scholarly journals Compactifications of manifolds with boundary

2018 ◽  
Vol 12 (04) ◽  
pp. 1073-1101
Author(s):  
Shijie Gu ◽  
Craig R. Guilbault
Keyword(s):  
Sufficient Conditions ◽  
Acta Math ◽  
Complete Characterization ◽  
Open Manifold ◽  
Important Special Case ◽  
Initial Work ◽  
Hilbert Cube Manifold ◽  
Necessary And Sufficient ◽  
Open Question ◽  
Completion Theorem

This paper is concerned with compactifications of high-dimensional manifolds. Siebenmann’s iconic 1965 dissertation [L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension greater than five, Ph.D. thesis, Princeton Univ. (1965), MR 2615648] provided necessary and sufficient conditions for an open manifold [Formula: see text] ([Formula: see text]) to be compactifiable by addition of a manifold boundary. His theorem extends easily to cases where [Formula: see text] is noncompact with compact boundary; however, when [Formula: see text] is noncompact, the situation is more complicated. The goal becomes a “completion” of [Formula: see text], i.e. a compact manifold [Formula: see text] containing a compactum [Formula: see text] such that [Formula: see text]. Siebenmann did some initial work on this topic, and O’Brien [G. O’Brien, The missing boundary problem for smooth manifolds of dimension greater than or equal to six, Topology Appl. 16 (1983) 303–324, MR 722123] extended that work to an important special case. But, until now, a complete characterization had yet to emerge. Here, we provide such a characterization. Our second main theorem involves [Formula: see text]-compactifications. An important open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann [T. A. Chapman and L. C. Siebenmann, Finding a boundary for a Hilbert cube manifold, Acta Math. 137 (1976) 171–208, MR 0425973] guarantee [Formula: see text]-compactifiability for a manifold [Formula: see text]. We cannot answer that question, but we do show that those conditions are satisfied if and only if [Formula: see text] is [Formula: see text]-compactifiable. A key ingredient in our proof is the above Manifold Completion Theorem — an application that partly explains our current interest in that topic, and also illustrates the utility of the [Formula: see text]-condition found in that theorem.

scholarly journals Multigenerator Gabor Frames on Local Fields

2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Gabor System ◽  
Necessary And Sufficient

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

On *-clean group rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Yuanlin Li ◽  
M. M. Parmenter ◽  
Pingzhi Yuan
Keyword(s):  
Local Ring ◽  
Prime Order ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Group Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Irreducible Factorization ◽  
Necessary And Sufficient

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

Stability of a thermoelastic mixture with second sound

2018 ◽  
Vol 24 (6) ◽  
pp. 1692-1706 ◽  
Author(s):  
Margareth S. Alves ◽  
Marcio V. Ferreira ◽  
Jaime E. Muñoz Rivera ◽  
O. Vera Villagrán
Keyword(s):  
Initial Data ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Second Sound ◽  
Dimensional Model ◽  
One Dimensional ◽  
The One ◽  
Necessary And Sufficient

We consider the one-dimensional model of a thermoelastic mixture with second sound. We give a complete characterization of the asymptotic properties of the model in terms of the coefficients of the model. We establish the necessary and sufficient conditions for the model to be exponential or polynomial stable and also the conditions for which there exist initial data for where the energy is conserved.

scholarly journals The Schur and (weak) Dunford-Pettis properties in Banach lattices

2002 ◽  
Vol 73 (2) ◽  
pp. 251-278 ◽  
Author(s):  
Anna Kamińska ◽  
Mieczysław Mastyło
Keyword(s):  
Orlicz Spaces ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Sequence Spaces ◽  
Banach Lattices ◽  
Schur Property ◽  
Atomic Measure ◽  
Orlicz Sequence Spaces ◽  
Necessary And Sufficient

AbstractWe study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that l1, c0 and l∞ are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class ∧. We also present examples of weighted Orlicz spaces with the Schur property which are not L1-spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.

Descent in Buildings (AM-190)

2017 ◽  
Author(s):  
Bernhard Mühlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Algebraic Solution ◽  
Affine Buildings ◽  
The Third ◽  
Final Volume ◽  
Geometric Context ◽  
Necessary And Sufficient ◽  
Open Question

This book begins with the resolution of a major open question about the local structure of Bruhat-Tits buildings. It then puts forward an algebraic solution into a geometric context by developing a general fixed point theory for groups acting on buildings of arbitrary type, giving necessary and sufficient conditions for the residues fixed by a group to form a kind of subbuilding or “form” of the original building. At the center of this theory is the notion of a Tits index, a combinatorial version of the notion of an index in the relative theory of algebraic groups. These results are combined at the end to show that every exceptional Bruhat-Tits building arises as a form of a “residually pseudo-split” building. The book concludes with a display of the Tits indices associated with each of these exceptional forms. This is the third and final volume of a trilogy that began with The Structure of Spherical Buildings and The Structure of Affine Buildings.

scholarly journals Real Representation Approach to Quaternion Matrix Equation Involving ϕ-Hermicity

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Xin Liu ◽  
Huajun Huang ◽  
Zhuo-Heng He
Keyword(s):  
Matrix Equation ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Quaternion Matrix ◽  
Real Representation ◽  
Quaternion Matrix Equation ◽  
The Matrix ◽  
Hermitian Solution ◽  
Necessary And Sufficient

For a quaternion matrix A, we denote by Aϕ the matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a nonstandard involution of quaternions. A is said to be ϕ-Hermitian or ϕ-skew-Hermitian if A=Aϕ or A=−Aϕ, respectively. In this paper, we give a complete characterization of the nonstandard involutions ϕ of quaternions and their conjugacy properties; then we establish a new real representation of a quaternion matrix. Based on this, we derive some necessary and sufficient conditions for the existence of a ϕ-Hermitian solution or ϕ-skew-Hermitian solution to the quaternion matrix equation AX=B. Moreover, we give solutions of the quaternion equation when it is solvable.

