L p (Ω, μ) if and only if B is an affinely independent subset of 𝓜(Ω, μ) (when 𝓜(Ω, μ) is considered as a real vector space). It follows that every two-valued (Schauder) basis of L p (Ω, μ) has strict p-negative type. For instance, for each p ∈ (0, ∞), the system of Walsh functions in L p [0,1] is seen to have strict p-negative type. The techniques developed in this paper also provide a systematic way to construct, for any p ∈ (2, ∞), subsets of L p (Ω, μ) that have p-negative type but not q-negative type for any q > p. Such sets preclude the existence of certain types of isometry into L p -spaces.

The aim of the paper is to present the binomial transformation formulae of Fibonacci numbers scaled by complex multipliers. Many of these new and nontrivial relations follow from the fundamental properties of the so-called delta-Fibonacci numbers defined by Wituła and Słota. The paper contains some original relations connecting the values of delta-Fibonacci numbers with the respective values of Chebyshev polynomials of the first and second kind.

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.

In this paper, we establish some Fejér-type inequalities for convex functions. They complement the results from the previous recent paper [Dragomir, S. S.—Milošević, D. S.——Sándor, J.: On some refinements of Hadamard’s inequalities and applications, Univ. Belgrad. Publ. Elek. Fak. Sci. Math. 4 (1993), 3–10].

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.

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.

In this paper, we first establish a singular ( 0 < β < n ${(0 ) Trudinger–Moser inequality on any bounded domain in ℝ n ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.1). Next, we prove the critical singular ( 0 < β < n ${(0 ) Trudinger–Moser inequality on any unbounded domain in ℝ n ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.2). Then, we set up a subcritical singular ( 0 < β < n ${(0 ) Trudinger–Moser inequality on any unbounded domain in ℝ n ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.3). Finally, we establish the subcritical nonsingular ( β = 0 ${(\beta=0}$ ) Trudinger–Moser inequality on any unbounded domain in ℝ n ${\mathbb{R}^{n}}$ with Lorentz–Sobolev norms (Theorem 1.5). The constants in all these inequalities are sharp. In [9], for the proof of Theorem 1.2 in the nonsingular case β = 0 ${\beta=0}$ , the following inequality was used (see [17]): u ∗ ⁢ ( r ) - u ∗ ⁢ ( r 0 ) ≤ 1 n ⁢ w n 1 / n ⁢ ∫ r r 0 U ⁢ ( s ) ⁢ s 1 / n ⁢ d ⁢ s s , $u^{\ast}(r)-u^{\ast}(r_{0})\leq\frac{1}{nw_{n}^{{1/n}}}\int_{r}^{r_{0}}U(s)s^{% {1/n}}\frac{ds}{s},$ where U ⁢ ( x ) ${U(x)}$ is the radial function built from | ∇ ⁡ u | ${|\nabla u|}$ on the level set of u, i.e., ∫ | u | > t | ∇ u | d x = ∫ 0 | { | u | > t } | U ( s ) d s . $\int_{|u|>t}\lvert\nabla u|\,dx=\int_{0}^{|\{|u|>t\}|}U(s)\,ds.$ The construction of such U uses the deep Fleming–Rishel co-area formula and the isoperimetric inequality and is highly nontrivial. Moreover, this argument will not work in the singular case 0 < β < n ${0 . One of the main novelties of this paper is that we can avoid the use of this deep construction of such a radial function U (see remarks at the end of the introduction). Moreover, our approach adapts the symmetrization-free argument developed in [19, 21, 23], where we derive the global inequalities on unbounded domains from the local inequalities on bounded domains using the level sets of the functions under consideration.

In this paper, a new relation for the set of interval-valued fuzzy relations is introduced. This relation is an interval order for the family of intervals and for the family of interval-valued fuzzy relations in a given set, it has the reflexivity property. Consequences of considering such a relation are studied in the context of operations on interval-valued fuzzy relations. A new transitivity property, namely possible T-transitivity is studied (pos-T-transitivity for short). This transitivity property is connected with the new relation proposed in this paper. Preservation of this type of transitivity by some operations is also discussed.

The aim of this paper is to introduce some new classes of functions termed as somewhat (μ, λ)-continuous functions, somewhat (μ, λ)-open functions and hardly (μ, λ)-open functions. Some properties of these newly defined functions are discussed in this paper. Relationships among these functions are also being studied.

