Since sum which is not necessarily commutative is defined in Omega-algebra A, then Omega-algebra A is called Omega-group. I also considered representation of Omega-group. Norm defined in Omega-group allows us to consider continuity of operations and continuity of representation.
In this paper, a new concept, the fuzzy rate of an operator in linear spaces is proposed for the very first time. Some properties and basic principles of it are studied. Fuzzy rate of an operator $B$ which is specific in a plane is discussed. As its application, a new fixed point existence theorem is proved.
We extend Hoeffdings lemma to general-state-space and not necessarily reversible Markov chains. Let ${X_i}_{i ge 1}$ be a stationary Markov chain with invariant measure $pi$ and absolute spectral gap $1-lambda$, where $lambda$ is defined as the operator norm of the transition kernel acting on mean zero and square-integrable functions with respect to $pi$. Then, for any bounded functions $f_i: x mapsto [a_i,b_i]$, the sum of $f_i(X_i)$ is sub-Gaussian with variance proxy $frac{1+lambda}{1-lambda} cdot sum_i frac{(b_i-a_i)^2}{4}$. This result differs from the classical Hoeffdings lemma by a multiplicative coefficient of $(1+lambda)/(1-lambda)$, and simplifies to the latter when $lambda = 0$. The counterpart of Hoeffdings inequality for Markov chains immediately follows. Our results assume none of countable state space, reversibility and time-homogeneity of Markov chains and cover time-dependent functions with various ranges. We illustrate the utility of these results by applying them to six problems in statistics and machine learning.
We extend Polyaks theorem on the convexity of joint numerical range from three to any number of quadratic forms on condition that they can be generated by three quadratic forms with a positive definite linear combination. Our new result covers the fundamental Diness theorem. As applications, we further extend Yuans lemma and S-lemma, respectively. Our extended Yuans lemma is used to build a more generalized assumption than that of Haeser (J. Optim. Theory Appl. 174(3): 641-649, 2017), under which the standard second-order necessary optimality condition holds at local minimizer. The extended S-lemma reveals strong duality of homogeneous quadratic optimization problem with two bilateral quadratic constraints.
We unify nonlinear Farkas lemma and S-lemma to a generalized alternative theorem for nonlinear nonconvex system. It provides fruitful applications in globally solving nonconvex non-quadratic optimization problems via revealing the hidden convexity.