A bstract We provide a new efficient diagrammatic tool, in the context of scattering equations, for computation covariant D -dimensional tree-level n -point amplitudes with pairs spinning massive particles using compact exponential numerators. discuss how this framework allows non-integer spin extensions recurrence relations developed integer spin. Our results facilitate on-going program genera...

A non-decreasing sequence of positive integer weights P = {p1, . . . , p2, pn} is called k-ordered if an intermediate sequence of weights produced by Huffman algorithm for initial sequence P on i-th step satisfies the following conditions: p (i) 2 = p (i) 3 , i = 0, k; p (i) 2 < p (i) 3 , i = k + 1, n − 3. Let T be a binary tree of size n and M = M(T ) be a set of such sequences of positive int...

The notion of transducer integer sequences is considered through a series of examples (the chosen examples are related to the Tower of Hanoi problem on 3 pegs). By definition, transducer integer sequences are integer sequences produced, under a suitable interpretation, by finite transducers encoding rational tree morphisms (length and prefix preserving transformations of words that have only fi...

A conic integer program is an integer programming problem with conic constraints. Conic integer programming has important applications in finance, engineering, statistical learning, and probabilistic integer programming. Here we study mixed-integer sets defined by second-order conic constraints. We describe general-purpose conic mixed-integer rounding cuts based on polyhedral conic substructure...

In Mixed Integer Linear Programming (MIP), a (strong) backdoor is ``small" subset of an instance's integer variables with the following property: in branch-and-bound procedure, instance can be solved to global optimality by branching only on backdoor. Constructing datasets pre-computed backdoors for widely used MIP benchmark sets or particular problem families enable new questions around novel ...

The sequential Branch and Bound Algorithm is the most established method for solving Mixed Integer and Discrete Programming Problems. It is based on the tree search of the possible subproblems of the original problem. There are two goals in carrying out a tree search, namely, (i) nding a good and ultimately the best integer solution, and (ii) to prove that the best solution has been found or no...

Let p be any fixed prime, n be any positive integer, ep(n) denotes the largest exponent of power p in n. That is, ep(n) = m, if p | n and p n. In problem 68 of [1], Professor F.Smarandache asked us to study the properties of the sequence {ep(n)}. About the elementary properties of this function, many scholars have studied it (see reference [2]-[7]), and got some useful results. For examples, Li...

We present a novel method for estimating tree-structured covariance matrices directly from observed continuous data. Specifically, we estimate a covariance matrix from observations of p continuous random variables encoding a stochastic process over a tree with p leaves. A representation of these classes of matrices as linear combinations of rank-one matrices indicating object partitions is used...