8,949 results
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2. A note on the paper 'Best proximity point results for $$p$$-proximal contractions'
- Author
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M. Gabeleh and J. Markin
- Subjects
Combinatorics ,Metric space ,Class (set theory) ,General Mathematics ,Fixed-point theorem ,Point (geometry) ,Mathematics - Abstract
Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of $$p$$ -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory.
- Published
- 2021
3. Corrigendum to a Paper by Charak and Laine
- Author
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Kuldeep Singh Charak and Ilpo Laine
- Subjects
Discrete mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Entire function ,Prime (order theory) ,Mathematics - Abstract
This is a corrigendum to our paper, “On a class of prime entire functions”, published in Acta Math. Sin., Engl. Ser., 25, 1647–1652 (2009).
- Published
- 2020
4. Sendov’s Conjecture: A Note on a Paper of Dégot
- Author
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T. P. Chalebgwa
- Subjects
Combinatorics ,Conjecture ,General Mathematics ,Sendov's conjecture ,Complex polynomial ,Unit distance ,Unit disk ,Critical point (mathematics) ,Mathematics - Abstract
Sendov’s conjecture states that if all the zeroes of a complex polynomial P(z) of degree at least two lie in the unit disk, then within a unit distance of each zero lies a critical point of P(z). In a paper that appeared in 2014, Degot proved that, for each a ∈ (0, 1), there exists an integer N such that for any polynomial P(z) with degree greater than N, if P(a) = 0 and all zeroes lie inside the unit disk, the disk |z − a| ≤ 1 contains a critical point of P(z). Based on this result, we derive an explicit formula N(a) for each a ∈ (0, 1) and, consequently obtain a uniform bound N for all a ∈ [α, β] where 0 < α < β < 1. This (partially) addresses the questions posed in Degot’s paper.
- Published
- 2020
5. On Variability and Interdependence of Local Porosity and Local Tortuosity in Porous Materials: a Case Study for Sack Paper
- Author
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Matthias Neumann, Eduardo Machado Charry, Volker Schmidt, and Karin Zojer
- Subjects
Statistics and Probability ,Geodesic ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Copula (linguistics) ,Sinuosity ,01 natural sciences ,Tortuosity ,010104 statistics & probability ,Goodness of fit ,Gumbel distribution ,Joint probability distribution ,0101 mathematics ,Porosity ,Mathematics - Abstract
The variability and interdependence of local porosity and local mean geodesic tortuosity, which is a measure for the sinuosity of shortest transportation paths, is investigated at the example of the microstructure in sack paper. By means of statistical image analysis, these two morphological characteristics are computed for several cutouts of 3D image data obtained by X-ray microcomputed tomography. Considering cutouts of different sizes allows us to study the influence of the sample size on the local variability of the considered characteristics. Moreover, the interdependence between local porosity and local mean geodesic tortuosity is quantified by modeling their joint distribution parametrically using Archimedean copulas. It turns out that the family of Gumbel copulas is an appropriate model type, which is formally validated by a goodness of fit test. Besides mean geodesic tortuosity, we consider further related morphological characteristics, describing the sinuosity of those shortest transportation paths, whose minimum diameter exceeds a predefined threshold. Moreover, we show that the copula approach investigated in this paper can also be used to quantify the negative correlation between local porosity and these modified versions of local mean geodesic tortuosity. Our results elucidate the impact of local porosity on various kinds of morphological characteristics, which are not experimentally accessible and which are important for local air permeance – a key property of sack paper.
- Published
- 2020
6. A Note on a Paper of Aivazidis, Safonova and Skiba
- Author
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M. M. Al-Shomrani, Adolfo Ballester-Bolinches, and A. A. Heliel
- Subjects
Subnormal subgroup ,Combinatorics ,Mathematics::Group Theory ,Finite group ,General Mathematics ,Mathematics - Abstract
The main result of this paper states that if $${\mathcal {F}}$$ is a subgroup-closed saturated formation of full characteristic, then the $${\mathcal {F}}$$ -residual of a K- $${\mathcal {F}}$$ -subnormal subgroup S of a finite group G is a large subgroup of G provided that the $${\mathcal {F}}$$ -hypercentre of every subgroup X of G containing S is contained in the $${\mathcal {F}}$$ -residual of X. This extends a recent result of Aivazidis, Safonova and Skiba.
- Published
- 2021
7. On a paper of Dressler and Van de Lune
- Author
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Pablo Andres Panzone
- Subjects
Combinatorics ,Lune ,General Mathematics ,Arithmetic function ,Natural number ,Prime (order theory) ,Mathematics - Abstract
If $$z\in {\mathbb {C}}$$ and $$1\le n$$ is a natural number then $$\begin{aligned} \sum _{d_1 d_2 =n} (1-z^{p_1})\cdots (1-z^{p_m}) z^{q_1 e_{1}+\cdots +q_i e_{i} }=1, \end{aligned}$$ where $$d_1=p_1^{r_1}\dots p_m^{r_m }$$ , $$d_2=q_1^{e_1}\dots q_i^{e_i }$$ are the prime decompositions of $$d_1, d_2$$ . This is one of the identities involving arithmetic functions that we prove using ideas from the paper of Dressler and van de Lune [3].
