A New Construction in Homological Algebra Buchbaum, D. A.; Rota, Gian-Carlo
Proceedings of the National Academy of Sciences - PNAS,
05/1994, Volume:
91, Issue:
10
Journal Article
Peer reviewed
Open access
We present a generalization of the classical bar construction with applications to resolutions of Weyl modules.
Let A and B be rings, U a (B,A)-bimodule, and T=A0UB the triangular matrix ring. In this paper, several notions in relative Gorenstein algebra over a triangular matrix ring are investigated. We first ...study how to construct w-tilting (tilting, semidualizing) over T using the corresponding ones over A and B. We show that when U is relative (weakly) compatible, we are able to describe the structure of GC-projective modules over T. As an application, we study when a morphism in T-Mod is a special GCP(T)-precover and when the class GCP(T) is a special precovering class. In addition, we study the relative global dimension of T. In some cases, we show that it can be computed from the relative global dimensions of A and B. We end the paper with a counterexample to a result that characterizes when a T-module has a finite projective dimension.
Secure Function Evaluation (SFE) has received recent attention due to the massive collection and mining of personal data, but remains impractical due to its large computational cost. Garbled Circuits ...(GC) is a protocol for implementing SFE which can evaluate any function that can be expressed as a Boolean circuit and obtain the result while keeping each party’s input private. Recent advances have led to a surge of garbled circuit implementations in software for a variety of different tasks. However, these implementations are inefficient, and therefore GC is not widely used, especially for large problems. This research investigates, implements, and evaluates secure computation generation using a heterogeneous computing platform featuring FPGAs. We have designed and implemented SIFO: secure computational infrastructure using FPGA overlays. Unlike traditional FPGA design, a coarse-grained overlay architecture is adopted which supports mapping SFE problems that are too large to map to a single FPGA. Host tools provided include SFE problem generator, parser, and automatic host code generation. Our design allows repurposing an FPGA to evaluate different SFE tasks without the need for reprogramming and fully explores the parallelism for any GC problem. Our system demonstrates an order of magnitude speedup compared with an existing software platform.
We introduce the notion of an ‘inverse property’ (IP) quandle C which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group GC. ...For a class of IP-quandles which we call ‘locally skew’, and when GC is finite, we show that the noncommutative de Rham cohomology H1(GC) is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset C⊆G∖{e} which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If C also generates G then we show that GC↠G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that this ‘covering map’ GC↠G is an isomorphism for all finite crystallographic reflection groups W with C the set of reflections, and that C is locally skew precisely in the simply laced case. This implies that H1(W)=k when W is simply laced, proving in particular a conjecture for Sn in Majid (2004) 12. We also consider C=ZP1∪ZP1 as a locally skew IP-quandle ‘Lie algebra’ of SL2(Z) and show that GC≅B3, the braid group on 3 strands. The map B3↠SL2(Z) which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map SL2(R)˜→SL2(R) to the inverse image of SL2(Z).
The canonical structure of DNA has four bases – Thymine (T), Adenine (A), Cytosine (C), and Guanine (G) – and DNA codes are regarded as words over the alphabet set Σ={A,C,G,T}, satisfying certain ...combinatorial conditions. Good DNA codes are desirable for DNA computation, DNA microarray technologies and molecular barcodes, etc. One of the main tasks in DNA code designing is to build more codewords and better GC-content for given fixed word length n. Existing heuristic methods work well for small n. In this paper, we present a systematic method for constructing good DNA codes for large n by using irreducible cyclic codes. Being different from traditional DNA constructions, our method is based on algebraic number theory rather than classical heuristic algorithms and the conventional coding theory. Furthermore, comparing with the traditional DNA codes, our codes have larger number of codewords and better GC-content. As far as we know, it is the very first time to utilize irreducible cyclic codes for constructing a type of DNA codes.
Greenberg’s well known conjecture, (GC) for short, asserts that the Iwasawa invariants
λ
and
μ
associated to the cyclotomic
Z
p
-extension of any totally real number field
F
should vanish. In his ...foundational 1976 paper, Greenberg has shown two necessary and sufficient conditions for (GC) to hold, in two seemingly opposite cases, when
p
is undecomposed, resp. totally decomposed in
F
. In this article we present an encompassing approach covering both cases and resting only on “ genus formulas ”, that is (roughly speaking) on formulas which express the order of the Galois (co-)invariants of certain modules along the cyclotomic tower. These modules are akin to class groups, and in the end we obtain several unified criteria, which naturally contain the particular conditions given by Greenberg.
Identifying geochemical patterns from backgrounds and generating associated mineralization remains challenging due to the complex structure of mineral deposits. To learn how to identify geochemical ...anomalies that are spatially associated with mineralization, we need in-depth knowledge of the dependence process. Quantitative association rules (QARs) are applied to discover remarkable relations and dependencies between attributes in a dataset, but it is difficult to generate relationships from geochemical data. In previous studies, no methodology to find association rules is proposed to deal with geochemical data problem, and the classical methods designed for Boolean and nominal attributes require previous discretization, which makes the whole process limited in processing complex data. In this paper, we proposed a hybrid method of graph clustering and quantitative association rules (GCQAR) as a new way of identifying significant geochemical patterns. Graph Clustering (GC) is used as partitioning paradigm because of its ability to handle large-scale datasets. The GC is based on modularity to effectively generate the groups of the graph, to avoid the over-partitioning, and to cover all the rules. In each partition, a set of geochemical quantitative association rules is produced. The results obtained in the experimental study performed on data collected in the field of Xiaoshan, Henan province, China. Our GCQAR has significant benefits in terms of recognition geochemical patterns compared to the traditional methods used in the field of geochemistry.
Multiple-wavelength optical orthogonal codes (MWOOCs) with autocorrelation sidelobes and cross-correlation values of both at most one were recently proposed for wavelength-time optical code-division ...multiple-access (O-CDMA) systems. The codes have cardinality as a quadratic function of the number of wavelengths and find applications in high bit-rate O-CDMA systems with broadband supercontinuum lasers, in which the number of available wavelengths is larger than the number of time slots. To support multimedia services with different bit-rate and quality-of-service requirements, a new class of multiple-length constant-weight MWOOCs with autocorrelation sidelobes of zero and cross correlations of at most one is constructed algebraically in this paper. The performance of these new codes in an O-CDMA system with double-media services is analyzed. In contrary to conventional single-length codes, our study shows that the performance of these multiple-length codes improves as the code length decreases. This unique property supports "prioritization" in O-CDMA.
The orbits of a real form G of a complex semisimple Lie group GC and those of the complexification KC of its maximal compact subgroup K acting on Z=GC/Q, a homogeneous, algebraic, GC-manifold, are ...finite. Consequently, there is an open G-orbit. Lower-dimensional orbits are on the boundary of the open orbit with the lowest dimensional one being closed. Induced action on the parameter space of certain compact geometric objects (cycles) related to the manifold in question has been characterized using duality relations between G- and KC-orbits in the case of an open G-orbit and more recently lower-dimensional G-orbits. We show that the parameter space associated with the unique closed G-orbit in Z agrees with that of the other orbits characterized as a certain explicitly defined universal domain.
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