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2. A Note on the Paper 'Fractional Order Pettis Integral Equations with Multiple Time Delay in Banach Spaces' by M. Benchohra and F.-Z. Mostefai
- Author
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Mieczysław Cichoń
- Subjects
Pettis integral ,Pure mathematics ,Weak topology ,General Mathematics ,Mathematical analysis ,Banach space ,Order (group theory) ,C0-semigroup ,Integral equation ,Strong topology (polar topology) ,Topology (chemistry) ,Mathematics - Abstract
On a recent paper Benchohra and Mostefai [2] presented some existence results for an integral equation of fractional order with multiple time delay in Banach spaces. In contrast to the classical case, when assumptions are expressed in terms of the strong topology, they considered another case, namely with the weak topology. It has some consequences for the proof. We present here some comments and corrections.
- Published
- 2015
3. A note on a paper by R. Heath-Brown: The density of rational points on curves and surfaces
- Author
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Niklas Broberg
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Family of curves ,Mathematics - Published
- 2004
4. Erratum to the paper: F. J. Gallego, B. P. Purnaprajna Projective normality and syzygies of algebraic surfaces
- Author
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Francisco Javier Gallego and Bangere P. Purnaprajna
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Algebraic surface ,Calculus ,Projective space ,Projective test ,Normality ,Mathematics ,media_common - Published
- 2000
5. REMARKS ON A PAPER BY BEG, LATIF AND MINHAS ABOUT FIXED POINTS
- Author
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Jacek Jachymski
- Subjects
Pure mathematics ,General Mathematics ,Locally convex topological vector space ,Mathematical analysis ,Fixed point ,Uniform space ,Mathematics - Published
- 1998
6. Correction to the paper: NevanlinnaCartan theory over function fields and a Diophantine equation
- Author
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Junjiro Noguchi
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Diophantine set ,Diophantine geometry ,Applied Mathematics ,General Mathematics ,Diophantine equation ,symbols ,Function (mathematics) ,Legendre's equation ,Thue equation ,Mathematics - Published
- 1998
7. ERRATUM TO THE PAPER: ON ORDERS OF SKEW ELEMENTS IN FINITE n-GROUPS
- Author
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M E Wanke-Jene and M. B. Wanke-Jakubowska
- Subjects
Pure mathematics ,General Mathematics ,Skew ,Mathematics - Published
- 1981
8. Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár
- Author
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Yongqiang Liu, Botong Wang, and Laurenţiu G. Maxim
- Subjects
Mathematics - Algebraic Geometry ,Pure mathematics ,Transformation (function) ,Applied Mathematics ,General Mathematics ,14F05, 14F35, 14F45, 32S60, 32L05, 58K15 ,Mathematics - Algebraic Topology ,Mathematics - Abstract
In their 2012 paper, Bobadilla and Koll\'ar studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer-Hopf conjecture in the complex projective setting., Comment: published/final version
- Published
- 2021
9. Sharp Hardy Identities and Inequalities on Carnot Groups
- Author
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Guozhen Lu, Nguyen Lam, and Joshua Flynn
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Statistical and Nonlinear Physics ,01 natural sciences ,03 medical and health sciences ,symbols.namesake ,0302 clinical medicine ,symbols ,030212 general & internal medicine ,0101 mathematics ,Carnot cycle ,Mathematics ,media_common - Abstract
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.
- Published
- 2021
10. On some higher order equations admitting meromorphic solutions in a given domain
- Author
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G. Barsegian and Fanning Meng
- Subjects
Work (thermodynamics) ,Pure mathematics ,Higher order equations ,Complex differential equation ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Simple (abstract algebra) ,0101 mathematics ,Value (mathematics) ,Complex plane ,Mathematics ,Meromorphic function - Abstract
This paper relates to a recent trend in complex differential equations which studies solutions in a given domain. The classical settings in complex equations were widely studied for meromorphic solutions in the complex plane. For functions in the complex plane, we have a lot of results of general nature, in particular, the classical value distributions theory describing numbers of a-points. Many of these results do not work for functions in a given domain. A recent principle of derivatives permits us to study the numbers of Ahlfors simple islands for functions in a given domain; the islands play, to some extend, a role similar to that of the numbers of simple a-points. In this paper, we consider a large class of higher order differential equations admitting meromorphic solutions in a given domain. Applying the principle of derivatives, we get the upper bounds for the numbers of Ahlfors simple islands of similar solutions.