- Published
- 2020
8. Note on a paper by Bordellès, Dai, Heyman, Pan and Shparlinski
- Author
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Jie Wu
- Subjects
Combinatorics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Very recently Bordelles, Dai, Heyman, Pan and Shparlinski studied asymptotic behaviour of the quantity $$\begin{aligned} S_f(x) := \sum _{n\leqslant x} f\left( \left[ \frac{x}{n}\right] \right) , \end{aligned}$$and established some asymptotic formulas for $$S_f(x)$$ under three different types of assumptions on f. In this short note we improve some of their results.
- Published
- 2019
9. On a paper of Erdös and Szekeres
- Author
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Mei-Chu Chang and Jean Bourgain
- Subjects
010101 applied mathematics ,Discrete mathematics ,Set (abstract data type) ,Partial differential equation ,Functional analysis ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics - Abstract
Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.
- Published
- 2018
10. Notes on the paper 'A note on pronormal p-subgroups of finite groups'
- Author
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Haoran Yu and Suli Liu
- Subjects
Discrete mathematics ,Lemma (mathematics) ,010505 oceanography ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,0105 earth and related environmental sciences ,Mathematics - Abstract
In this short note, we show that Theorem 4.3 of Liu and Yu (Monatshefte Math 195:173–176, 2021) is a consequence of Lemma 2 of Ballester-Bolinches and Esteban-Romero (J Aust Math Soc 75:181–191, 2003).
- Published
- 2021
11. Comments on the paper 'Asymptotic behavior for a fourth-order parabolic equation involving the Hessian. Z. Angew. Math. Phys., (2018) 69: 147'
- Author
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Jun Zhou and Hang Ding
- Subjects
Hessian matrix ,symbols.namesake ,Fourth order ,Applied Mathematics ,General Mathematics ,symbols ,General Physics and Astronomy ,Applied mathematics ,Finite time ,Mathematics ,Energy functional ,Blowing up - Abstract
In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).
- Published
- 2019
12. Remark on the paper 'On products of Fourier coefficients of cusp forms'
- Author
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Yuk-Kam Lau, Deyu Zhang, and Yingnan Wang
- Subjects
Cusp (singularity) ,Discrete group ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,02 engineering and technology ,01 natural sciences ,Cusp form ,Combinatorics ,Integer ,Product (mathematics) ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Fourier series ,Mathematics - Abstract
Let a(n) be the Fourier coefficient of a holomorphic cusp form on some discrete subgroup of \(SL_2({\mathbb R})\). This note is to refine a recent result of Hofmann and Kohnen on the non-positive (resp. non-negative) product of \(a(n)a(n+r)\) for a fixed positive integer r.
- Published
- 2016
13. Addendum to the paper 'Linearly topologized modules over a discrete valuation ring'
- Author
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Patricia Couto G. Mauro and Dinamérico P. Pombo
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Addendum ,0101 mathematics ,01 natural sciences ,Discrete valuation ring ,Linear equation ,Mathematics - Abstract
For any discrete valuation ring R, any R-linear mapping u from an R-module E into an R-module F and any \(y_0\in F\), a necessary and sufficient condition for the solvability of the equation \(u(x)=y_0\) is established, and an application of this result is presented.
- Published
- 2016
14. On a paper by Yuri G. Zarhin
- Author
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Elmer Rees
- Subjects
Pure mathematics ,Polynomial ,General Mathematics ,Direct proof ,Algebraic geometry ,Complex quadratic polynomial ,Mathematics - Abstract
In a recent paper, (Math Notes 91(3–4): 508–516, 2012) Zarhin proved that each member of a naturally defined family of linear maps \({\mathbb {C}}^n \rightarrow {\mathbb {C}}^n\) has co-rank one. We present a direct proof of Zarhin’s result about complex polynomials with distinct roots; it is rather similar to that of Appendix by Vik.S. Kulikov to Zarhin’s paper but we give explicit constants. We also discuss the case of a polynomial with multiple roots.
- Published
- 2015
15. On D.Y. Gao and X. Lu paper 'On the extrema of a nonconvex functional with double-well potential in 1D'
- Author
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Constantin Zălinescu
- Subjects
021103 operations research ,Applied Mathematics ,General Mathematics ,0211 other engineering and technologies ,General Physics and Astronomy ,Double-well potential ,02 engineering and technology ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Maxima and minima ,35J20, 35J60, 74G65, 74S30 ,Optimization and Control (math.OC) ,FOS: Mathematics ,Preprint ,0101 mathematics ,Constant (mathematics) ,Mathematics - Optimization and Control ,Subspace topology ,Mathematics - Abstract
The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3
- Published
- 2017
16. A Note on Recent Papers by Grafakos and Teschl, and Estrada
- Author
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Adam Nowak and Krzysztof Stempak
- Subjects
Hankel transform ,Partial differential equation ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Function (mathematics) ,Transplantation ,symbols.namesake ,Radial function ,Fourier transform ,Fourier analysis ,symbols ,Analysis ,Mathematics - Abstract
We indicate how recent results of Grafakos and Teschl (J Fourier Anal Appl 19:167–179, 2013), and Estrada (J Fourier Anal Appl 20:301–320, 2014), relating the Fourier transform of a radial function in $$\mathbb R^n$$ and the Fourier transform of the same function in $$\mathbb R^{n+2}$$ and $$\mathbb R^{n+1}$$ , respectively, are located within known results on transplantation for Hankel transforms.