A complete characterization of the existence of rational functional solutions with infinite support bases

2016 ◽  
Vol 15 (09) ◽  
pp. 1650178 ◽  
Author(s):  
Lan Nguyen
Keyword(s):  
Rational Function ◽  
Characteristic Zero ◽  
Polynomial Solution ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Polynomial Solutions ◽  
Function Solution ◽  
Necessary And Sufficient

It is known that there exist polynomial solutions [Formula: see text] with infinite support base [Formula: see text], of certain functional equations arising from quantum arithmetics, which cannot be constructed from quantum integers. A description of the necessary and sufficient conditions on a set of primes [Formula: see text] for the existence of a polynomial solution, with field of coefficients of characteristic zero and support base [Formula: see text], which cannot be constructed from quantum integers is also known, leading to the classification of the set of polynomial solutions. In his papers on quantum arithmetics, Melvyn Nathanson raises a question concerning the classification of the possibly non-trivially broader set of solutions, namely the set of rational function solutions. It is not known at the time that the set of rational function solutions is more than just the set of ratio of polynomial solutions. However, it is now known that there are infinitely many rational function solutions [Formula: see text], with support base [Formula: see text] and field of coefficients of characteristic zero, which are not ratios of polynomial solutions with the same support base, even in the purely cyclotomic case. Thus, a natural question that should be asked in order to classify the set of rational function solutions, is: If polynomial solutions are replaced by merely rational function solutions, what would the necessary and sufficient conditions be on the support base [Formula: see text]? In this paper, we give a complete description of the necessary and sufficient conditions on the set of primes [Formula: see text] for the existence of a rational function solution, with field of coefficients of characteristic zero and support base [Formula: see text], which cannot be constructed from quantum integers.

scholarly journals Eventual domination for linear evolution equations

2021 ◽  
Author(s):  
Jochen Glück ◽  
Delio Mugnolo
Keyword(s):  
Evolution Equations ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Abstract Theory ◽  
Important Special Case ◽  
Linear Evolution ◽  
Necessary And Sufficient ◽  
Adjoint Operators ◽  
Linear Evolution Equations ◽  
Special Case

AbstractWe consider two $$C_0$$ C 0 -semigroups on function spaces or, more generally, Banach lattices and give necessary and sufficient conditions for the orbits of the first semigroup to dominate the orbits of the second semigroup for large times. As an important special case we consider an $$L^2$$ L 2 -space and self-adjoint operators A and B which generate $$C_0$$ C 0 -semigroups; in this situation we give criteria for the existence of a time $$t_1 \ge 0$$ t 1 ≥ 0 such that $$e^{tB} \ge e^{tA}$$ e tB ≥ e tA for all subsequent times $$t\ge t_1$$ t ≥ t 1 . As a consequence of our abstract theory, we obtain many surprising insights into the behaviour of various second and fourth order differential operators.

scholarly journals Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type

2020 ◽  
Vol 18 (1) ◽  
pp. 1814-1829
Author(s):  
Simin Liu ◽  
Hongyan Xu ◽  
Yongqin Cui ◽  
Pan Gong
Keyword(s):  
Half Plane ◽  
Analytic Functions ◽  
Sufficient Conditions ◽  
Acta Math ◽  
General Function ◽  
Stieltjes Transform ◽  
The Right ◽  
Necessary And Sufficient ◽  
Formula Type ◽  
Stieltjes Transforms

Abstract In this paper, we introduce the concept of the perfect ϕ \phi -type to describe the growth of the maximal molecule of Laplace-Stieltjes transform by using the more general function than the usual. Based on this concept, we investigate the approximation and growth of analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms convergent in the half plane and obtain some results about the necessary and sufficient conditions on analytic functions F ( s ) F(s) defined by Laplace-Stieltjes transforms with perfect ϕ \phi -type, which are some generalizations and improvements of the previous results given by Kong [On generalized orders and types of Laplace-Stieltjes transforms analytic in the right half-plane, Acta Math. Sin. 59A (2016), 91–98], Singhal and Srivastava [On the approximation of an analytic function represented by Laplace-Stieltjes transformations, Anal. Theory and Appl. 31 (2015), 407–420].

scholarly journals CENTER PROBLEM FOR THIRD-ORDER ODEs

2013 ◽  
Vol 23 (05) ◽  
pp. 1350078 ◽  
Author(s):  
ADAM MAHDI
Keyword(s):  
Center Manifold ◽  
Sufficient Conditions ◽  
Positive Answer ◽  
Necessary And Sufficient Conditions ◽  
Center Problem ◽  
Local Center ◽  
Third Order ◽  
Quadratic Systems ◽  
Necessary And Sufficient ◽  
Open Question

We obtain necessary and sufficient conditions for the existence of a center on a local center manifold for three 4-parameter families of quadratic systems on ℝ3. We also give a positive answer to an open question posed in [Dias & Mello, 2010] related to similar systems.

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