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Page 973 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 12, Issue 6 [page]

1961 American Mathematical Society. Proceedings of the American Mathematical Society  
Let V and V’ be real or complex linear spaces and F a separating family of linear functionals on V’.  ...  Denote by #’ the weak-F topology on V’; and let S be a subset of V which spans it linearly, S’ a t/-compact subset of V’, $ the set of linear transformations of V into V’ which map S into S’, and ¢ the  ... 

Page 501 of Mathematical Reviews Vol. 57, Issue 2 [page]

1979 Mathematical Reviews  
R. 3810 Some new linear topological spaces (of operators), with trivial and non-trivial duals. Nanta Math. 10 (1977), no. 1, 72-76.  ...  Two compatible locally convex (LC) topologies 7, 7, on E are termed G-equivalent if the spaces P(E, 7), G) and L((E, 72), G) of continuous linear maps coincide.  ... 

Comparison of ReLU and linear saturated activation functions in neural network for universal approximation

Dominik Stursa, Petr Dolezel
2019 2019 22nd International Conference on Process Control (PC19)  
This paper aims to compare two types of linear based functions -symmetric linear saturated function and the rectifier linear unit (ReLU) function as activation functions of the feedforward neural network  ...  Topologies with one hidden layer and the combination of defined quantities of hidden layer neurons in the feedforward neural network are used.  ...  To be more specific, a comparison of symmetric linear saturated and rectified linear unit activation functions for the purposes of nonlinear system approximation is examined.  ... 
doi:10.1109/pc.2019.8815057 fatcat:dcl5l4dglrewtjpnjk6t346goq

Linear-size approximations to the vietoris-rips filtration

Donald R. Sheehy
2012 Proceedings of the 2012 symposuim on Computational Geometry - SoCG '12  
Use packing arguments to get linear size. Union trick eliminates zigzag. The Future Distance to a measure? Other types of simplification.  ...  Use packing arguments to get linear size. Union trick eliminates zigzag. The Future Distance to a measure? Other types of simplification. Construct the net-tree in O(n log n) time.  ...  Persistence Diagrams describe the changes in topology corresponding to changes in scale. d ∞ B = max i |p i − q i | ∞ Bottleneck Distance In approximate persistence diagrams, birth and death times differ  ... 
doi:10.1145/2261250.2261286 dblp:conf/compgeom/Sheehy12 fatcat:a2hiocle7zdktpybotlj7uitwe

Scaling the Topology of Symmetric, Second-Order Planar Tensor Fields [chapter]

Xavier Tricoche, Gerik Scheuermann, Hans Hagen
2003 Mathematics and Visualization  
This results in a simplified topology and a clarified depiction, though globally maintaining the qualitative properties of the original data.  ...  The whole process can be seen as a scaling of the topological information that neglects details of small scale and only retains their structural aspect in the large.  ...  Furthermore, we would like to thank Tom Bobach, Holger Burbach, Stefan Clauss, Jan Frey, Aragorn Rockstroh, René Schätzl and Thomas Wischgoll for their programming efforts.  ... 
doi:10.1007/978-3-642-55787-3_10 fatcat:nyzpi3wnrfhx3hgeigjtjlxf74

Approximation Properties of Locally Convex Spaces and the Problem of Uniqueness of the Trace of a Linear Operator [article]

Grigory L. Litvinov
2011 arXiv   pre-print
A number of approximation-type properties of locally convex spaces and relations between these properties are considered in Section 3.  ...  We prove that the uniqueness problem has a positive solution if E possesses the "bounded approximation property." Preliminary information and results are presented in Section 2.  ...  Approximation-type properties of locally convex spaces 3.1.  ... 
arXiv:1108.1721v1 fatcat:7hqa4knnwzfwbay2xwwxwi7dwy

Page 4967 of Mathematical Reviews Vol. , Issue 83m [page]

1983 Mathematical Reviews  
Studying the duality between Lp.s. and locally convex linear topological spaces (1.c.t.s.)  ...  Author’s summary: “We consider some classes of linear pseudo- topological spaces (1.p.s.) which are important both in the theory itself and in its applications.  ... 