- Published
- 2014
17. Some comments on the paper: Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959
- Author
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Michelle Pierri and Donal O'Regan
- Subjects
Discrete mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Controllability ,Algebra ,0101 mathematics ,Differential (mathematics) ,Mathematics - Abstract
The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.
- Published
- 2016
18. Fractional Factorials and Prime Numbers (A Remark on the Paper 'On Prime Values of Some Quadratic Polynomials')
- Author
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A. N. Andrianov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Prime element ,01 natural sciences ,Prime k-tuple ,Prime (order theory) ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Prime factor ,Unique prime ,0101 mathematics ,Fibonacci prime ,Prime power ,Sphenic number ,Mathematics - Abstract
Congruences mod p for a prime p and partial products of the numbers 1,…, p − 1 are obtained. Bibliography: 2 titles.
- Published
- 2016
19. Erratum to our paper On the divergence of Fourier series of functions in several variables
- Author
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Larry Gogoladze and V. Tsagareishvili
- Subjects
General Mathematics ,Mathematical analysis ,Divergence (statistics) ,Fourier series ,Mathematics - Published
- 2015
20. On the Paper 'A Note on Spaces of Absolutely Convergent Fourier Transforms' by Björn G.Walther (this Issue). Letter to the Editor
- Author
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S. V. Kislyakov
- Subjects
Algebra ,symbols.namesake ,Letter to the editor ,Partial differential equation ,Fourier transform ,Fourier analysis ,Applied Mathematics ,General Mathematics ,Mathematics education ,symbols ,Absolute convergence ,Analysis ,Mathematics - Published
- 2014
21. Biased Adjusted Poisson Ridge Estimators-Method and Application
- Author
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Pär Sjölander, Muhammad Qasim, Muhammad Amin, B. M. Golam Kibria, and Kristofer Månsson
- Subjects
Mean squared error ,General Mathematics ,Maximum likelihood ,General Physics and Astronomy ,Regression estimator ,Poisson distribution ,Modified almost unbiased ridge estimators ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,Statistics ,Poisson regression ,0101 mathematics ,Mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Estimator ,Mean square error ,General Chemistry ,Ridge (differential geometry) ,Poisson ridge regression ,Multicollinearity ,Maximum likelihood estimator ,symbols ,General Earth and Planetary Sciences ,General Agricultural and Biological Sciences ,Research Paper - Abstract
Månsson and Shukur (Econ Model 28:1475–1481, 2011) proposed a Poisson ridge regression estimator (PRRE) to reduce the negative effects of multicollinearity. However, a weakness of the PRRE is its relatively large bias. Therefore, as a remedy, Türkan and Özel (J Appl Stat 43:1892–1905, 2016) examined the performance of almost unbiased ridge estimators for the Poisson regression model. These estimators will not only reduce the consequences of multicollinearity but also decrease the bias of PRRE and thus perform more efficiently. The aim of this paper is twofold. Firstly, to derive the mean square error properties of the Modified Almost Unbiased PRRE (MAUPRRE) and Almost Unbiased PRRE (AUPRRE) and then propose new ridge estimators for MAUPRRE and AUPRRE. Secondly, to compare the performance of the MAUPRRE with the AUPRRE, PRRE and maximum likelihood estimator. Using both simulation study and real-world dataset from the Swedish football league, it is evidenced that one of the proposed, MAUPRRE ($$ \hat{k}_{q4} $$ k ^ q 4 ) performed better than the rest in the presence of high to strong (0.80–0.99) multicollinearity situation.
- Published
- 2020
22. An efficient characterization of submodular spanning tree games
- Author
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Zhuan Khye Koh, Laura Sanità, and Discrete Mathematics
- Subjects
FOS: Computer and information sciences ,Computer Science::Computer Science and Game Theory ,05C57 Games on graphs (graph-theoretic aspects) ,Class (set theory) ,Theoretical computer science ,Discrete Mathematics (cs.DM) ,QA75 Electronic computers. Computer science ,General Mathematics ,Open problem ,05C05 Trees ,0211 other engineering and technologies ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Outcome (game theory) ,Convexity ,Submodular set function ,Set (abstract data type) ,91A12 Cooperative games ,Computer Science - Computer Science and Game Theory ,Mathematics ,021103 operations research ,Spanning tree ,Full Length Paper ,ComputingMilieux_PERSONALCOMPUTING ,05C05 TREES, 05C57 GAMES ON GRAPHS (GRAPH-THEORETIC ASPECTS), 91A12 COOPERATIVE GAMES ,010201 computation theory & mathematics ,Game theory ,Software ,Computer Science and Game Theory (cs.GT) ,Computer Science - Discrete Mathematics - Abstract
Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.
- Published
- 2020
23. Special Ulrich bundles on regular Weierstrass fibrations
- Author
-
Joan Pons-Llopis and Rosa M. Miró-Roig
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Short paper ,Elliptic surfaces ,Ulrich bundles ,01 natural sciences ,Mathematics::Algebraic Geometry ,Simple (abstract algebra) ,0103 physical sciences ,Weierstrass fibrations ,Rank (graph theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.
