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This article first introduces the concept of a general pseudo-homogeneous triangular norm. It then gives some properties of general pseudo-homogeneous triangular norms. Finally, it characterizes all general pseudo-homogeneous triangular norms completely., Comment: 18

This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f: [0,1]\rightarrow [0,1]$ is a strictly increasing function and $f^{(-1)}:[0,1]\rightarrow[0,1]$ is the pseudo-inverse of $f$. We prove that the associativity of functions $T$ only depends on the range of $f$, which is used to give a sufficient and necessary condition for the function $T$ being associative when the triangular norm $F$ is an ordinal sum of triangular norms and an ordinal sum of triangular subnorms in the sense of A. H. Clifford, respectively. These results finally are applied for describing classes of triangular norms generated by strictly increasing functions., Comment: 24

Associativity of a two-place function $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,\infty]^2\rightarrow[0,\infty]$ is an associative function, $f: [0,1]\rightarrow [0,\infty]$ is a monotone function which satisfies either $f(x)=f(x^{+})$ when $f(x^{+})\in \mbox{Ran}(f)$ or $f(x)\neq f(y)$ for any $y\neq x$ when $f(x^{+})\notin \mbox{Ran}(f)$ for all $x\in[0,1]$ and $f^{(-1)}:[0,\infty]\rightarrow[0,1]$ is a pseudo-inverse of $f$ depends only on properties of the range of $f$. The necessary and sufficient conditions for the $T$ to be associative are presented by applying the properties of the monotone function $f$., Comment: 16

We study real resonances and embedded eigenvalues of the Kramers--Fokker--Planck operator with a long-range potential. We prove that thresholds are only possible accumulation points of eigenvalues and that the limiting absorption principle holds true for energies outside an exceptional set. We also prove that the eigenfunctions associated with discrete eigenvalues decay exponentially and those associated with embedded non-threshold ones decay polynomially.

This article pays attention to the $\alpha$-migrativity of 2-uninorms with $\alpha\in [0,1]$ deeply. It describes the $\alpha$-migrativity of 2-uninorms completely, which generalizes and unifies some current existing characterizations for $\alpha$-migrativity of triangular norm, triangular conorm, uninorms, nullnorms, uni-nullnorms and null-uninorms, respectively., Comment: 14

In this article, we first give the characterizations of quasi-homogeneous aggregation functions, which show us that quasi-homogeneous aggregation functions are classified into three classes. We then introduce the concept of triple generator of quasi-homogeneous aggregation function, which is applied to construct a quasi-homogeneous aggregation function., Comment: 15

This article focuses on minimax programming problems subject to addition-{\L}ukasiewicz fuzzy relational inequalities. We first establish two necessary and sufficient conditions that a solution of the fuzzy relational inequalities is a minimal one and explore the existence condition of the unique minimal solution. We also supply an algorithm to search for minimal solutions of the fuzzy relational inequalities starting from a given solution. We then apply minimal solutions of the fuzzy relational inequalities to the minimax programming problems for searching optimal solutions. We provide two algorithms to solve a kind of single variable optimization problems, and obtain the greatest optimal solution. The algorithm for finding minimal solutions of a given solution are also used for searching minimal optimal solutions., Comment: 25

This book provides a systematic study of spectral and scattering theory for many-body Schr\"odinger operators at two-cluster thresholds. While the two-body problem (reduced after separation of the center of mass motion to a one-body problem at zero energy) is a well-studied subject, the literature on the many-body problem is sparse. However our analysis covers for example the system of three particles interacting by Coulomb potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue $\lambda_0$. In general we address the question: How does scattering quantities for the many-body atomic and molecular models behave in the limit when the total energy approaches a fixed two-cluster threshold $\lambda_0$? This includes mapping properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross-section and absence of transmission from one channel to another in the small inter-cluster kinetic energy region. Our principal tools are the Feshbach-Grushin dimension reduction method and spectral analysis based on a certain Mourre estimate. Additional topics (of independent interest) are the limiting absorption principle, micro-local resolvent estimates, Rellich and Sommerfeld type theorems and asymptotics of the limiting resolvents at thresholds. While all of these features are fairly well-understood for two-body Schr\"odinger operators, they are poorly understood in the many-body case even for two-cluster thresholds. It is the goal of the book to remedy this point. The mathematical physics field under study is very rich, and there are many open problems, several of them stated explicitly in the book for the interested reader.

In order to guarantee the downloading quality requirements of users and improve the stability of data transmission in a BitTorrent-like peer-to-peer file sharing system, this article deals with eigenproblems of addition-min algebras. First, it provides a sufficient and necessary condition for a vector being an eigenvector of a given matrix, and then presents an algorithm for finding all eigenvalues and eigenvectors of a given matrix. It further proposes a sufficient and necessary condition for a vector being a constrained eigenvector of a given matrix and supplies an algorithm for computing all the constrained eigenvectors and eigenvalues of a given matrix. This article finally discusses the supereigenproblem of a given matrix and presents an algorithm for obtaining the maximum constrained supereigenvalue and depicting the feasible region of all the constrained supereigenvectors for a given matrix. It also gives some examples for illustrating the algorithms, respectively., Comment: 21

