Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point $z$ if either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\leq 2$, $1\leq q\leq \infty$, or the $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\leq 3$, $1 \leq q < \infty$, or the $L^{p,q}_{x,t}$-norm of the gradient of the vorticity with $3/p+2/q\leq 4$, $1 \leq q$, $1 \leq p$, is sufficiently small near $z$

Tropical linear algebra with the Lukasiewicz T-norm

The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over this semiring can be developed in the usual way. We observe that any problem of the max-Lukasiewicz linear algebra can be equivalently formulated as a problem of the tropical (max-plus) linear algebra. Based on this equivalence, we develop a theory of the matrix powers and the eigenproblem over the max-Lukasiewicz semiring.Comment: 27 page

A note on drastic product logic

The drastic product $*_D$ is known to be the smallest $t$-norm, since $x *_D y = 0$ whenever $x, y < 1$. This $t$-norm is not left-continuous, and hence it does not admit a residuum. So, there are no drastic product $t$-norm based many-valued logics, in the sense of [EG01]. However, if we renounce standard completeness, we can study the logic whose semantics is provided by those MTL chains whose monoidal operation is the drastic product. This logic is called ${\rm S}_{3}{\rm MTL}$ in [NOG06]. In this note we justify the study of this logic, which we rechristen DP (for drastic product), by means of some interesting properties relating DP and its algebraic semantics to a weakened law of excluded middle, to the $\Delta$ projection operator and to discriminator varieties. We shall show that the category of finite DP-algebras is dually equivalent to a category whose objects are multisets of finite chains. This duality allows us to classify all axiomatic extensions of DP, and to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure

Some properties concerning the quasi-inverse of a T-norm

Some properties of the quasi-inverse operators are presented. They are basic tools in order to reduce complex expressions involving several of such operators. An effective calculation for the quasi-inverse of a continuous t-norm is also provided.Peer ReviewedPostprint (published version

An extension method for t-norms on subintervals to t-norms on bounded lattices

summary:In this paper, a construction method on a bounded lattice obtained from a given t-norm on a subinterval of the bounded lattice is presented. The supremum distributivity of the constructed t-norm by the mentioned method is investigated under some special conditions. It is shown by an example that the extended t-norm on $L$ from the t-norm on a subinterval of $L$ need not be a supremum-distributive t-norm. Moreover, some relationships between the mentioned construction method and the other construction methods in the literature are presented

Fuzzy Association Rules pada Recommender System berbasis Collaborative Filtering

ABSTRAKSI: Pengembangan teknologi yang sangat cepat pada akhir dekade ini telah menyebabkan informasi yang terlalu banyak. Collaborative Filtering(CF) adalah teknik yang cukup potensial pada recommender systems, yaitu dengan menyediakan rekomendasi secara personal kepada user berdasarkan minat user dan user lain yang memiliki keminatan yang sama. Tugas akhir ini mengimplementasikan dan menganalisis user-based collaborative filtering recommender system, yang menerapkan fuzzy association rule mining. Teknik fuzzy association rule mining digunakan untuk mengolah nilai numerik. Tujuan dari tugas akhir ini adalah untuk membandingkan t-norm manakah yang paling akurat untuk di implementasikan pada recommender system. Selain itu, parameter yang digunakan adalah rasio antara training dan test set,minimum confidence, dan jumlah rule yang diukur dengan MAE dan precision Prediksi yang dihasilkan dengan t-norm = min(x,y) memiliki nilai MAE yang lebih kecil dibanding dengan prediksi menggunakan t-norm = xy. Hal ini dikarenakan perhitungan t-norm xy menghasilkan jumlah rule yang lebih kecil, sehingga collaborative user yang dihasilkan lebih sedikit. Sebaliknya dengan menggunakan t-norm = xy menghasilkan precision yang lebih baik daripada tnorm = min(x,y). Hal ini disebabkan t-norm = xy menghasilkan rule yang lebih sedikit sehingga memperbesar nilai precisionKata Kunci : recommender system, collaborative filtering, fuzzy association ruleABSTRACT: The rapid development of Internet technologies in recent decades has imposed a heavy information burden on users. Collaborative Filtering (CF) is the most promising technique in recommender systems, providing personalized recommendations to users based on their previously expressed preferences and those of other similar users. This final project implemented and analysed the user-based collaborative filtering recommender system, wich applies fuzzy association rule mining. Fuzzy association rule mining techniques is used to deal numerical values. The aim of this final project is to compare which one of t-norm is most accurate to be implemented on recommender system. Moreover, Parameters which are used are ratio between training and test set, minimum confidence, and number of rules. These are measured by MAE and Precision. Prediction produced by t-norm = min(x,y) is less error than prediction produced by t-norm=xy. This is because calculation by t-norm = xy producing less number of rules, so it will cause less number of collaborative user. On the other hand, recommendation by t-norm = xy is more preferable then t-norm = min(x,y). This is caused by less rules is generated by t-norm = xy which increases precision value.Keyword: recommender system, collaborative filtering, fuzzy association rul