- Published
- 2020
11. Admissible Galois Structures on the categories dual to some varieties of universal algebras
- Author
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Dali Zangurashvili
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Galois theory ,Commutative ring ,Amalgamation property ,01 natural sciences ,010101 applied mathematics ,Differential algebra ,0101 mathematics ,Variety (universal algebra) ,Abelian group ,Commutative property ,Unit (ring theory) ,Mathematics - Abstract
The subject of the paper is suggested by G. Janelidze and motivated by his earlier result giving a positive answer to the question posed by S. MacLane whether the Galois theory of homogeneous linear ordinary differential equations over a differential field (which is Kolchin–Ritt theory and an algebraic version of Picard–Vessiot theory) can be obtained as a particular case of G. Janelidze’s Galois theory in categories. One ground category in the Galois structure involved in this theory is dual to the category of commutative rings with unit, and another one is dual to the category of commutative differential rings with unit. In the present paper, we apply the general categorical construction, the particular case of which gives this Galois structure, by replacing “commutative rings with unit” by algebras from any variety V \mathscr{V} of universal algebras satisfying the amalgamation property and a certain condition (of the syntactical nature) for elements of amalgamated free products which was introduced earlier, and replacing “commutative differential rings with unit” by V \mathscr{V} -algebras equipped with additional unary operations which satisfy some special identities to construct a new Galois structure. It is proved that this Galois structure is admissible. Moreover, normal extensions with respect to it are characterized in the case where V \mathscr{V} is any of the following varieties: abelian groups, loops and quasigroups.
- Published
- 2020
12. Positive Solutions for Systems of Quasilinear Equations with Non-homogeneous Operators and Weights
- Author
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Marta García-Huidobro, Satoshi Tanaka, and Raúl Manásevich
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,Non homogeneous ,010102 general mathematics ,Statistical and Nonlinear Physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
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.
- Published
- 2020
13. The Kobayashi–Royden metric on punctured spheres
- Author
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Junqing Qian and Gunhee Cho
- Subjects
Rational number ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Exponential function ,Bell polynomials ,010101 applied mathematics ,Metric (mathematics) ,Backslash ,SPHERES ,0101 mathematics ,Asymptotic expansion ,Mathematics - Abstract
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.
- Published
- 2020
14. Bell–Sheffer exponential polynomials of the second kind
- Author
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Paolo Ricci and Pierpaolo Natalini
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Exponential polynomial ,Mathematics - Abstract
In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers and several integer sequences related to them have been studied. In the present paper, new sets of Bell–Sheffer polynomials are introduced. Connections with Bell numbers are shown.
- Published
- 2020
15. Sugihara algebras and Sugihara monoids: Multisorted dualities
- Author
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Hilary A. Priestley and Leonardo Manuel Cabrer
- Subjects
Pure mathematics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system R-mingle.This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications.
- Published
- 2020
16. A new characterization of a proper type B semigroup
- Author
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Zhi Pei, Chunhua Li, and Baogen Xu
- Subjects
type b semigroup ,Pure mathematics ,20m10 ,Mathematics::Operator Algebras ,Semigroup ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,e-unitary ,proper ,0102 computer and information sciences ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,06f05 ,010201 computation theory & mathematics ,q-semigroup ,QA1-939 ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2020
17. Determinants of two kinds of matrices whose elements involve sine functions
- Author
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Michał Różański
- Subjects
11c20 ,15a06 ,Pure mathematics ,40a05 ,lcsh:Mathematics ,General Mathematics ,fourier series ,010102 general mathematics ,determinant ,lcsh:QA1-939 ,01 natural sciences ,sine matrix ,010101 applied mathematics ,Alternating series ,alternating series ,Sine ,0101 mathematics ,42a05 ,Fourier series ,Mathematics - Abstract
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.