This paper deals with the resolutions of fuzzy relation equations with addition-min composition. When the fuzzy relation equations have a solution, we first propose an algorithm to find all minimal solutions of the fuzzy relation equations and also supply an algorithm to find all maximal solutions of the fuzzy relation equations, which will be illustrated, respectively, by numeral examples. Then we prove that every solution of the fuzzy relation equations is between a minimal solution and a maximal one, so that we describe the solution set of the fuzzy relation equations completely., Comment: 19

In this article, we first characterize pseudocomplemented inductive modular lattices by using their two 0-sublattices. Then we use two 0-sublattices of a subgroup lattice to describe all locally cyclic abelian groups. In particular, we show that a locally cyclic abelian group can be characterized by its three subgroups., Comment: 9

In this article, we present two methods to construct 2-uninorms on bounded lattices by using additive generators, which are further used for inducing uninorms, nullnorms, uni-nullnorms and null-uninorms, respectively. We also provide some examples for illustrating the constructing methods of 2-uninorms., Comment: 23

In this work, we prove an optimal global-in-time L p --L q estimate for solutions to the Kramers-Fokker-Planck equation with short range potential in dimension three. Our result shows that the decay rate as t $\rightarrow$ +$\infty$ is the same as the heat equation in x-variables and the divergence rate as t $\rightarrow$ 0 + is related to the sub-ellipticity with loss of 1/3 derivatives of the Kramers-Fokker-Planck operator.

This paper mainly investigates the distributive laws between extended nullnorms and uninorms on fuzzy truth values under the condition that the nullnorm is conditionally distributive over the uninorm. It presents the distributive laws between the extended nullnorm and t-conorm, and the left and right distributive laws between the extended generalization nullnorm and uninorm, where a generalization nullnorm is an operator from the class of aggregation operators with absorbing element that generalizes a nullnorm., Comment: 20

In 2010, G\'{a}bor Cz\'{e}dli and E. Tam\'{a}s Schmidt mentioned that the best cover-preserving embedding of a given semimodular lattice is not known yet [A cover-preserving embedding of semimodular lattices into geometric lattices, Advances in Mathematics 225 (2010) 2455-2463]. That is to say: What are the geometric lattices $G$ such that a given finite semimodular lattice $L$ has a cover-preserving embedding into $G$ with the smallest $|G|$? In this paper, we propose an algorithm to calculate all the best extending cover-preserving geometric lattices $G$ of a given semimodular lattice $L$ and prove that the length and the number of atoms of every best extending cover-preserving geometric lattice $G$ equal the length of $L$ and the number of non-zero join-irreducible elements of $L$, respectively. Therefore, we comprehend the best cover-preserving embedding of a given semimodular lattice., Comment: 26

For quickly decreasing potentials with one position variable, the first threshold zero is always a resonance of the Kramers-Fokker-Planck operator. In this article we study low-energy spectral properties of the operator and calculate large time asymptotics of solutions in terms of the Maxwellian.

In this article, we study a class of non-selfadjoint Schr{\"o}dinger operators H which are perturbation of some model operator H 0 satisfying a weighted coercive assumption. For the model operator H 0 , we prove that the derivatives of the resolvent satisfy some Gevrey estimates at threshold zero. As application, we establish large time expansions of semigroups e --tH and e --itH for t > 0 with subexponential time-decay estimates on the remainder, including possible presence of zero eigenvalue and real resonances.

This paper first gives a necessary and sufficient condition that a lattice $L$ can be represented as the collection of all up-sets of a poset. Applying the condition, it obtains a necessary and sufficient condition that a lattice can be embedded into the lattice $L$ such that all infima, suprema, the top and bottom elements are preserved under the embedding by defining a monotonic operator on a poset. This paper finally shows that the quotient of the set of the monotonic operators under an equivalence relation can be naturally ordered and it is a lattice if $L$ is a finite distributive lattice., Comment: 15

This paper deals with conditions under which the quotient of $L$-fuzzy up-sets forms a complete lattice by using terminologies of closure operators. It first gives a condition that a family of some subsets of a nonempty set can be represented by $L$-fuzzy up-sets, which is then used to formulate a necessary and sufficient condition under which the quotient of $L$-fuzzy up-sets forms a complete lattice. This paper finally shows that the quotient of a kind of $L$-fuzzy up-sets is isomorphic to an interval generated by an $L$-fuzzy up-set., Comment: 13

This paper primarily studies monomial ideals by their associated lcm-lattices. It first introduces notions of weak coordinatizations of finite atomic lattices which have weaker hypotheses than coordinatizations and shows the characterizations of all such weak coordinatizations. It then defines a finite super-atomic lattice in $\mathcal{L}(n)$, investigates the structures of $\mathcal{L}(n)$ by their super-atomic lattices and proposes an algorithm to calculate all the super-atomic lattices in $\mathcal{L}(n)$. It finally presents a specific labeling of finite atomic lattice and obtains the conditions that the specific labelings of finite atomic lattices are the weak coordinatizations or the coordinatizations by using the terminology of super-atomic lattices., Comment: 27pages. arXiv admin note: text overlap with arXiv:1009.1430 by other authors