Плюригармонические определимые функции в некоторых $o$-минимальных расширениях вещественного поля

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

Solving Nonlinear Wave Equation Based on Topology

A method of solving nonlinear wave equation based on topology is proposed. Firstly, the characteristics of stochastic graph and Scaleless network are compared, and their topological characteristics are analyzed. Because of the existence of a few axis nodes, Scaleless networks have higher average aggregation than those with the same number of airport nodes and connected stochastic graphs. According to the topological structure of nonlinear wave equation, the first-order integral method is used to solve the nonlinear wave equation. According to the first integration, the threshold range is set, and the solution flow is designed in line with the division theorem. The topology of the network is analyzed according to the node degree, aggregation coefficient and reciprocity of the network, so as to verify and analyze. The experimental results show that the application of this method is 98%, which is still effective for the hyperbolic development equation of the same type.

Some properties of the polynomially bounded o-minimal expansions of the real field and of some quasianalytic local rings

In this paper, we study the Weierstrass division theorem over the rings of smooth germs that are definable in an arbitrary polynomially bounded o-minimal expansion of the real field by giving some criteria for satisfying this theorem. Afterwards, we study some topological properties of some quasianalytic subrings of the ring of smooth germs for the $(x_1)$-adic topology by showing that these rings are separable metric spaces. Also, we cite a criterion for their completeness with respect to the $(x_1)$-adic topology.

Local zero estimates and effective division in rings of algebraic power series

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact, we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally, we apply these results to prove a gap theorem for power series which are remainders of the Grauert–Hironaka–Galligo Division Theorem.

Reduced Complexity Linearity Improved Threshold Quantized Comparator Based Flash ADC

This paper describes a standard cell-based new approach of comparator design for flash ADC. Conventional flash ADC comparator consumes up to 60% of the power due to resistive ladder network and analog comparators. Threshold inverter quantized (TIQ) comparators reported earlier have improved speed and provide low-power, low-voltage operation. But they need feature size variation and have non-linearity issues. Here, a new standard cell comparator is proposed which retains all advantages of TIQ comparator and provides improved linearity with reduced hardware complexity. A 4-bit ADC designed using the proposed comparator requires 206 minimum-sized transistors and provides large area saving compared to previously proposed designs. Thermometer code is partitioned using algebraic division theorem. This conversion is used for mathematical modeling and complexity reduction of decoder circuit using semi-parallel organization of comparators. Circuit is designed using 90 nm technology which exhibits satisfactory performance even in process variation.

Skoda division of line bundle sections and pseudo-division

We first present a Skoda-type division theorem for holomorphic sections of line bundles on a projective variety which is essentially the most general, compared to previous ones. Then we revisit Geometric Effective Nullstellensatz and observe that even this general Skoda division is far from sufficient to yield stronger GEN such as ‘vanishing order [Formula: see text] division’, which could be used for finite generation of section rings by the basic finite generation lemma. To resolve this problem, we develop a notion of pseudo-division and show that it can replace the usual division in the finite generation lemma. We also give a vanishing order 1 pseudo-division result when the line bundle is ample.

The Investigation of Exact Solutions for the Appropriate Type of the Dispersive Long Wave Equation

Improved (G'/G)-expansion and first integral methods are used to construct exact solutions of the 2+1-dimensional Eckhaus-type extension of the dispersive long wave equation. The (G'/G)-expansion method is based on the assumptions that the travelling wave solutions can be expressed by a polynomial in (G'/G) and the first integral method is based on the theory of commutative algebra in which Division Theorem is of concern. It is worth mentioning that these methods are used for different systems and those two different systems can both be reduced to a system that will be mentioned in this paper. To recapitulate, this investigation has resulted in the exact solutions of the given systems.