A Note on the Generalized Nonlinear Vector Variational-Like Inequality Problem

Journal of Function Spaces ◽

10.1155/2021/4488217 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Ankit Gupta ◽

Satish Kumar ◽

Ratna Dev Sarma ◽

Pankaj Kumar Garg ◽

Reny George

Keyword(s):

Function Space ◽

Topological Properties ◽

Kkm Theorem ◽

Topological Approach ◽

Inequality Problem ◽

Space Topology ◽

Nonlinear Vector

In this paper, we discuss two variants of the generalized nonlinear vector variational-like inequality problem. We provide their solutions by adopting topological approach. Topological properties such as compactness, closedness, and net theory are used in the proof. The admissibility of the function space topology and KKM-Theorem have played important role in proving the results.

Abstract Köthe spaces. IV

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100042547 ◽

1968 ◽

Vol 64 (1) ◽

pp. 45-52

Author(s):

D. H. Fremlin

Keyword(s):

Tensor Product ◽

Function Space ◽

Function Spaces ◽

Riesz Space ◽

Natural Order ◽

Order Structure ◽

Topological Properties ◽

Projective Tensor Product ◽

Köthe Spaces ◽

Projective Tensor

In this paper I investigate the completed projective tensor product of two perfect Riesz spaces, and show how a natural order structure on this renders it also a perfect Riesz space. Sections 7–14 contain interesting order-topological properties of this tensor product. Finally, section 15 describes how the tensor product of function spaces may be represented as a function space, in the manner of (1 b).

On Certain Topological Properties of Normed Space Valued Null Function Space c0 (S, (E, || . || ), xi, u) and Its Paranormed Structure

Nepal Journal of Science and Technology ◽

10.3126/njst.v15i1.12028 ◽

2015 ◽

Vol 15 (1) ◽

pp. 121-128

Author(s):

Narayan Prasad Pahari

Keyword(s):

Functional Analysis ◽

Function Space ◽

Normed Space ◽

Function Spaces ◽

Sequence Spaces ◽

Basic Sequence ◽

Topological Properties ◽

New Class ◽

And Function ◽

Class Of Functions

The aim of this paper is to introduce and study a new class c0 (S, (E, || . || ), ξ, u) of normed space E valued functions which will generalize some of the well known basic sequence spaces and function spaces studied in Functional Analysis.. Beside the investigation pertaining to the linear paranormed structure of the class c0 ( S, (E, || . || ), ξ, u ) when topologized it with suitable natural paranorm , our primarily interest is to explore the conditions pertaining the containment relation of the class c0 (S, (E, || . || ), ξ, u) in terms of different ξ and u so that such a class of functions is contained in or equal to another class of similar nature.DOI: http://dx.doi.org/10.3126/njst.v15i1.12028Nepal Journal of Science and TechnologyVol. 15, No.1 (2014) 121-128

Betti splitting from a topological point of view

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501169 ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050116

Author(s):

Davide Bolognini ◽

Ulderico Fugacci

Keyword(s):

Monomial Ideal ◽

Betti Numbers ◽

Simplicial Complexes ◽

Point Of View ◽

Topological Spaces ◽

Topological Properties ◽

Topological Approach ◽

Graded Betti Numbers ◽

Alexander Duality ◽

Splitting Condition

A Betti splitting [Formula: see text] of a monomial ideal [Formula: see text] ensures the recovery of the graded Betti numbers of [Formula: see text] starting from those of [Formula: see text] and [Formula: see text]. In this paper, we introduce an analogous notion for simplicial complexes, using Alexander duality, proving that it is equivalent to a recursive splitting condition on links of some vertices. We provide results ensuring the existence of a Betti splitting for a simplicial complex [Formula: see text], relating it to topological properties of [Formula: see text]. Among other things, we prove that orientability for a manifold without boundary is equivalent to the admission of a Betti splitting induced by the removal of a single facet. Taking advantage of our topological approach, we provide the first example of a monomial ideal which admits Betti splittings in all characteristics but with characteristic-dependent resolution. Moreover, we introduce new numerical descriptors for simplicial complexes and topological spaces, useful to deal with questions concerning the existence of Betti splitting.