- Published
- 2019
18. Generalized Riesz potentials of functions in Morrey spaces L (1,ϕ;κ)(G) over non-doubling measure spaces
- Author
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Yoshihiro Sawano, Masaki Shigematsu, and Tetsu Shimomura
- Subjects
010104 statistics & probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,01 natural sciences ,Measure (mathematics) ,Mathematics - Abstract
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.
- Published
- 2019
19. On commutator Krylov transitive and commutator weakly transitive Abelian p-groups
- Author
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Andrey R. Chekhlov and Peter V. Danchev
- Subjects
010101 applied mathematics ,Pure mathematics ,Transitive relation ,law ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Commutator (electric) ,0101 mathematics ,Abelian group ,01 natural sciences ,Mathematics ,law.invention - Abstract
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.
- Published
- 2019
20. Stabilizers in EQ-algebras
- Author
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Wei Wang, Xiao Yun Cheng, Mei Wang, and Jun Tao Wang
- Subjects
0209 industrial biotechnology ,Pure mathematics ,(fuzzy) prefilter ,lcsh:Mathematics ,General Mathematics ,08a72 ,02 engineering and technology ,lcsh:QA1-939 ,eq-algebra ,020901 industrial engineering & automation ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,(fuzzy) stabilizer ,fuzzy congruence relation ,03e72 ,Mathematics - Abstract
The main goal of this paper is to introduce the notion of stabilizers in EQ-algebras and develop stabilizer theory in EQ-algebras. In the paper, we introduce (fuzzy) left and right stabilizers and investigate some related properties of them. Then, we discuss the relations among (fuzzy) stabilizers, (fuzzy) prefilters (filters) and (fuzzy) co-annihilators. Also, we obtain that the set of all prefilters in a good EQ-algebra forms a relative pseudo-complemented lattice, where Str(F, G) is the relative pseudo-complemented of F with respect to G. These results will provide a solid algebraic foundation for the consequence connectives in higher fuzzy logic.
- Published
- 2019
21. On the bounded approximation property on subspaces of ℓ p when 0 < p < 1 and related issues
- Author
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Félix Cabello Sánchez, Jesús M. F. Castillo, and Yolanda Moreno
- Subjects
Pure mathematics ,Approximation property ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Linear subspace ,Mathematics - Abstract
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.
- Published
- 2019
22. Sharp bounds of Fekete-Szegő functional for quasi-subordination class
- Author
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Shashi Kant and Prem Pratap Vyas
- Subjects
Subordination (linguistics) ,Pure mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,subordination ,30c45 ,01 natural sciences ,010101 applied mathematics ,univalent functions ,fekete-szegő coefficients ,QA1-939 ,0101 mathematics ,quasi-subordination ,Mathematics - Abstract
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.
- Published
- 2019
23. Augmented, free and tensor generalized digroups
- Author
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Raúl Velásquez, José Gregorio Rodríguez-Nieto, and Olga Salazar-Diaz
- Subjects
20n99 ,Pure mathematics ,Semidirect product ,20e06 ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,digroups ,group actions ,Group action ,free and tensor groups ,Tensor (intrinsic definition) ,QA1-939 ,20a05 ,20e34 ,semidirect product ,0101 mathematics ,20b10 ,Mathematics ,20a10 - Abstract
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.
- Published
- 2019
24. On the equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow
- Author
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Yongjia Zhang
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Ricci flow ,02 engineering and technology ,01 natural sciences ,020901 industrial engineering & automation ,Bounded function ,Mathematics::Differential Geometry ,0101 mathematics ,Entropy (arrow of time) ,Mathematics - Abstract
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.
