In this paper, we will be working with rings and ideals of real analytic functions. Given x ∈ R, let Ax be the ring of real-valued functions f : R → R that are analytic at x. When x = 0 is the origin, it is useful to think of A0 as a subring of the formal power series ring R[[ω1, . . . , ωd]] = R[[ω]] which consists of power series which are convergent in some neighborhood of the origin. For al...

‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...

In pointfree topology, a continuous real function on a frame L is a map L(R) → L from the frame of reals into L. The discussion of continuous real functions with possibly infinite values can be easily brought to pointfree topology by replacing the frame L(R) with the frame of extended reals L ( R ) (i.e. the pointfree counterpart of the extended real line R = R ∪ {±∞}). One can even deal with a...

In the early sixties, Amice ([Am1], [Am2]) studied the space of K-valued, locally analytic functions on Z Z p and formulated a complete description of its dual, the ring of K-valued, locally Q p-analytic distributions on Z Z p , when K is a complete subfield of C p. She found an isomorphism between the ring of distributions and the space of global functions on a rigid variety over K parameteriz...

Brain-inspired spiking neural networks (SNNs) have recently drawn more and attention due to their event-driven energy-efficient characteristics. The integration of storage computation paradigm on neuromorphic hardwares makes SNNs much different from Deep Neural Networks (DNNs). In this paper, we argue that may not benefit the weight-sharing mechanism, which can effectively reduce parameters imp...

We give a proposal for future development of the model theory of valued fields. We also summarize recent results on p-adic numbers. Let K be a valued field with a valuation map v : K → G ∪ {∞} to an ordered group 1 G; this is a map satisfying (i) v(x) = ∞ if and only if x = 0; (ii) v(xy) = v(x) + v(y) for all x, y ∈ K; (iii) v(x + y) ≥ min{v(x), v(y)} for all x, y ∈ K. We write R for the valuat...

the purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of sturm-liouville problem with equation of the form w??=(?2f(x)?r(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in c2(a,b which has a single zero (so called turning point) at point 0x=x and r(x) is a continuously differentiable function. ...