- Published
- 2019
24. On sums of squares of primes and a k-th power of prime
- Author
-
Zhixin Liu and Rui Zhang
- Subjects
Discrete mathematics ,010505 oceanography ,General Mathematics ,010102 general mathematics ,Short paper ,Sander ,01 natural sciences ,Prime (order theory) ,Power (physics) ,Integer ,Congruence (manifolds) ,0101 mathematics ,0105 earth and related environmental sciences ,Mathematics - Abstract
In this short paper, we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of two squares of primes and a k-th power of prime for any integer $$k \ge 3$$ . Our results improve the recent results due to Brudern (in: Sander, Steuding, Steuding (eds) From arithmetic to zeta-functions, Springer, Cham 2016). The similar method can be also applied to some related questions in this direction, and this can improve the previous results.
- Published
- 2018
25. Tikhonov regularization of a second order dynamical system with Hessian driven damping
- Author
-
Szilárd László, Radu Ioan Boţ, and Ernö Robert Csetnek
- Subjects
Hessian matrix ,General Mathematics ,0211 other engineering and technologies ,Dynamical Systems (math.DS) ,02 engineering and technology ,Dynamical system ,01 natural sciences ,Hessian-driven damping ,90C26 ,Tikhonov regularization ,symbols.namesake ,34G25, 47J25, 47H05, 90C26, 90C30, 65K10 ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Dynamical Systems ,Mathematics - Optimization and Control ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,47J25 ,47H05 ,010102 general mathematics ,Hilbert space ,90C30 ,Function (mathematics) ,Convex optimization ,Optimization and Control (math.OC) ,Second order dynamical system ,34G25 ,symbols ,Fast convergence methods ,Convex function ,Software - Abstract
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm.
- Published
- 2020
26. Global optimization in Hilbert space
- Author
-
Benoît Chachuat, Boris Houska, Engineering & Physical Science Research Council (EPSRC), and Commission of the European Communities
- Subjects
Technology ,Optimization problem ,Mathematics, Applied ,0211 other engineering and technologies ,CONVEX COMPUTATION ,010103 numerical & computational mathematics ,02 engineering and technology ,ELLIPSOIDS ,01 natural sciences ,90C26 ,93B40 ,Convergence analysis ,0102 Applied Mathematics ,Branch-and-lift ,CUT ,Mathematics ,65K10 ,021103 operations research ,Full Length Paper ,Operations Research & Management Science ,0103 Numerical and Computational Mathematics ,Bounded function ,Physical Sciences ,symbols ,49M30 ,Calculus of variations ,INTEGRATION ,SET ,Complexity analysis ,Complete search ,Operations Research ,General Mathematics ,APPROXIMATIONS ,Set (abstract data type) ,symbols.namesake ,Applied mathematics ,ALGORITHM ,0101 mathematics ,INTERSECTION ,Global optimization ,0802 Computation Theory and Mathematics ,Science & Technology ,Infinite-dimensional optimization ,Hilbert space ,Computer Science, Software Engineering ,Constraint (information theory) ,Computer Science ,Software - Abstract
We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.
- Published
- 2017
27. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
- Author
-
Oswaldo Lezama and Claudia Gallego
- Subjects
Hermite polynomials ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Short paper ,Skew ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,symbols ,Kronecker's theorem ,Finitely-generated abelian group ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.
- Published
- 2015
28. A Remark on the Local Cohomology Modules of a Union of Disjoint Matroids
- Author
-
Cong Minh Nguyen and Minh Cong Nguyen
- Subjects
Discrete mathematics ,Ideal (set theory) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Short paper ,0102 computer and information sciences ,Disjoint sets ,Local cohomology ,01 natural sciences ,Matroid ,Simplicial complex ,010201 computation theory & mathematics ,0101 mathematics ,Mathematics - Abstract
Let I be the Stanley–Reisner ideal of a simplicial complex Δ. In this short paper, we shall give a formula of vanishing of the local cohomology modules for S/I(r) in the case Δ is a union of disjoint matroids, where I(r) is the rth symbolic power of I. As an application, we will improve a previous result in Minh and Nakamura (Nagoya Math. J. 213, 127–140, 2014) for the k-Buchsbaumness of S/I(r).
- Published
- 2015
29. An introduction to functional data analysis and a principal component approach for testing the equality of mean curves
- Author
-
Gregory Rice and Lajos Horváth
- Subjects
Weak convergence ,General Mathematics ,Principal component analysis ,Statistics ,Paper version ,Functional data analysis ,Simple random sample ,Null hypothesis ,Mathematics - Abstract
We give an introduction to functional data analysis, with examples, and provide a brief review of the literature. We explain how principal component analysis (PCA) can be used to transform curves into finite dimensional data. An application of PCA is developed to test for the equality of the means of several populations (functional analysis of variance). Asymptotics are derived under the null hypothesis that the populations have the same mean curves. The selection of the basis for the projections and the power of the test is discussed for simple random samples and stationary time series samples of curves. We review the part of the literature which is needed to establish the validity of the PCA method. Two data sets, magnetogram records and stock returns, are used to illustrate the applicability of our limit results.
- Published
- 2015
30. Smoothness of Generalized Solutions of the Neumann Problem for a Strongly Elliptic Differential-Difference Equation on the Boundary of Adjacent Subdomains
- Author
-
D. A. Neverova
- Subjects
Statistics and Probability ,Smoothness (probability theory) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Neumann boundary condition ,Boundary (topology) ,Differential difference equations ,General Medicine ,Mathematics - Abstract
This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.