- Published
- 2018
25. Solution of the Ulam stability problem for Euler–Lagrange k-quintic mappings
- Author
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Syed Abdul Mohiuddine, Abdullah Alotaibi, and John Michael Rassias
- Subjects
Linear map ,symbols.namesake ,Pure mathematics ,Quadratic equation ,Probability theory ,General Mathematics ,Functional equation ,Euler's formula ,symbols ,Type (model theory) ,Stability (probability) ,Mathematics ,Quintic function - Abstract
The “oldest quartic” functional equation f ( x + 2 y ) + f ( x - 2 y ) = 4 [ f ( x + y ) + f ( x - y ) ] - 6 f ( x ) + 24 f ( y ) f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]-6f(x)+24f(y) was introduced and solved by the second author of this paper (see J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glas. Mat. Ser. III 34(54) 1999, 2, 243–252). Similarly, an interesting “quintic” functional equation was introduced and investigated by I. G. Cho, D. Kang and H. Koh, Stability problems of quintic mappings in quasi-β-normed spaces, J. Inequal. Appl. 2010 2010, Article ID 368981, in the following form: 2 f ( 2 x + y ) + 2 f ( 2 x - y ) + f ( x + 2 y ) + f ( x - 2 y ) = 20 [ f ( x + y ) + f ( x - y ) ] + 90 f ( x ) . 2f(2x+y)+2f(2x-y)+f(x+2y)+f(x-2y)=20[f(x+y)+f(x-y)]+90f(x). In this paper, we generalize this “Cho–Kang–Koh equation” by introducing pertinent Euler–Lagrange k-quintic functional equations, and investigate the “Ulam stability” of these new k-quintic functional mappings.
- Published
- 2018
26. Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II
- Author
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Sungmun Cho
- Subjects
Pure mathematics ,Residue (complex analysis) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Hermitian matrix ,Mathematics - Abstract
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)} .
- Published
- 2018
27. Homogeneous Finsler spaces and the flag-wise positively curved condition
- Author
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Ming Xu and Shaoqiang Deng
- Subjects
Mathematics - Differential Geometry ,22E46, 53C30 ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Flag (linear algebra) ,010102 general mathematics ,Lie group ,Space (mathematics) ,01 natural sciences ,Differential Geometry (math.DG) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,Compact Lie algebra ,Tangent space ,Mathematics::Differential Geometry ,0101 mathematics ,Invariant (mathematics) ,010306 general physics ,Hopf conjecture ,Mathematics - Abstract
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
- Published
- 2018
28. Maia type fixed point theorems for Ćirić-Prešić operators
- Author
-
Margareta-Eliza Balazs
- Subjects
primary 54h25 ,Pure mathematics ,two metrics ,General Mathematics ,ćirić-prešić ,Fixed-point theorem ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,secondary 47h10 ,maia ,010101 applied mathematics ,fixed point ,QA1-939 ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2018
29. A general fixed point theorem for two pairs of mappings satisfying a mixed implicit relation
- Author
-
Valeriu Popa and Alina-Mihaela Patriciu
- Subjects
010101 applied mathematics ,Pure mathematics ,Relation (database) ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,0101 mathematics ,Fixed point ,01 natural sciences ,Mathematics - Abstract
In this paper a general fixed point theorem for mappings with a new type of common limit range property satisfying a mixed implicit relation is proved. In the last part of the paper, as application, some fixed point results for mappings satisfying contractive conditions of integral type and for φ-contractive mappings are obtained.
- Published
- 2018
30. On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain
- Author
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G. Barsegian and Wenjun Yuan
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Domain (mathematical analysis) ,Mathematics ,Meromorphic function - Abstract
The value distribution and, in particular, the numbers of a-points, have not been studied for meromorphic functions which are solutions of some complex differential equations in a given domain. Instead, the numbers of good a-points and Ahlfors islands, which play to a certain extend a role similar to that of the numbers of a-points, have been considered in some recent papers. In this paper, we consider meromorphic functions in a given domain, which are the solutions of some higher order equations and largely generalize the solutions of Painlevé equations 3–6. We give the upper bounds for the numbers of good a-points and Ahlfors islands of similar solutions.
- Published
- 2018
31. On the formula of Cohen–Vogt relatively pointed topological semi-simplicial sets
- Author
-
Leonard Mdzinarishvili
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In the papers [1] and [6], for an inverse sequence of pointed topological spaces and fibrations preserving the base points E = E 1 ← p 1 E 2 ← p 2 ⋯ ← p m E m + 1 , E=E_{1}\xleftarrow{p_{1}}E_{2}\xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}E_{m+1}, there exists an exact sequence * → lim ← ( 1 ) [ X , Ω E m ] → [ X , lim ← E ] → lim ← ( 1 ) [ X , E m ] → * . *\rightarrow{\varprojlim}^{(1)}[X,\Omega E_{m}]\rightarrow[X,\varprojlim E]% \rightarrow{\varprojlim}^{(1)}[X,E_{m}]\rightarrow*. In the present paper, for an inverse sequence of pointed topological semi-simplicial sets and fibrations preserving base points E ¯ = E ¯ ← p 1 1 E ¯ ← p 2 2 ⋯ ← p m E ¯ ← m + 1 ⋯ , \underline{E}=\underline{E}{}_{1}\xleftarrow{p_{1}}\underline{E}{}_{2}% \xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}\underline{E}{}_{m+1}\xleftarrow{% \hphantom{p_{1}}}\cdots, an analogous formula is proved.