Page 4261 of Mathematical Reviews Vol. , Issue 87h [page]

1987 Mathematical Reviews  
In Part 1, Cristescu presents some classes of linear operators on ordered linear spaces of various types, especially topological ordered linear spaces.  ...  Kajiwara, On the trace on C*-crossed products (Japanese) (pp. 110-117); Hideo Takemoto, Approximate innerness of positive linear maps of fac- tors of type II (pp. 118-127); Masamichi Hamana, Injective  ... 

A Weierstrass approximation theorem for topological vector spaces

Vincent J Bruno
1984 Journal of Approximation Theory  
Before stating the main theorem let us make the following definition: A sequence of linear operators (t,) on a topological vector space has the Grothendieck approximation propety if and only if (1) each  ...  Let E and F be real Hausdorff topological vector spaces.  ...  Before stating the main theorem let us make the following definition: A sequence of linear operators (t,) on a topological vector space has the Grothendieck approximation propety if and only if (1) each  ... 
doi:10.1016/0021-9045(84)90049-2 fatcat:kf77dmajpnbvhciotcv4sejnha

On the computational algorithms for time-lag optimal control problems

Kar Hung Wong
1985 Bulletin of the Australian Mathematical Society  
Chapters I I , I I I , IV, and V deal with problems of the f i r s t type and Chapter VI deals with problems of the second type.  ...  The type of convergence of this algorithm is as follows:  ...  of L^ and in the almost everywhere topology.  ... 
doi:10.1017/s0004972700009989 fatcat:p46unlbyunaapma2t3l6cpi2jy

Countable approximation of topological G -manifolds: Compact Lie groups G

Qayum Khan
2018 Topology and its Applications  
(Compact) topological G-manifolds have the G-homotopy type of (finite-dimensional) countable G-CW complexes (2.5).  ...  This partly generalizes Elfving's theorem for locally linear G-manifolds [Elf96], wherein the Lie group G is linear (such as compact).  ...  Elmar Vogt (Freie U Berlin) excited me further into classical topology (ANRs and Lemma 4.1).  ... 
doi:10.1016/j.topol.2017.11.001 fatcat:un7tc5b7lvar5hftlg3q6r32mm

Page 124 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 59, Issue 2 [page]

1953 American Mathematical Society. Bulletin of the American Mathematical Society  
Both of these are special cases of linear topological spaces, spaces which are linear and have a topology in which addi- tion and multiplication by real numbers is continuous, and which are determined  ...  For ¥* we have two types of weak convergence, that induced by % and that by ¥**.  ... 

On some generalizations of Newton non-degeneracy for hypersurface singularities

Dmitry Kerner
2010 Journal of the London Mathematical Society  
Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton  ...  For such singularities the classical formulas for the Milnor number and the zeta function of the Nnd hypersurface are generalized.  ...  Assume also that the directional approximations are of linear type [Kerner06] , i.e. any representative of the type of C (i) can be brought to Γ i by linear transformations only.  ... 
doi:10.1112/jlms/jdq011 fatcat:qttuutxdvbdhtbz5xkeoqndwhm

Topology Aware Stream Surfaces

Dominic Schneider, Wieland Reich, Alexander Wiebel, Gerik Scheuermann
2010 Computer graphics forum (Print)  
We present an algorithm that allows stream surfaces to recognize and adapt to vector field topology.  ...  Standard stream surface algorithms either refine the surface uncontrolled near critical points which slows down the computation considerably and may lead to a poor surface approximation.  ...  Acknowledgments We wish to thank Markus Rütten from German Aerospace Center (DLR) in Göttingen for the Delta Wing dataset, and Ronald Peikert and VA Tech for the Draft Tube dataset.  ... 
doi:10.1111/j.1467-8659.2009.01672.x fatcat:3yqi2mmhcnevngqs2y7mui54zu

Page 39 of American Mathematical Society. Proceedings of the American Mathematical Society Vol. 11, Issue 1 [page]

1960 American Mathematical Society. Proceedings of the American Mathematical Society  
The answer is affirmative for the following types of domain and range spaces: (i) both A and B are Hilbert spaces (see [2, p. 204]), (ii) both A and B are C[a, 6] (see [2, p. 222] or [3]), (iii) there  ...  Can T be approximated arbitrarily close in norm by bounded linear transformations whose ranges are finite dimen- sional (see [1, p. 49])?  ... 
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