An alternative proof of Granger’s Representation Theorem forI(1) systems through Jordan matrices

1998 ◽  
Vol 7 (2) ◽  
pp. 111-127 ◽  
Author(s):  
Fragiskos Archontakis
Keyword(s):  
Representation Theorem ◽  
Alternative Proof ◽  
Jordan Matrices

Shepherdson’s Representation Theorems

1992 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Representation Theorem ◽  
Recursive Function ◽  
Function Theory ◽  
Peano Arithmetic ◽  
Alternative Proof ◽  
Binary Relations ◽  
Representation Theorems ◽  
Strong Separation ◽  
Weak Separation ◽  
Th 1

We have already remarked that at the time of Gödel’s proof, the only known way of showing the set P* of Peano Arithmetic to be representable in P.A. involved the assumption of ω-consistency. Well, in 1960, A. Ehrenfeucht and S. Feferman showed that all Σ1-sets can be represented in all simply consistent axiomatizable extensions of the system (R). Hence, all Σ1-sets can be shown to be representable in P.A. under the weaker assumption that P.A. is simply consistent. Their proof combined a Rosser-type argument with a celebrated result in recursive function theory due to John Myhill which goes beyond the scope of this volume. Very shortly after, however, John Shepherdson [1961] found an extremely ingenious alternative proof that is more direct and which we study in this chapter. [In our sequel to this volume, we compare Shepherdson’s proof with the original one. The comparison is of interest in that the two methods are very different and the proofs generalize in different directions which are apparently incomparable in strength.] We recall that for each n > 1, a system S is called a Rosser system for n-ary relations if for any Σ1-relations R1(x1,..., xn) and R2(X1, ..., xn), the relation R1 — R2 is separable from R2 — R1 in S. We wish to prove the following theorem and its corollary (Th. 1 below). Theorem S1—Shepherdson’s Representation Theorem. If S is a simply consistent axiomatizable Rosser system for binary relations (n-ary relations for n = 2), then all Σ1-sets are representable in S. Theorem 1—Ehrenfeucht, Feferman. All Σ1-sets are representable in every consistent axiomatizable extension of the system (R). Shepherdson’s Lemma and Weak Separation For emphasis, we will now sometimes write “strongly separates” for “separates”. We will say that a formula F(v1) weakly separates A from B in S if F(v1) represents some superset of A disjoint from B, We showed in the last chapter (Lemma 1) that strong separation implies weak separation provided that the system S is consistent. We also say that a formula F(v1,. .. ,vn) weakly separates a relation R I (x1 , . .. ,xn) from .R2(x1,... ,xn) if F(v1, ..., vn) represents some relation R’(x1,. .. ,xn) such that R1 ⊆ R1’ and R1 is disjoint from -R2.

scholarly journals A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Tomás Pérez Becerra ◽  
Juan Alberto Escamilla Reyna ◽  
Daniela Rodríguez Tzompantzi ◽  
Jose Jacobo Oliveros Oliveros ◽  
Khaing Khaing Aye
Keyword(s):  
Representation Theorem ◽  
Type Theorem ◽  
Alternative Proof ◽  
Bilinear Operator ◽  
Riesz Representation Theorem ◽  
Stieltjes Integral ◽  
Henstock Integral ◽  
Riesz Representation ◽  
Vector Valued Functions ◽  
Vector Valued

Using a bounded bilinear operator, we define the Henstock-Stieltjes integral for vector-valued functions; we prove some integration by parts theorems for Henstock integral and a Riesz-type theorem which provides an alternative proof of the representation theorem for real functions proved by Alexiewicz.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

scholarly journals A short note on extreme points of certain polytopes

2020 ◽  
Vol 8 (1) ◽  
pp. 36-39
Author(s):  
Lei Cao ◽  
Ariana Hall ◽  
Selcuk Koyuncu
Keyword(s):  
Short Proof ◽  
Convex Polytopes ◽  
Short Note ◽  
Convex Polytope ◽  
Extreme Points ◽  
Alternative Proof ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.

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