- Published
- 2018
32. Rational homology and homotopy of high-dimensional string links
- Author
-
Paul Arnaud Songhafouo Tsopméné and Victor Turchin
- Subjects
Homotopy group ,Pure mathematics ,Conjecture ,Hochschild homology ,Direct sum ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Codimension ,Homology (mathematics) ,Mathematics::Algebraic Topology ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Isotopy ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
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}} .
- Published
- 2018
33. On Φ-Dedekind, Φ-Prüfer and Φ-Bezout modules
- Author
-
Shahram Motmaen and Ahmad Yousefian Darani
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we introduce some classes of R-modules that are closely related to the classes of Prüfer, Dedekind and Bezout modules. Let R be a commutative ring with identity and set ℍ = { M ∣ M is an R -module and Nil ( M ) is a divided prime submodule of M } . \mathbb{H}=\bigl{\{}M\mid M\text{ is an }R\text{-module and }\mathrm{Nil}(M)% \text{ is a divided prime submodule of }M\bigr{\}}. For an R-module M ∈ ℍ {M\in\mathbb{H}} , set T = ( R ∖ Z ( R ) ) ∩ ( R ∖ Z ( M ) ) {T=(R\setminus Z(R))\cap(R\setminus Z(M))} , 𝔗 ( M ) = T - 1 M {\mathfrak{T}(M)=T^{-1}M} and P = ( Nil ( M ) : R M ) {P=(\mathrm{Nil}(M):_{R}M)} . In this case, the mapping Φ : 𝔗 ( M ) → M P {\Phi:\mathfrak{T}(M)\to M_{P}} given by Φ ( x / s ) = x / s {\Phi(x/s)=x/s} is an R-module homomorphism. The restriction of Φ to M is also an R-module homomorphism from M into M P {M_{P}} given by Φ ( x ) = x / 1 {\Phi(x)=x/1} for every x ∈ M {x\in M} . A nonnil submodule N of M is said to be Φ-invertible if Φ ( N ) {\Phi(N)} is an invertible submodule of Φ ( M ) {\Phi(M)} . Moreover, M is called a Φ-Prüfer module if every finitely generated nonnil submodule of M is Φ-invertible. If every nonnil submodule of M is Φ-invertible, then we say that M is a Φ-Dedekind module. Furthermore, M is said to be a Φ-Bezout module if Φ ( N ) {\Phi(N)} is a principal ideal of Φ ( M ) {\Phi(M)} for every finitely generated submodule N of the R-module M. The paper is devoted to the study of the properties of Φ-Prüfer, Φ-Dedekind and Φ-Bezout R-modules.
- Published
- 2018
34. Fourier transforms of powers of well-behaved 2D real analytic functions
- Author
-
Michael Greenblatt
- Subjects
Class (set theory) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Newton polygon ,Function (mathematics) ,01 natural sciences ,Subclass ,symbols.namesake ,Fourier transform ,Mathematics - Classical Analysis and ODEs ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,010307 mathematical physics ,42B20 ,0101 mathematics ,Analytic function ,Mathematics - Abstract
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
- Published
- 2017
35. Donaldson–Thomas invariants versus intersection cohomology of quiver moduli
- Author
-
Markus Reineke and Sven Meinhardt
- Subjects
Pure mathematics ,Mathematics::Algebraic Geometry ,Conjecture ,Intersection ,Intersection homology ,Applied Mathematics ,General Mathematics ,Quiver ,Closure (topology) ,Invariant (mathematics) ,Moduli ,Moduli space ,Mathematics - Abstract
The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.