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.

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.

This paper studies the existence of nodal solutions for a class of boundary value problems that arise in population dynamics models. These problems are degenerate in the sense that the nonlinear term vanishes in a subinterval of the underlying region. In contrast to the non-degenerate case, in these degenerate situations the structure of the set of nodal solutions might consist of several components, making the analysis of this problem more complicated. This paper provides an initial step towards the solution of this general problem.

In this paper we complete our work started in [31], where the present paper was announced; in order to set a unified theory of the irrigation problem. The main result of the paper is the equivalence of the various formulations introduced so far as well as a new one introduced here. To this aim we introduce several geometric and analytical concepts which are essential for reaching our final goal even if they may deserve an intrinsic interest in themselves.

It is well known that the linear stability of Lagrangian elliptic equilateral triangle homographic solutions in the classical planar three-body problem depends on the mass parameter β = 27(m1m2 + m2m3 + m3m1)/(m1 + m2 + m3)2 ∈ [0, 9] and the eccentricity ℯ ∈ [0, 1). We are not aware of any existing analytical method which relates the linear stability of these solutions to these two parameters directly in the full rectangle [0, 9] × [0, 1), besides perturbation methods for ℯ > 0 small enough, blow-up techniques for e sufficiently close to 1, and numerical studies. In the recent paper [5], X. Hu, S. Sun and the author introduced a new rigorous analytical method to study the linear stability of these solutions in terms of the two parameters in the full (β, ℯ) range [0, 9] × [0, 1) via the ω-index theory of symplectic paths for ω belonging to the unit circle of the complex plane, and the theory of linear operators. This is an expository article on [5]. In Section 1 of this paper we give a brief review on the history of this stability problem, and describe recent results obtained in [5]. Then in Section 2, after reviewing the ω-index theory in [15], the ω-index decreasing property of the elliptic Lagragian solutions in β for fixed ℯ ∈ [0, 1) proved in [5] is introduced, then the existence of three curves located from left to right in the parameter rectangle [0, 9] × [0, 1) is explained. Among these three curves, two are precisely the −1 degeneracy curves and the third one is the left boundary curve of the hyperbolic subregion in [0, 9] × [0, 1). Then in Section 3, it is explained why the linear stability pattern of such an elliptic Lagrangian solution qβ,ℯ changes if and only if the parameter (β, ℯ) passes through one of these three curves. Some open problems on related topics are suggested at the end of this paper.

The purpose of the paper is to study the uniqueness of meromorphic functions sharing a nonzero polynomial. The results of the paper improve and generalize the recent results due to X. B. Zhang and J. F. Xu [19]. We also solve an open problem as posed in the last section of [19].

In this paper, we give some sufficient conditions for f : X → H to be a diffeomorphism, where X is a Banach space and H is a Hilbert space. The proof of the result is based on the mountain pass theorem. Using this result, in the final part of the paper, we prove an existence theorem for some class of integro-differential equations.

This paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y.Y. Li, O. Rey in [1], and O. Rey in [16] for Euclidean domains. While the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

We consider the problem of making statistical inference about the mean of a normal distribution based on a random sample of quantized (digitized) observations. This problem arises, for example, in a measurement process with errors drawn from a normal distribution and with a measurement device or process with a known resolution, such as the resolution of an analog-to-digital converter or another digital instrument. In this paper we investigate the effect of quantization on subsequent statistical inference about the true mean. If the standard deviation of the measurement error is large with respect to the resolution of the indicating measurement device, the effect of quantization (digitization) diminishes and standard statistical inference is still valid. Hence, in this paper we consider situations where the standard deviation of the measurement error is relatively small. By Monte Carlo simulations we compare small sample properties of the interval estimators of the mean based on standard approach (i.e. by ignoring the fact that the measurements have been quantized) with some recently suggested methods, including the interval estimators based on maximum likelihood approach and the fiducial approach. The paper extends the original study by Hannig et al. (2007).

