‎this paper presents two main results that the singular values of the hadamard product of normal matrices $a_i$ are weakly log-majorized by the singular values of the hadamard product of $|a_{i}|$ and the singular values of the sum of normal matrices $a_i$ are weakly log-majorized by the singular values of the sum of $|a_{i}|$‎. ‎some applications to these inequalities are also given‎. ‎in addi...

LP is a recently introduced formalism that extends answer set programs by adopting the log-linear weight scheme of Markov Logic. This paper investigates the relationships between LP and two other extensions of answer set programs: weak constraints to express a quantitative preference among answer sets, and P-log to incorporate probabilistic uncertainty. We present a translation of LP into progr...

We prove that the log Hodge de Rham spectral sequences of certain proper smooth schemes Cartier type in characteristic $$p>0$$ degenerate at $$E_1$$ . also Kodaira vanishings for them hold when they are projective. formulate weak Lefschetz conjecture crystalline cohomologies and it is true cases.