Identifying the interacting positions of a protein using Boolean learning and support vector machines

It is known that in the three-dimensional structure of a protein, certain amino acids can interact with each other in order to provide structural integrity or aid in its catalytic function. If these positions are mutated the loss of this interaction usually leads to a non-functional protein. Directed evolution experiments, which probe the sequence space of a protein through mutations in search for an improved variant, frequently result in such inactive sequences. In this work, we address the use of machine learning algorithms, Boolean learning and support vector machines (SVMs), to find such pairs of amino acid positions. The recombination method of imparting mutations was simulated to create in silico sequences that were used as training data for the algorithms. The two algorithms were combined together to develop an approach that weighs the structural risk as well as the empirical risk to solve the problem. This strategy was adapted to a multi-round framework of experiments where the data generated in the present round is used to design experiments for the next round to improve the generated library, as well as the estimation of the interacting positions. It is observed that this strategy can greatly improve the number of functional variants that are generated as well as the average number of mutations that can be made in the library.     

The fraction of large random trees representing a given Boolean function in implicational logic

We consider the logical system of Boolean expressions built on the single connector of implication and on positive literals. Assuming all expressions of a given size to be equally likely, we prove that we can define a probability distribution on the set of Boolean functions expressible in this system. Then we show how to approximate the probability of a function f when the number of variables grows to infinity, and that this asymptotic probability has a simple expression in terms of the complexity of f. We also prove that most expressions computing any given function in this system are “simple”, in a sense that we make precise. The probability of all read‐once functions of a given complexity is also evaluated in this model. At last, using the same techniques, the relation between the probability of a function and its complexity is also obtained when random expressions are drawn according to a critical branching process. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

一类4次旋转对称布尔函数的汉明重量和非线性度

旋转对称布尔函数在密码学中具有重要的应用价值.本文研究了一类特殊4次的旋转对称布尔函数的快速求值及其汉明重量的递归关系,通过将该函数分解成数个子函数,并利用这些函数的傅里叶变换值的递归关系,证明了其汉明重量与非线性度相等.本文的结果和处理指数和的方法对于进一步研究Cusick的一个猜想可能有帮助.     

Spline R-Function and Applications in FEM

R-function is a widely used tool when considering objects obtained through the Boolean operations start from simple base primitives. However, there is square root operation in the representation. Considering that the use of splines will facilitate the calculations within the CAD system, in this paper, we propose a system of R-functions represented in spline form called Spline R-function (SR). After transforming the function ranges of two base primitives to a new coordinate system, a series of sign constraints following a specific Boolean operation are derived and the spline R-function can be formulated as a piecewise function. Representation of SR in both Bézier form and B-spline form have been given. Among which the Bézier ordinates are determined with the help of the B-net method through setting up a series of relations according to the sign constraints and properties of R-functions. The construction processes for both Boolean intersection and union operations with different smoothness are discussed in detail. Numerical experiments are conducted to show the potential of the proposed spline R-function.     

Minimal prime ideals of reduced rings

Let R be a reduced ring with Q its Martindale symmetric ring of quotients, and let B be the complete Boolean algebra of all idempotents in C, where C is the extended centroid of R. It is proved that every minimal prime ideal of R must be of the form mQ∩R for some maximal ideal m of B but the converse is in general not true. In addition, if R is centrally closed or has only finitely many minimal prime ideals, then the converse also holds. By applying the explicit expression, many properties of minimal prime ideals of reduced rings are realized more easily.     

A refinement of Cusick-Cheon bound for the second order binary Reed-Muller code

We prove a stronger form of the conjectured Cusick-Cheon lower bound for the number of quadratic balanced Boolean functions. We also prove various asymptotic results involving B(k,m), the number of balanced Boolean functions of degree ≤k in m variables, in the case k=2. Finally, we connect our results for k=2 with the (still unproved) conjectures of Cusick-Cheon for the functions B(k,m) with k>2.     

Influences of monotone Boolean functions

Recently, Keller and Pilpel conjectured that the influence of a monotone Boolean function does not decrease if we apply to it an invertible linear transformation. Our aim in this short note is to prove this conjecture.