- Published
- 2022
31. Order 3 symplectic automorphisms on K3 surfaces
- Author
-
Alice Garbagnati and Yulieth Prieto Montañez
- Subjects
Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Order (ring theory) ,Automorphism ,01 natural sciences ,Cohomology ,14J28, 14J50 ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics ,Symplectic geometry - Abstract
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141
- Published
- 2021
32. Additive subgroups generated by noncommutative polynomials
- Author
-
Tsiu-Kwen Lee
- Subjects
Combinatorics ,Ring (mathematics) ,Polynomial ,General Mathematics ,Unital ,Image (category theory) ,Structure (category theory) ,Ideal (ring theory) ,Algebra over a field ,Noncommutative geometry ,Mathematics - Abstract
Let R be an algebra. Given a noncommutative polynomial f, let f(R) stand for the additive subgroup of R generated by the image of f. For a unital or an affine algebra R, $$S_k(R)$$ is completely determined for any standard polynomial $$S_k$$ when R is generated by $$S_k(R)$$ as an ideal. Motivated by Bresar’s paper [Adv. Math. 374 (2020), 107346, 21 pp] and Robert’s paper [J. Oper. Theory 75 (2016), 387–408], under certain conditions we also prove that f(R) is equal to either [R, R] or the whole ring R. We obtain these results by studying the structure of Lie ideals L of a ring R whenever R is generated by [R, L] as an ideal.
- Published
- 2021
33. Characterization on transcendental entire solutions of certain types of non-linear generalized delay-differential equations
- Author
-
Tania Biswas and Abhijit Banerjee
- Subjects
Pure mathematics ,Nonlinear system ,General Mathematics ,Content (measure theory) ,Delay differential equation ,Transcendental number ,Characterization (mathematics) ,Mathematics - Abstract
In this paper, we study on the existence of transcendental entire solutions of certain non-linear generalized delay-differential equations. In this respect we have improved a recent result of Wang et al. (Turk J Math 43:941–954, 2019). Also at the time of investigating the solutions of shift equations we have improved as well as extended an earlier result due to Latreuch (Mediterr J Math 14:1–16, 2017). A handful number of examples have been exhibited by us relevant to the content of the paper to show that each case as demonstrated in the conclusions of the theorems actually occurs. At last we raise a question for future investigations.
- Published
- 2021
34. Navier-Stokes equations under slip boundary conditions: Lower bounds to the minimal amplitude of possible time-discontinuities of solutions with two components in L∞(L3)
- Author
-
Hugo Beirão da Veiga and Jiaqi Yang
- Subjects
Combinatorics ,Amplitude ,General Mathematics ,Boundary (topology) ,Slip (materials science) ,Boundary value problem ,Classification of discontinuities ,Navier–Stokes equations ,Omega ,Mathematics ,Bar (unit) - Abstract
The main purpose of this paper is to extend the result obtained by Beirao da Veiga (2000) from the whole-space case to slip boundary cases. Denote by u two components of the velocity u. To fix ideas set ū = (u1,u2, 0) (the half-space) or $${\boldsymbol{\bar u}} = {\hat u_1}{\hat e_1} + {\hat u_2}{\hat e_2}$$ (the general boundary case (see (7.1))). We show that there exists a constant K, which enjoys very simple and significant expressions such that if at some time τ ∈ (0,T) one has $$\lim {\sup _{t \to \tau - 0}}\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3 < \left\| {{\boldsymbol{\bar u}}(\tau )} \right\|_{{L^3}(\Omega )}^3 + K$$ , then u is continuous at τ with values in L3(Ω). Roughly speaking, the above norm-discontinuity of merely two components of the velocity cannot occur for steps’ amplitudes smaller than K. In particular, if the above condition holds at each τ ∈ (0,T), the solution is smooth in (0,T) × Ω. Note that here there is no limitation on the width of the norms $$\left\| {{\boldsymbol{\bar u}}(t)} \right\|_{{L^3}(\Omega )}^3$$ . So K is independent of these quantities. Many other related results are discussed and compared among them. This is a second main aim of this paper. New results are proved in Sections 5–7.
- Published
- 2021
35. On a Lotka-Volterra Competition Diffusion Model with Advection
- Author
-
Qi Wang
- Subjects
Advection ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Quantitative Biology::Populations and Evolution ,Statistical physics ,Diffusion (business) ,Constant (mathematics) ,Stability (probability) ,Competition (biology) ,Competitive Lotka–Volterra equations ,media_common ,Mathematics - Abstract
In this paper, the author focuses on the joint effects of diffusion and advection on the dynamics of a classical two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. For comparison purposes, the two species are assumed to have identical competition abilities throughout this paper. The results explore the condition on the diffusion and advection rates for the stability of former species. Meanwhile, an asymptotic behavior of the stable coexistence steady states is obtained.