- Published
- 2017
36. Fejér-type inequalities (II)
- Author
-
Shiow-Ru Hwang, Kuei-Lin Tseng, and Sever S Dragomir
- Subjects
010101 applied mathematics ,Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Mathematics ,media_common - Abstract
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].
- Published
- 2017
37. Homoclinic solutions for second order Hamiltonian systems with general potentials
- Author
-
Ziheng Zhang, Honglian You, and Rong Yuan
- Subjects
010101 applied mathematics ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Order (group theory) ,Homoclinic orbit ,0101 mathematics ,01 natural sciences ,Mathematics ,Hamiltonian system - Abstract
In this paper we are concerned with the existence of infinitely many homoclinic solutions for the following second order non-autonomous Hamiltonian systems u ¨ t − L t u t + ∇ W t , u t = 0 $$ \ddot u\left( t \right) - L\left( t \right)u\left( t \right) + \nabla W\left( {t,u\left( t \right)} \right) = 0$$ (HS) where t ∈ ℝ, L ∈ C(ℝ, ℝ n 2 ) is a symmetric and positive definite matrix for all t ∈ ℝ, W ∈ C 1(ℝ × ℝ n , ℝ) and ∇W(t,u) is the gradient of W at u. The novelty of this paper is that, assuming that L meets some coercive condition and the potential W is of the form W(t, u) = W 1(t, u) + W2(t, u), for the first time we show that (HS) possesses two different sequences of infinitely many homoclinic solutions via the Fountain theorem and the dual Fountain theorem such that the corresponding energy functional of (HS) goes to infinity and zero, respectively. Some recent results in the literature are generalized and significantly improved.
- Published
- 2016
38. Sharp Singular Trudinger–Moser Inequalities in Lorentz–Sobolev Spaces
- Author
-
Guozhen Lu and Hanli Tang
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,Lorentz transformation ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Statistical and Nonlinear Physics ,01 natural sciences ,Sobolev inequality ,010101 applied mathematics ,Sobolev space ,symbols.namesake ,symbols ,Birnbaum–Orlicz space ,0101 mathematics ,Mathematics ,media_common - Abstract
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.
- Published
- 2016
39. States with values in the Łukasiewicz groupoid
- Author
-
Pavel Pták and Milan Matoušek
- Subjects
Pure mathematics ,010201 computation theory & mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,Topology ,01 natural sciences ,Mathematics - Abstract
In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let ∗ stand for the Łukasiewicz t-norm on [0, 1]2. Let us consider the operation ⋄ on [0, 1] by setting x ⋄ y = (x ⊥ ∗y ⊥)⊥ ∗ (x∗y)⊥, where x ⊥ = 1−x. Let us call the triple L = ([0, 1], ⋄, 1) the Łukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z 2-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as “quantum logics” in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state.
- Published
- 2016
40. On Certain Types of Functions via Generalized Open Sets
- Author
-
Bishwambhar Roy
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Open set ,020201 artificial intelligence & image processing ,02 engineering and technology ,0101 mathematics ,Lambda ,01 natural sciences ,Mathematics - Abstract
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.
- Published
- 2016
41. The Asaeda–Haagerup fusion categories
- Author
-
Noah Snyder, Pinhas Grossman, and Masaki Izumi
- Subjects
Class (set theory) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Center (group theory) ,01 natural sciences ,Subfactor ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Morita equivalence ,Symmetry (geometry) ,Orbifold ,Quotient ,Mathematics - Abstract
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.
- Published
- 2016
42. Fermat curves and a refinement of the reciprocity law on cyclotomic units
- Author
-
Tomokazu Kashio
- Subjects
Fermat's Last Theorem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Reciprocity law ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
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.
