The objective of this memory is to give a purely algebraic proof of the Sauer- Shelah-Perles Lemma (inspired by the elegant proof in [FrPa,1983]), based only in duality in the Q−algebra Q[Vn] of polynomial functions de_ned on the zero-dimensional algebraic variety of subsets of the set [n] := {1, 2, . . . , n}. In fact, two di_erent proofs of this lemma will be given. Furthermore, we prove how several other classical results from Combinatorics are particular examples of a Trace (Inversion) Formula in _nite Q−algebras. For instance, one of this results is the general form of the Inclusion-Exclusion Principle (both with direct and reverse order associated to subsets inclusion). This approach also allows us to show a basis of the space of null t−designs, which di_ers from the one described in Theorem 4 of [DeFr,1982]. All results are still true if we replace Q[Vn] by K[Vn], where K is a perfect _eld of characteristic di_erent from 2. This memory has then the underlying purpose of connecting two _elds of mathematical knowledge that are not usually connected, at least not in this form.
The skyrocketing market value of cryptocurrencies has prompted more investors to pour funds into cryptocurrencies to seek asset hedging. However, the anonymity of blockchain makes cryptocurrency naturally a tool of choice for criminals to commit smart contract scams. Consequently, smart contract scam detection is particularly critical for investors to avoid economic loss. Previous methods mainly leverage specific code logic of smart contracts and/or design rules based on abnormal transaction behaviors for scam detection. Although these methods gain success at detecting particular scams, they perform worse when applied to scams with highly similar codes. Besides, well-designed decision rules rely on expert knowledge and tedious data collection steps, which causes poor flexibility. To combat these challenges, we consider the problem of smart contract scam detection via mining topological features of account interaction information that dynamically evolves. We adopt interactive features extracted from dynamic interaction information of accounts and propose a framework named TTG-SCSD to utilize the features and Topological Data Analysis for smart contract scams detection. The TTG-SCSD constructs discrete dynamic interaction graphs for each contract and designs interactive features that characterize account behaviors. The features are modeled combined with a topology quantification mechanism to capture contract intentions in transactions. Experimental results on real-world transaction datasets from Ethereum show that TTG-SCSD obtains better generalizability and improves the performance of the bare versions of the comparison methods.
Let k be a field, let R be a ring of polynomials in a finite number of variables over k, let D be the ring of k-linear differential operators of R and let f be a non-zero element of R. It is well-known that R_f, with its natural D-module structure, has finite length in the category of D-modules. We give a characteristic-free proof of this fact. To the best of our knowledge this is the first characteristic-free proof.