- Published
- 2021
36. On the squeezing function for finitely connected planar domains
- Author
-
Oliver Roth and Pavel Gumenyuk
- Subjects
Pure mathematics ,Conjecture ,conformal mapping ,Mathematics - Complex Variables ,General Mathematics ,Conformal map ,Annulus (mathematics) ,Function (mathematics) ,Primary: 30C75, Secondary: 30C35, 30C85 ,Function problem ,Planar ,squeezing function ,Simple (abstract algebra) ,FOS: Mathematics ,finitely connected domain ,Complex Variables (math.CV) ,extremal problem ,Loewner differential equation ,Mathematics - Abstract
In a recent paper, Ng, Tang and Tsai (Math. Ann. 2020) have found an explicit formula for the squeezing function of an annulus via the Loewner differential equation. Their result has led them to conjecture a corresponding formula for planar domains of any finite connectivity stating that the extremum in the squeezing function problem is achieved for a suitably chosen conformal mapping onto a circularly slit disk. In this paper we disprove this conjecture. We also give a conceptually simple potential-theoretic proof of the explicit formula for the squeezing function of an annulus which has the added advantage of identifying all extremal functions., Comment: Version 2: (1) a statement on the history of the notion of squeezing function has been corrected; (2) a new reference [5] (F. Deng: Levi's problem, convexity, and squeezing functions on bounded domains) has been added; (3) a small technical issue with numbering of equations has been resolved
- Published
- 2021
37. Distinguishing regular and singular black holes in modified gravity
- Author
-
Ahmadjon Abdujabbarov, Javlon Rayimbaev, Wen-Biao Han, and Aleksandra Demyanova
- Subjects
Gravity (chemistry) ,Field (physics) ,General Mathematics ,Quasiperiodic function ,Precession ,Radial motion ,Astrophysics ,Radius ,Schwarzschild radius ,Specific relative angular momentum ,Mathematics - Abstract
This paper is devoted to investigate the possible ways of distinguishing regular and singular black holes (BHs) in modified gravity (MOG) called regular MOG (RMOG) and Schwarzschild MOG (SMOG) BHs through observational data from twin peak quasiperiodic oscillations (QPOs) which are generated by test particles in stable orbits around the BHs. The presence of MOG field causes to sufficiently the mpeak in effective potential for a radial motion of test particles. The effect of MOG parameter on specific angular momentum and energy has also studied. As a main part of the paper, we focus on investigations of QPOs around SMOG and RMOG BHs in RP model and the relations of upper and lower frequencies of twin peak QPOs in SMOG and RMOG BH models together with extreme rotating Kerr and Schwarzschild BH. Moreover, possible parameters for the central BHs of the objects GRS J1915 + 105 and XTE 1550 – 564 have also obtained numerically in the relativistic precession (RP) model. Finally, we provide comparisons of the innermost stable circular orbit (ISCO) and the orbits where twin peak QPOs with the ratio 3:2 taken place and show that QPOs can not be generated at/inside ISCO and there is a correlation between the radius of ISCO and QPO orbits.
- Published
- 2021
38. Distinguished limits and drifts: between nonuniqueness and universality
- Author
-
Vladimir A. Vladimirov
- Subjects
Mathematical and theoretical biology ,Number theory ,General Mathematics ,Mathematical analysis ,Universality (philosophy) ,Ode ,Inverse ,Uniqueness ,Focus (optics) ,Mathematics ,Variable (mathematics) - Abstract
This paper deals with a version of the two-timing method which describes various ‘slow’ effects caused by externally imposed ‘fast’ oscillations. Such small oscillations are often called vibrations and the research area can be referred as vibrodynamics. The governing equations represent a generic system of first-order ODEs containing a prescribed oscillating velocity $${\varvec{u}}$$ , given in a general form. Two basic small parameters stand in for the inverse frequency and the ratio of two time-scales; they appear in equations as regular perturbations. The proper connections between these parameters yield the distinguished limits, leading to the existence of closed systems of asymptotic equations. The aim of this paper is twofold: (i) to clarify (or to demystify) the choices of a slow variable, and (ii) to give a coherent exposition which is accessible for practical users in applied mathematics, sciences and engineering. We focus our study on the usually hidden aspects of the two-timing method such as the uniqueness or multiplicity of distinguished limits and universal structures of averaged equations. The main result is the demonstration that there are two (and only two) different distinguished limits. The explicit instruction for practically solving ODEs for different classes of $${\varvec{u}}$$ is presented. The key roles of drift velocity and the qualitatively new appearance of the linearized equations are discussed. To illustrate the broadness of our approach, two examples from mathematical biology are shown.
- Published
- 2021
39. Meromorphic functions of restricted hyper-order sharing one or two sets with its linear C-shift operator
- Author
-
A. Banerjee and A. Roy
- Subjects
Fermat's Last Theorem ,Discrete mathematics ,Operator (computer programming) ,Conjecture ,Corollary ,General Mathematics ,Order (group theory) ,Uniqueness ,Shift operator ,Mathematics ,Meromorphic function - Abstract
In this paper, in the light of weighted sharing of sets, we investigate the possible uniqueness of meromorphic function of restricted hyper order with its linear c-shift operator. Our first two theorems improve a number of earlier results. Our last theorem together with a corollary improves and extends a result due to Li, Lu and Xu [14]. Most importantly, our another corollary deducted from the last theorem not only provides an answer to an open question posed by Liu [16] but also noticeably improves two results of Chen and Chen [4]. A number of examples have been exhibited by us pertinent with the content of the paper. At the penultimate section which is also the application part of our result, we extend a recent result of Liu, Ma and Zhai [17]. Finally, presenting two examples, we conjecture that one of our result may hold for a larger class of functions and we place it as an open question for future investigations.
