ENUMERATING LOCAL PERMUTATION POLYNOMIALS OVER RESIDUE CLASS RINGS

The universal binary neurons are a central point of this Chapter. The mathematical models of the universal binary neuron over the field of the complex numbers, the residue class ring and the finite field are considered. A notion of the P-realizable Boolean function, which may be considered as a generalization of a notion of the threshold Boolean function, is introduced. It is shown that the implementation with a single neuron the input/output mapping described by non-threshold Boolean functions is possible, if weights are complex, and activation function of the neuron is a function of the argument of the weighted sum (similar to multi-valued neuron). It is also possible to define an activation function on the residue class ring and the finite field in such a way that the implementation of non-threshold Boolean functions will be possible on the single neuron with weights from these sets. P-realization of multiple-valued function over finite algebras and residue class ring in particular is also considered. The general features of P-realizable Boolean functions are considered, also as the synthesis of universal binary neuron.

Permutation polynomials on matrices

Letbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base”x. In particular, we define and characterize digit representations for the ringS:=((x−1,y−1))/f((x−1,y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.

Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if f ( x ) f(x) and g ( x ) g(x) are integral polynomials of degree n ≥ 2 n \geq 2 and p is a prime exceeding ( n 2 − 3 n + 4 ) 2 {({n^2} - 3n + 4)^2} for which f and g are both permutation polynomials of the finite field F p {F_p} , then their difference h = f − g h = f - g cannot be such that h ( x ) = c x h(x) = cx for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in F p {F_p} and t is the degree of h, then t ≥ 3 n / 5 t \geq 3n/5 and, provided t ≤ n − 3 t \leq n - 3 , t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.

Let G k be the de Bruijn graph with 2k vertices. When reproduction labels from a 4-letter alphabet A 4 are assigned linearly to the edges of G k using a three row binary generating matrix, a 2k-state trellis code results which is suitable for encoding a source with alphabet A 4 at a rate of one bit per source sample with respect to the Hamming fidelity criterion. There is a special type of code which results in this way called systematic code. A systematic code belongs to three sets of codes, each set isomorphic to a residue class ring of polynomials over GF(2). A systematic code sweeps out a cycle in each of its three residue class rings via repeated multiplication by the residue class of polynomial x. The local coordinates of a systematic code are non-negative integer parameters which describe how the code is positioned in each of its three cycles. Our main result is that a systematic code is uniquely determined by its local coordinates, except in rare instances. The paper concludes with some preliminary work indicating how a code with good distortion performance in coding a memoryless source with alphabet A 4 can be found in relatively small sets of systematic codes whose local coordinates are of a certain type.

We give an upper bound for the Stanley depth of the edge ideal of a complete k-partite hypergraph and as an application we give an upper bound for the Stanley depth of a monomial ideal in a polynomial ring S. We also give a lower and an upper bound for the cyclic module S/I associated to the complete k-partite hypergraph.

From the 19th century, the theory of permutation polynomial over finite fields, that are arose in the work of Hermite and Dickson, has drawn general attention. Permutation polynomials over finite fields are an active area of research due to their rising applications in mathematics and engineering. The last three decades has seen rapid progress on the research on permutation polynomials due to their diverse applications in cryptography, coding theory, finite geometry, combinatorics and many more areas of mathematics and engineering. For this reason, the study of permutation polynomials is important nowadays. In this chapter, we propose some new problems in connection to permutation polynomials over finite fields by the help of prime numbers.

Synthesis of multifunction and universal logic elements over residue class rings. I

Given a positive integer [see formula in PDF] and the residue class ring [see formula in PDF], we set [see formula in PDF] to be the group of units in[see formula in PDF], i.e., [see formula in PDF]. Let [see formula in PDF] be the number of solutions of[see formula in PDF] with [see formula in PDF]. In this note, we determine an explicit expression of [see formula in PDF]. This extends the results of Sun and Yang in 2014.

Several modules M over algebras with straightening law A have a structure which is similar to the structure of A itself: M has a system of generators endowed with a natural partial order, a standard basis over the ring B of coefficients, and the multiplication A × M → A satisfies a “straightening law”. We call them modules with straightening law, briefly MSLs. In section 1 we recall the notion of an algebra with straightening law together with those examples which will be important in the sequel. Section 2 contains the basic results on MSLs, whereas section 3 is devoted to examples: (i) powers of certain ideals and residue class rings with respect to them, (ii) “generic” modules defined by generic, alternating or symmetric matrices of indeterminates, (iii) certain modules related to differentials and derivations of determinantal rings. The essential homological invariant of a module is its depth. We discuss how to compute the depth of an MSL in section 4. The main tool are filtrations related to the MSL structure. The last section contains a natural strengthening of the MSL axioms which under certain circumstances leads to a straightening law on the symmetric algebra. The main examples of such modules are the “generic” modules defined by generic and alternating matrices. The notion of an MSL was introduced by the author in [Br.3] and discussed extensively during the workshop. The main differences of this survey to [Br.3] are the more detailed study of examples and the treatment of the depth of MSLs which is almost entirely missing in [Br.3]

On additive properties of two sets in a residue class ring

Residue Class Rings

Generalized bilinear forms graphs and MRD codes over a residue class ring

For a given polynomial f and an ideal I of a polynomial ring K[X] over a field K, we give a necessary and sufficient condition for I to have a smallest ideal extension J such that f is invertible in the residue class ring K[X]/J. If the condition holds, J is shown to be the saturation ideal I:∞. We also show primary decompositions of the ideals I:∞ and I+9fm;:, where m is a natural number such that I:∞ = I:fm, all together forms a primary decomposition of I with no modification. These observations are especially useful for polynomial rings with parameters.

In the paper, we present a new method to study the differential properties of addition modulo 2n using the formulas of coefficients of sum of 2-adic integers.