Normal subgroups in the group of column-finite infinite matrices

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

Encoding Two-Dimensional Range Top-k Queries

We 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.

Fixed Point Processing of the SAR Back Projection Algorithm on FPGA

Time-domain back projection (BP) is a widely known method used in Synthetic Aperture Radar (SAR) image formation. Despite its advantages over other image formation algorithms, the BP method is hindered due to its computational complexity and its requirement of higher number of operations and processing power. Recently, Field Programmable Gate Array (FPGA) devices have been used for BP acceleration mainly due to their parallel processing capabilities, reconfigurability, scalability, and low power requirement. This paper presents a new Fixed-point based BP (FxBP) design for FPGA devices and a Floating-point based BP (FlBP) design to compare performance. Both designs are developed with N-Dimensional Range (NDR) structure and Single Work Item (SWI) structure using OpenCL. The FPGA performance is evaluated using a FPGA performance metric (FPM). It is shown that FxBP-NDR and FxBP-SWI designs generate high quality back projected images compared to FlBP designs, while saving 16.87 % and 42.54 % on logic resources and gaining 17.90 % and 91.62 % on FPGA performance in NDR and SWI, respectively. Obtained results clearly indicate that FPGA devices perform significantly better with FxBP designs compared to FlBP designs, even with hardened FPUs.

Fixed Point Processing of the SAR Back Projection Algorithm on FPGA

Time-domain back projection (BP) is a widely known method used in Synthetic Aperture Radar (SAR) image formation. Despite its advantages over other image formation algorithms, the BP method is hindered due to its computational complexity and its requirement of higher number of operations and processing power. Recently, Field Programmable Gate Array (FPGA) devices have been used for BP acceleration mainly due to their parallel processing capabilities, reconfigurability, scalability, and low power requirement. This paper presents a new Fixed-point based BP (FxBP) design for FPGA devices and a Floating-point based BP (FlBP) design to compare performance. Both designs are developed with N-Dimensional Range (NDR) structure and Single Work Item (SWI) structure using OpenCL. The FPGA performance is evaluated using a FPGA performance metric (FPM). It is shown that FxBP-NDR and FxBP-SWI designs generate high quality back projected images compared to FlBP designs, while saving 16.87 % and 42.54 % on logic resources and gaining 17.90 % and 91.62 % on FPGA performance in NDR and SWI, respectively. Obtained results clearly indicate that FPGA devices perform significantly better with FxBP designs compared to FlBP designs, even with hardened FPUs.