- Published
- 2021
40. Approximation of functions of H$$\ddot{o}$$lder class and solution of ODE and PDE by extended Haar wavelet operational matrix
- Author
-
Priya Kumari and Shyam Lal
- Subjects
Combinatorics ,Approximation theory ,Wavelet ,Partial differential equation ,Exact solutions in general relativity ,General Mathematics ,Ode ,Estimator ,Interval (mathematics) ,Haar wavelet ,Mathematics - Abstract
In this paper, extended H $$\ddot{o}$$ lder class $$H_\alpha ^{(w)}[0,\mu )$$ is considered. This class is the generalization of H $$\ddot{o}$$ lder class $$H_\alpha [0,\mu )$$ . Three new estimators $$E_N^{(1)}(f), E_N^{(2)}(f)$$ and $$E_N^{(3)}(f)$$ of functions of classes $$H_\alpha [0,\mu )$$ and $$H_\alpha ^{(w)}[0,\mu )$$ have been obtained. These estimators are best in approximation of functions by wavelet methods. The estimators obtained in this paper and the solution of ordinary and partial differential equations by extended Haar wavelet operational matrix method in the interval $$[0,\mu )$$ and its comparison with exact solution for different values of $$\mu$$ are the significant achievement of this research paper in approximation theory as well as Wavelet Analysis.
- Published
- 2021
41. On boundary-value problems for semi-linear equations in the plane
- Author
-
Vladimir Gutlyanskiĭ, Vladimir Ryazanov, O.V. Nesmelova, and A.S. Yefimushkin
- Subjects
Statistics and Probability ,Dirichlet problem ,Sobolev space ,Pure mathematics ,Harmonic function ,Applied Mathematics ,General Mathematics ,Neumann boundary condition ,Hölder condition ,Boundary value problem ,Type (model theory) ,Analytic function ,Mathematics - Abstract
The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Holder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form $$ {\partial}_{\overline{z}}f+ af+b\overline{f}=c, $$ where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that $$ {\partial}_{\overline{z}}f=g $$ with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type $$ {\partial}_{\overline{z}}f(z)=h(z)q\left(f(z)\right). $$ On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.
- Published
- 2021
42. Kolmogorov’s theory of turbulence and its rigorous 1d model
- Author
-
Sergei Kuksin
- Subjects
Physics::Fluid Dynamics ,Nonlinear Sciences::Chaotic Dynamics ,Number theory ,Relation (database) ,Turbulence ,General Mathematics ,Content (measure theory) ,Algebra over a field ,Burgers' equation ,Mathematical physics ,Mathematics - Abstract
This paper is a synopsis of the recent book [9]. The latter is dedicated to the stochastic Burgers equation as a model for 1d turbulence, and the paper discusses its content in relation to the Kolmogorov theory of turbulence.
- Published
- 2021
43. Quantitative subspace theorem and general form of second main theorem for higher degree polynomials
- Author
-
Duc Quang Si
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Subspace theorem ,General Mathematics ,Algebraic geometry ,Diophantine approximation ,Algebraic number field ,Nevanlinna theory ,11J68, 32H30, 11J25, 11J97, 32A22 ,Number theory ,FOS: Mathematics ,Number Theory (math.NT) ,Projective variety ,Meromorphic function ,Mathematics - Abstract
This paper deals with the quantitative Schmidt's subspace theorem and the general from of the second main theorem, which are two correspondence objects in Diophantine approximation theory and Nevanlinna theory. In this paper, we give a new below bound for Chow weight of projective varieties defined over a number field. Then, we apply it to prove a quantitative version of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety. Finally, we apply this new below bound for Chow weight to establish a general form of second main theorem in Nevanlinna theory for meromorphic mappings into projective varieties intersecting hypersurfaces in subgeneral position with a short proof. Our results improve and generalize the previous results in these directions., Comment: 21 pages. arXiv admin note: text overlap with arXiv:math/0408381 by other authors
- Published
- 2021
44. Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions
- Author
-
Bin Han and Ran Lu
- Subjects
FOS: Computer and information sciences ,Pure mathematics ,Information Theory (cs.IT) ,Computer Science - Information Theory ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,42C40, 42C15, 41A25, 41A35, 65T60 ,010103 numerical & computational mathematics ,Spectral theorem ,Trigonometric polynomial ,01 natural sciences ,Hermitian matrix ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Spline (mathematics) ,Wavelet ,Factorization ,FOS: Mathematics ,0101 mathematics ,Vector-valued function ,Mathematics - Abstract
Generalizing wavelets by adding desired redundancy and flexibility, framelets (i.e., wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multiscale representation, fast framelet transforms for numerical efficiency, and redundancy for robustness. However, it is a challenging problem to study and construct multivariate nonseparable framelets, mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices. Moreover, all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment, and framelets derived from refinable vector functions are barely studied yet in the literature. In this paper, we circumvent the above difficulties through the approach of quasi-tight framelets, which behave almost identically to tight framelets. Employing the popular oblique extension principle (OEP), from an arbitrary compactly supported M-refinable vector function ϕ with multiplicity greater than one, we prove that we can always derive from ϕ a compactly supported multivariate quasi-tight framelet such that: (i) all the framelet generators have the highest possible order of vanishing moments; (ii) its associated fast framelet transform has the highest balancing order and is compact. For a refinable scalar function ϕ (i.e., its multiplicity is one), the above item (ii) often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived from ϕ satisfying item (i). We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices. Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter, which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets. This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders. This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.