- Published
- 2016
43. A Linking/S1-equivariant Variational Argument in the Space of Dual Legendrian Curves and the Proof of the Weinstein Conjecture on S3 'in the Large'
- Author
-
Abbas Bahri
- Subjects
Pure mathematics ,Argument ,General Mathematics ,Dimension (graph theory) ,Equivariant map ,Weinstein conjecture ,Statistical and Nonlinear Physics ,Space (mathematics) ,Stable manifold ,Dual (category theory) ,Mathematics - Abstract
Let α be a contact form on S3, let ξ be its Reeb vector-field and let v be a non-singular vector-field in ker α. Let Cβ be the space of curves x on S3 such ẋ = aξ + bv, ȧ = 0, a ≩ 0. Let L+, respectively L−, be the set of curves in Cβ such that b ≥ 0, respectively b ≤ 0. Let, for x ∈ Cβ, J(x) = ∫0 1 αx(ẋ)dt. The framework of the present paper has been introduced previously in eg [3]. We establish in this paper that some cycles (an infinite number of them, indexed by odd integers, tending to ∞) in the S 1-equivariant homology of Cβ, relative to L+ ∪ L− and to some specially designed ”bottom set”, see section 4, are achieved in the Morse complex of (J,Cβ) by unions of unstable manifolds of critical points (at infinity)which must include periodic orbits of ξ; ie unions of unstable manifolds of critical points at infinity alone cannot achieve these cycles. At the odd indexes (2k−1) = 1+(2k−2), 1 for the linking, (2k−2) for the S1-equivariance, we find that the equivariant contributions of a critical point at infinity to L+ and to L− are fundamentally asymmetric when compared to those of a periodic orbit [5]. The topological argument of existence of a periodic orbit for ξ turns out therefore to be surprisingly close, in spirit, to the linking/equivariant argument of P. Rabinowitz in [12]; e.g. the definition of the ”bottom sets” of section 4 can be related in part to the linking part in the argument of [12]. The objects and the frameworks are strikingly different, but the original proof of [12] can be recognized in our proof, which uses degree theory, the Fadell-Rabinowitz index [8] and the fact that πn+1(Sn) = Z2, n ≥ 3. We need of course to prove, in our framework, that these topological classes cannot be achieved by critical points at infinity only, periodic orbits of ξ excluded, and this is the fundamental difficulty. The arguments hold under the basic assumption that no periodic orbit of index 1 connects L+ and L−. It therefore follows from the present work that either a periodic orbit of index 1 connects L+ and L− (as is probably the case for all three dimensional over-twisted [8] contact forms, see the work of H. Hofer [10], the periodic orbit found in [10] should be of index 1 in the present framework); or (with a flavor of exclusion in either/or) a linking/ equivariant variational argument a la P. Rabinowitz [12] can be put to work. Existence of (possibly multiple) periodic orbits of ξ, maybe of high Morse index, follows then. Therefore, to a certain extent, the present result runs, especially in the case of threedimensional over-twisted [8] contact forms, against the existence of non-trivial algebraic invariants defined by the periodic orbits of ξ and independent of what ker α and/or α are.
- Published
- 2015
44. On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces
- Author
-
Aldona Dutkiewicz
- Subjects
Pure mathematics ,General Mathematics ,Eberlein–Šmulian theorem ,Aubin–Lions lemma ,Banach space ,Interpolation space ,Finite-rank operator ,Banach manifold ,Lp space ,C0-semigroup ,Mathematics - Abstract
In this paper we prove an existence theorem for ordinary differential equations in Banach spaces. The main assumptions in our results, formulated in terms of the Kuratowski measure of noncompactness, are motivated by the paper [CONSTANTIN, A.: On Nagumo’s theorem, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 41-44].
- Published
- 2015
45. Nodal Solutions for a Class of Degenerate Boundary Value Problems
- Author
-
Paul H. Rabinowitz and Julián López-Gómez
- Subjects
010101 applied mathematics ,Class (set theory) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Degenerate energy levels ,Statistical and Nonlinear Physics ,Boundary value problem ,0101 mathematics ,NODAL ,01 natural sciences ,Mathematics - Abstract
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.
- Published
- 2015
46. Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals
- Author
-
Antara Bhar and Manjul Gupta
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Pure mathematics ,General Mathematics ,Topological tensor product ,Lorentz transformation ,Banach space ,Finite-rank operator ,symbols.namesake ,symbols ,Interpolation space ,Dual polyhedron ,Birnbaum–Orlicz space ,Lp space ,Mathematics - Abstract
In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M (X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M (X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.