A Topological Approach in the Extended Fraenkel-Mostowski Model of Set Theory

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/aicu-2013-0029 ◽

2014 ◽

Vol 60 (2) ◽

pp. 261-277

Author(s):

Andrei Alexandru ◽

Gabriel Ciobanu

Keyword(s):

Set Theory ◽

Continuous Functions ◽

New Order ◽

Scott Topology ◽

Topological Properties ◽

Arbitrary Group ◽

Topological Approach ◽

Lattices Of Subgroups ◽

Group A ◽

Algebraic Domains

Abstract Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroups lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remains also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.

Finite approximation of stably compact spaces

Applied General Topology ◽

10.4995/agt.2002.2063 ◽

2002 ◽

Vol 3 (2) ◽

pp. 197 ◽

Cited By ~ 4

Author(s):

M.B. Smyth ◽

J. Webster

Keyword(s):

Function Space ◽

Directed Graphs ◽

New Method ◽

Hausdorff Spaces ◽

Topological Graphs ◽

Topological Approach ◽

Compact Spaces ◽

Finite Directed Graphs ◽

Finite Approximation ◽

Inverse Spectra

<p>Finite approximation of spaces by inverse sequences of graphs (in the category of so-called topological graphs) was introduced by Smyth, and developed further. The idea was subsequently taken up by Kopperman and Wilson, who developed their own purely topological approach using inverse spectra of finite T<sub>0</sub>-spaces in the category of stably compact spaces. Both approaches are, however, restricted to the approximation of (compact) Hausdorff spaces and therefore cannot accommodate, for example, the upper space and (multi-) function space constructions. We present a new method of finite approximation of stably compact spaces using finite stably compact graphs, which when the topology is discrete are simply finite directed graphs. As an extended example, illustrating the problems involved, we study (ordered spaces and) arcs.</p>

Electron Microscopy of Nucleic Acids

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100051529 ◽

1974 ◽

Vol 32 ◽

pp. 302-303

Author(s):

Norman Davidson

Keyword(s):

Electron Microscopy ◽

Nucleic Acids ◽

Basic Protein ◽

Molecular Biologist ◽

Topological Properties ◽

Protein Film ◽

Polynucleotide Chain

The basic protein film technique for mounting nucleic acids for electron microscopy has proven to be a general and powerful tool for the working molecular biologist in characterizing different nucleic acids. It i s possible to measure molecular lengths of duplex and single-stranded DNAs and RNAs. In particular, it is thus possible to as certain whether or not the nucleic acids extracted from a particular source are or are not homogeneous in length. The topological properties of the polynucleotide chain (linear or circular, relaxed or supercoiled circles, interlocked circles, etc. ) can also be as certained.

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010018135x ◽

1990 ◽

Vol 48 (1) ◽

pp. 520-521

Author(s):

Neng-Yu Zhang ◽

Bruce F. McEwen ◽

Joachim Frank

Keyword(s):

Function Space ◽

Convex Sets ◽

Electron Tomography ◽

Spinal Ganglion ◽

Fourier Space ◽

Missing Information ◽

Voltage Electron ◽

High Voltage Electron Microscope ◽

Energy Bound ◽

Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

Topological properties of cellular structures based on the staggered packing of prisms

Journal de Physique ◽

10.1051/jphys:01989005007072500 ◽

1989 ◽

Vol 50 (7) ◽

pp. 725-731 ◽

Cited By ~ 8

Author(s):

M. A. Fortes

Keyword(s):

Cellular Structures ◽

Topological Properties

Modification of Fuzzy c-Means Method Using a Nonlinear Vector Criterion

Journal of Automation and Information Sciences ◽

10.1615/jautomatinfscien.v42.i12.20 ◽

2010 ◽

Vol 42 (12) ◽

pp. 13-21

Author(s):

Anatoliy F. Bulat ◽

Elena M. Kiseleva ◽

Sergey A. Pichugov ◽

Oleg B. Blyuss

Keyword(s):

Vector Criterion ◽

Fuzzy C Means ◽

Nonlinear Vector

Finite-order Integration Weights Can be Dangerous

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2007-0015 ◽

2007 ◽

Vol 7 (3) ◽

pp. 239-254 ◽

Cited By ~ 3

Author(s):

I.H. Sloan

Keyword(s):

Function Space ◽

Finite Order ◽

Small Subset ◽

Worst Case ◽

Integration Rule ◽

Dimensional Lattice ◽

3 Dimensional ◽

Sum Of Functions ◽

Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.