- Published
- 2021
45. Notes on Functional Integration
- Author
-
A. V. Ivanov
- Subjects
Statistics and Probability ,Algebra ,Applied Mathematics ,General Mathematics ,Product (mathematics) ,Path integral formulation ,Loop space ,Functional derivative ,Functional integration ,Orthonormal basis ,Space (mathematics) ,Special class ,Mathematics - Abstract
The paper is devoted to the construction of an “integral” on an infinite-dimensional space, combining the approaches proposed previously and at the same time the simplest. A new definition of the construction and study its properties on a special class of functionals is given. An introduction of a quasi-scalar product, an orthonormal system, and applications in physics (path integral, loop space, functional derivative) are proposed. In addition, the paper contains a discussion of generalized functionals.
- Published
- 2021
46. A Generalized Self-Adaptive Algorithm for the Split Feasibility Problem in Banach Spaces
- Author
-
Pongsakorn Sunthrayuth and Truong Minh Tuyen
- Subjects
Algebra ,Sequence ,General Mathematics ,Convergence (routing) ,Banach space ,Self adaptive ,Operator norm ,Mathematics - Abstract
In this paper, we propose a generalized self-adaptive method for solving the multiple-set split feasibility problem in the framework of certain Banach spaces. Under some suitable conditions, we prove the strong convergence of the sequence generated by our method with a new way to select the step-sizes without prior knowledge of the operator norm. Several numerical experiments to illustrate the convergence behavior are presented. The results presented in this paper improve and extend the corresponding results in the literature.
- Published
- 2021
47. Geometric law for numbers of returns until a hazard under ϕ-mixing
- Author
-
Fan Yang and Yuri Kifer
- Subjects
Hazard (logic) ,Pure mathematics ,Mixing (mathematics) ,Dynamical systems theory ,Integrable system ,law ,General Mathematics ,Order (ring theory) ,Geometric distribution ,Sequence space ,Mathematics ,Cylinder (engine) ,law.invention - Abstract
We consider a ϕ-mixing shift T on a sequence space Ω and study the number of returns {Tkω ∈ U} to a union U of cylinders of length n until the first return {Tkω ∈ V} to another union V of cylinder sets of length m. It turns out that if probabilities of the sets U and V are small and of the same order, then the above number of returns has approximately geometric distribution. Under appropriate conditions we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymptotical behavior of numbers of returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a “hole”.
- Published
- 2021
48. A description via second degree character of a family of quasi-symmetric forms
- Author
-
Mohamed Khalfallah and Imed Ben Salah
- Subjects
Pure mathematics ,Character (mathematics) ,Degree (graph theory) ,Differential equation ,General Mathematics ,Order (ring theory) ,Semiclassical physics ,Riemann–Stieltjes integral ,Function (mathematics) ,Connection (algebraic framework) ,Mathematics - Abstract
The purpose of this paper is to give, through the second degree character, new characterizations of a part of the family of quasi-symmetric forms. In fact, thanks to the Stieltjes function and also the moments, we give necessary and sufficient conditions for a regular form to be at the same time of the second degree, quasi-symmetric and semiclassical one of class two. We focus our attention not only on the link between all these forms and the Jacobi forms $${{\mathcal {T}}}_{p, q}={{\mathcal {J}}}(p-1/2, q-1/2), \; p, q\in {\mathbb {Z}},~p+q\ge 0$$ but also on their connection with the Tchebychev form of the first kind $${{\mathcal {T}}}={\mathcal J}\left( -1/2, -1/2\right) $$ . The paper concludes by explicitly giving their characteristic elements of the structure relation and of the second order differential equation, which leads to interesting electrostatic models.
- Published
- 2021
49. A note on quasi-bi-slant submanifolds of Sasakian manifolds
- Author
-
Rajendra Prasad and Sandeep Kumar Verma
- Subjects
Pure mathematics ,Generalization ,Computer Science::Computer Vision and Pattern Recognition ,General Mathematics ,Mathematics::History and Overview ,Metric (mathematics) ,Structure (category theory) ,Mathematics::Differential Geometry ,Object (computer science) ,Mathematics::Symplectic Geometry ,Computer Science::Computers and Society ,Mathematics - Abstract
The object of the present paper is to study the notion of quasi-bi-slant submanifolds of almost contact metric manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant, and quasi-hemi-slant submanifolds. We study and characterize quasi-bi-slant submanifolds of Sasakian manifolds and provide non-trivial examples to signify that the structure presented in this paper is valid. Furthermore, the integrability of distributions and geometry of foliations are researched. Moreover, we characterize quasi-bi-slant submanifolds with parallel canonical structures.
- Published
- 2021
50. On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary
- Author
-
Denis Borisov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,Mathematical analysis ,Spectrum (functional analysis) ,Boundary (topology) ,Function (mathematics) ,symbols.namesake ,Amplitude ,Dirichlet boundary condition ,symbols ,Flat band ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude.
- Published
- 2021
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