- Published
- 2014
47. Variational Methods on Finite Dimensional Banach Spaces and Discrete Problems
- Author
-
Giuseppina D'Aguì, Pasquale Candito, and Gabriele Bonanno
- Subjects
Pure mathematics ,Picard–Lindelöf theorem ,Functional analysis ,General Mathematics ,Multiplicity results ,NONLINEAR DIFFERENCE-EQUATIONS ,MOUNTAIN PASS THEOREM ,MULTIPLE SOLUTIONS ,Mountain pass theorem ,Eberlein–Šmulian theorem ,Banach space ,Statistical and Nonlinear Physics ,C0-semigroup ,Mathematics - Abstract
In this paper, existence and multiplicity results for a class of second-order difference equations are established. In particular, the existence of at least one positive solution without requiring any asymptotic condition at infinity on the nonlinear term is presented and the existence of two positive solutions under a superlinear growth at infinity of the nonlinear term is pointed out. The approach is based on variational methods and, in particular, on a local minimum theorem and its variants. It is worth noticing that, in this paper, some classical results of variational methods are opportunely rewritten by exploiting fully the finite dimensional framework in order to obtain novel results for discrete problems.
- Published
- 2014
48. Cerone’s Generalizations of Steffensen’s Inequality
- Author
-
Ksenija Smoljak, Josip Pečarić, and Anamarija Perušić
- Subjects
Pure mathematics ,Inequality ,Generalization ,General Mathematics ,media_common.quotation_subject ,Calculus ,Monotonic function ,Type (model theory) ,Steffensen inequality ,monotonic functions ,weaker conditions ,exponential convexity ,Stolarsky type means ,Mathematics ,media_common - Abstract
In this paper, generalizations of Steffensen’s inequality with bounds involving any two subintervals motivated by Cerone’s generalizations are given. Furthermore, weaker conditions for Cerone’s generalization as well as for new generalizations obtained in this paper are given. Moreover, functionals defined as the difference between the left-hand and the right-hand side of these generalizations are studied and new Stolarsky type means related to them are obtained.
- Published
- 2014
49. On the stability of some properties of partial algebras under powers
- Author
-
N. Chaisansuk, Josef Šlapal, and S. Leeratanavalee
- Subjects
power of partial algebras ,Pure mathematics ,Property (philosophy) ,diagonal and commutative partial algebras ,General Mathematics ,Subalgebra ,Non-associative algebra ,Partial algebra ,Stability (probability) ,Algebra ,Quadratic algebra ,reflexive ,Interior algebra ,partial algebra ,Nest algebra ,Mathematics - Abstract
In the paper, we study eight properties of partial algebras most of which are related to diagonality. For each of the properties, we give sufficient conditions under which this property is preserved by powers of partial algebras. In the paper, we study eight properties of partial algebras most of which are related to diagonality. For each of the properties, we give sufficient conditions under which this property is preserved by powers of partial algebras.
- Published
- 2014
50. Constant mean curvature surfaces in hyperbolic 3-space via loop groups
- Author
-
Josef Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi, and Technische Universität München, Faculty of Mathematics
- Subjects
Mathematics - Differential Geometry ,Surface (mathematics) ,Pure mathematics ,Minimal surface ,Mean curvature ,Euclidean space ,Applied Mathematics ,General Mathematics ,Type (model theory) ,Space (mathematics) ,ddc ,Loop (topology) ,Differential Geometry (math.DG) ,FOS: Mathematics ,Constant (mathematics) ,Mathematics - Abstract
In hyperbolic 3-space $\mathbb{H}^3$ surfaces of constant mean curvature $H$ come in three types, corresponding to the cases $0 \leq H < 1$, $H = 1$, $H > 1$. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space $\mathbb{E}^3$ with H=0 and $H \neq 0$, respectively. These surface classes have been investigated intensively in the literature. For the case $0 \leq H < 1$ there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstra{\ss} type representation for surfaces of constant mean curvature in $\mathbb{H}^3$ with particular emphasis on the case of mean curvature $0\leq H < 1$. In particular, the generalized Weierstra{\ss} type representation presented in this paper enables us to construct simultaneously minimal surfaces (H=0) and non-minimal constant mean curvature surfaces ($0, Comment: 37 pages, 4 figures. v3: Various typos fixed. v4: Proposition D.1 has been fixed
- Published
- 2014
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