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)

In this paper, we study the following Schrödinger–Poisson equations: { - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u + K ⁢ ( x ) ⁢ ϕ ⁢ u = | u | p - 2 ⁢ u , x ∈ ℝ 3 , - ε 2 ⁢ Δ ⁢ ϕ = K ⁢ ( x ) ⁢ u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi u% =\lvert u\rvert^{p-2}u,&\hskip 10.0ptx&\displaystyle\in\mathbb{R}^{3},\\ &\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2},&\hskip 10.0ptx&% \displaystyle\in\mathbb{R}^{3},\end{aligned}\right. where p ∈ ( 4 , 6 ) {p\in(4,6)} , ε > 0 {\varepsilon>0} is a parameter, and V and K are nonnegative potential functions which satisfy the critical frequency conditions in the sense that inf ℝ 3 ⁡ V = inf ℝ 3 ⁡ K = 0 {\inf_{\mathbb{R}^{3}}V=\inf_{\mathbb{R}^{3}}K=0} . By using a penalization method, we show the existence of multi-bump solutions for the above problem, with several local maximum points whose corresponding values are of different scales with respect to ε → 0 {\varepsilon\rightarrow 0} . Moreover, under suitable local assumptions on V and K, we prove the uniqueness of multi-bump solutions concentrating around zero points of V and K via the local Pohozaev identity.

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.

In this paper, we study the nonlinear Klein–Gordon systems arising from relativistic physics and quantum field theories { u t ⁢ t - u x ⁢ x + b ⁢ u + ε ⁢ v + f ⁢ ( t , x , u ) = 0 , v t ⁢ t - v x ⁢ x + b ⁢ v + ε ⁢ u + g ⁢ ( t , x , v ) = 0 , \left\{\begin{aligned} \displaystyle{}u_{tt}-u_{xx}+bu+\varepsilon v+f(t,x,u)&\displaystyle=0,\\ \displaystyle v_{tt}-v_{xx}+bv+\varepsilon u+g(t,x,v)&\displaystyle=0,\end{aligned}\right. where u , v u,v satisfy the Dirichlet boundary conditions on spatial interval [ 0 , π ] [0,\pi] , b > 0 b>0 and f , g f,g are 2 ⁢ π 2\pi -periodic in 𝑡. We are concerned with the existence, regularity and asymptotic behavior of time-periodic solutions to the linearly coupled problem as 𝜀 goes to 0. Firstly, under some superlinear growth and monotonicity assumptions on 𝑓 and 𝑔, we obtain the solutions ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) with time period 2 ⁢ π 2\pi for the problem as the linear coupling constant 𝜀 is sufficiently small, by constructing critical points of an indefinite functional via variational methods. Secondly, we give a precise characterization for the asymptotic behavior of these solutions, and show that, as ε → 0 \varepsilon\to 0 , ( u ε , v ε ) (u_{\varepsilon},v_{\varepsilon}) converge to the solutions of the wave equations without the coupling terms. Finally, by careful analysis which is quite different from the elliptic regularity theory, we obtain some interesting results concerning the higher regularity of the periodic 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

In this paper, we consider the following Schrödinger–Poisson system with perturbation: { - Δ ⁢ u + u + λ ⁢ ϕ ⁢ ( x ) ⁢ u = | u | p - 2 ⁢ u + g ⁢ ( x ) , x ∈ ℝ 3 , - Δ ⁢ ϕ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} \displaystyle-\Delta u+u+\lambda\phi(x)u&\displaystyle=% |u|^{p-2}u+g(x),&&\displaystyle x\in\mathbb{R}^{3},\\ \displaystyle-\Delta\phi&\displaystyle=u^{2},&&\displaystyle x\in\mathbb{R}^{3% },\end{aligned}\right. where λ > 0 {\lambda>0} , p ∈ ( 3 , 6 ) {p\in(3,6)} and the radial general perturbation term g ⁢ ( x ) ∈ L p p - 1 ⁢ ( ℝ 3 ) {g(x)\in L^{\frac{p}{p-1}}(\mathbb{R}^{3})} . By establishing a new abstract perturbation theorem based on the Bolle’s method, we prove the existence of infinitely many radial solutions of the above system. Moreover, we give the asymptotic behaviors of these solutions as λ → 0 {\lambda\to 0} . Our results partially solve the open problem addressed in [Y. Jiang, Z. Wang and H.-S. Zhou, Multiple solutions for a nonhomogeneous Schrödinger–Maxwell system in ℝ 3 \mathbb{R}^{3} , Nonlinear Anal. 83 2013, 50–57] on the existence of infinitely many solutions of the Schrödinger–Poisson system for p ∈ ( 2 , 4 ] {p\in(2,4]} and a general perturbation term g.