Lower Bound to the nth Eigenvalue of the Helmholtz Equation Over Two-Dimensional Regions of Arbitrary Shape

1969 ◽  
Vol 36 (3) ◽  
pp. 630-631 ◽  
Author(s):  
David Pnueli
Keyword(s):  
Lower Bound ◽  
Helmholtz Equation ◽  
Arbitrary Shape ◽  
Two Dimensional

A method is presented to compute a lower bound to the nth eigenvalue of the Helmholtz equation over two-dimensional regions. The shape of the regions is arbitrary and only their area need be known.

Lower Bounds to the Nth Eigenvalue of the Helmholz Equation Over Three-Dimensional Regions of Arbitrary Shape

1974 ◽  
Vol 41 (3) ◽  
pp. 819-820
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Helmholtz Equation ◽  
Lower Bounds ◽  
Arbitrary Shape ◽  
Three Dimensional ◽  
Helmholz Equation

A method is presented to compute a lower bound to the nth eigenvalue of the Helmholtz equation over three-dimensional regions. The shape of the regions is arbitrary and only their volume need be known.

scholarly journals Encoding Two-Dimensional Range Top-k Queries

Algorithmica ◽  
2021 ◽  
Author(s):  
Seungbum Jo ◽  
Rahul Lingala ◽  
Srinivasa Rao Satti
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Total Order ◽  
Two Dimensional ◽  
Information Theoretic ◽  
Cartesian Tree ◽  
Dimensional Range

AbstractWe consider the problem of encoding two-dimensional arrays, whose elements come from a total order, for answering $${\text{Top-}}{k}$$ Top- k queries. The aim is to obtain encodings that use space close to the information-theoretic lower bound, which can be constructed efficiently. For an $$m \times n$$ m × n array, with $$m \le n$$ m ≤ n , we first propose an encoding for answering 1-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, whose query range is restricted to $$[1 \dots m][1 \dots a]$$ [ 1 ⋯ m ] [ 1 ⋯ a ] , for $$1 \le a \le n$$ 1 ≤ a ≤ n . Next, we propose an encoding for answering for the general (4-sided) $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries that takes $$(m\lg {{(k+1)n \atopwithdelims ()n}}+2nm(m-1)+o(n))$$ ( m lg ( k + 1 ) n n + 2 n m ( m - 1 ) + o ( n ) ) bits, which generalizes the joint Cartesian tree of Golin et al. [TCS 2016]. Compared with trivial $$O(nm\lg {n})$$ O ( n m lg n ) -bit encoding, our encoding takes less space when $$m = o(\lg {n})$$ m = o ( lg n ) . In addition to the upper bound results for the encodings, we also give lower bounds on encodings for answering 1 and 4-sided $${\textsf {Top}}{\text {-}}k{}$$ Top - k queries, which show that our upper bound results are almost optimal.

A new lower bound for the non-oriented two-dimensional bin-packing problem

2007 ◽  
Vol 35 (3) ◽  
pp. 365-373 ◽  
Author(s):  
François Clautiaux ◽  
Antoine Jouglet ◽  
Joseph El Hayek
Keyword(s):  
Lower Bound ◽  
Bin Packing ◽  
Packing Problem ◽  
Two Dimensional ◽  
Bin Packing Problem

D-C Network-Analyzer Determination of Fluid-Flow Pattern in a Centrifugal Impeller

1947 ◽  
Vol 14 (2) ◽  
pp. A113-A118
Author(s):  
C. Concordia ◽  
G. K. Carter
Keyword(s):  
Fluid Flow ◽  
Flow Pattern ◽  
Arbitrary Shape ◽  
Centrifugal Impeller ◽  
Network Analyzer ◽  
Two Dimensional ◽  
Special Shape ◽  
Electrical Method ◽  
Incompressible Ideal Fluid

Abstract The objects of this paper are, first, to describe an electrical method of determining the flow pattern for the flow of an incompressible ideal fluid through a two-dimensional centrifugal impeller, and second, to present the results obtained for a particular impeller. The method can be and has been applied to impellers with blades of arbitrary shape, as distinguished from analytical methods which can be applied directly only to blades of special shape (1).

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