On the polynomiality and asymptotics of moments of sizes for random(n,dn ± 1)-core partitions with distinct parts

Amdeberhan’s conjectures on the enumeration,the average size,and the largest size of(n,n+1)-core partitions with distinct parts have motivated many research on this topic.Recently,Straub(2016)and Nath and Sellers(2017)obtained formulas for the numbers of(n,dn-1)-and(n,dn+1)-core partitions with distinct parts,respectively.Let X_(s,t) be the size of a uniform random(s,t)-core partition with distinct parts when s and t are coprime to each other.Some explicit formulas for the k-th moments E[X_(n,n+1)^(k)]and E[X_(2 n+1,2 n+3)^(k)]were given by Zaleski and Zeilberger(2017)when k is small.Zaleski(2017)also studied the expectation and higher moments of X_(n,dn-1) and conjectured some polynomiality properties concerning them in ar Xiv:1702.05634.Motivated by the above works,we derive several polynomiality results and asymptotic formulas for the k-th moments of X_(n,dn+1) and X_(n,dn-1) in this paper,by studying theβ-sets of core partitions.In particular,we show that these k-th moments are asymptotically some polynomials of n with degrees at most 2 k,when d is given and n tends to infinity.Moreover,when d=1,we derive that the k-th moment E[X_(n,n+1)^(k)]of X_(n,n+1) is asymptotically equal to(n^(2)/10)^(k)when n tends to infinity.The explicit formulas for the expectations E[X_(n,dn+1)]and E[X_(n,dn-1)]are also given.The(n,dn-1)-core case in our results proves several conjectures of Zaleski(2017)on the polynomiality of the expectation and higher moments of X_(n,dn-1).

Calculation and analysis of losses of magnetic-valve controllable reactor

Magnetic-valve controllable reactor(MCR)has characteristics of DC bias and different types of magnetic flux density in the magnetic circuit and winding current distortion.These characteristics not only lead to loss calculation method of MCR different from that of power transformer,but also make it more difficult to calculate the core loss and wingding loss of MCR accurately.Our study combines core partition method with dynamic inverse J-A model to calculate the core loss of MCR.The winding loss coefficient of MCR is proposed,which takes into account the influence of harmonics and magnetic flux leakage on the winding loss of MCR.The result shows that the proposed core loss calculation method and winding loss coefficient are effective and correct for the loss calculation of MCR.