and Applied Analysis 3 3. Main Results Throughout this paper, we will consider lattices as complete lattices,X will denote space, and μ is a membership function of any fuzzy set X. Definition 3.1. If m : σ L → R ∪ {∞} satisfies the following properties, then m is called a lattice measure on the lattice σ-algebra σ L . i m ∅ L0. ii For all f, g ∈ σ L such that m f , m g ≥ L0 : f ≤ g ⇒ m f ≤ m g ...

This paper proposes an infeasible interior-point algorithm with full-Newton step for linear programming, which is an extension of the work of Roos (SIAM J. Optim., 16(4):1110–1136, 2006). We introduce a kernel function in the algorithm. For p ∈ [0, 1), the polynomial complexity can be proved and the result coincides with the best result for infeasible interior-point methods, that is, O(n log n/ε).

We expect this conjecture to be false (for sufficiently large d) when E does have complex multiplication. Nevertheless, our main result is that Conjecture 1.1 at least holds on average. For A,B > 0, let SA,B denote the set of elliptic curves with Weierstrass equations y = x + ax+ b with a, b ∈ Z, |a| ≤ A and |b| ≤ B. For an elliptic curve E and x > 0, let π E(x) denote the number of primes p ≤ ...

In this paper we study primal-dual path-following algorithms for the second-order cone programming (SOCP) based on a family of directions that is a natural extension of the Monteiro-Zhang (MZ) family for semidefinite programming. We show that the polynomial iteration-complexity bounds of two well-known algorithms for linear programming, namely the short-step path-following algorithm of Kojima e...

In this paper, we made an extension to the convergence analysis of the dynamics of two-layered bias-free networks with one ReLU output. We took into consideration two popular regularization terms: the `1 and `2 norm of the parameter vector w, and added it to the square loss function with coefficient λ/2. We proved that when λ is small, the weight vector w converges to the optimal solution ŵ (wi...

In this paper we propose a complete axiomatization of the bisimilarity distance of Desharnais et al. for the class of finite labelled Markov chains. Our axiomatization is given in the style of a quantitative extension of equational logic recently proposed by Mardare, Panangaden, and Plotkin (LICS’16) that uses equality relations t ≡ε s indexed by rationals, expressing that “t is approximately e...

We expect this conjecture to be false (for sufficiently large d) when E does have complex multiplication. Nevertheless, our main result is that Conjecture 1.1 at least holds on average. For A,B > 0, let SA,B denote the set of elliptic curves with Weierstrass equations y = x + ax + b with a, b ∈ Z, |a| ≤ A and |b| ≤ B. For an elliptic curve E and x > 0, let π E(x) denote the number of primes p ≤...

We consider two basic types of coarse-graining: the Ehrenfests’ coarsegraining and its extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining based on uncertainty of dynamical models and ε-motions (orbits). Non-technical discussion of basic notions and main coarse-graining theorems are presented: the theorem about entropy overproduction for the Ehrenfests’ c...

We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of Sylvester-Uhlmann type to achieve this. A main tool is an extension of the Nakamura-Uhlmann intertwinin...

Given a set F of vertices of a connected graph G, we study the problem of testing the connectivity of G−F in polynomial time with respect to |F | and the maximum degree ∆ of G. We present two approaches. The first algorithm for this problem runs in O ( |F |∆2ε−1 log(|F |∆ε−1) ) time for every graph G with vertex expansion at least ε > 0. The other solution, designed for the class of graphs with...