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Automatic generation of epsilon-delta proofs of continuity [chapter]

Michael Beeson
1998 Lecture Notes in Computer Science  
As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilon-delta arguments.  ...  tapestries of proofs.  ...  epsilon-delta proofs.  ... 
doi:10.1007/bfb0055903 fatcat:gmfhmyitcfez5jevchvzddoddm

A singular quasilinear diffusion equation in $L^{1}$

Ralph E. SHOWALTER
1984 Journal of the Mathematical Society of Japan  
PROOF. If $u$ is such a strong solution then it is the limit in $C(O, T;H^{-1})$ of solutions $u_{\epsilon}$ of (4) .  ...  But $\epsilon\alpha(\cdot)$ and $\epsilon\beta(\cdot)$ belong to $\mathcal{M}$ , so we need only check for $\epsilon=1$ , and this follows from the first part of the proof, with $\gamma=\alpha(I+\beta)  ... 
doi:10.2969/jmsj/03620177 fatcat:yheo6rxe5rahhld2jehnwdgyei

A Generalized Definition of Caputo Derivatives and Its Application to Fractional ODEs

Lei Li, Jian-Guo Liu
2018 SIAM Journal on Mathematical Analysis  
The group property can then be verified using the semigroup property and the fact that g \gamma \ast g - \gamma = \delta . 2.2. Time-continuous fractional calculus.  ...  Proof. Using the above facts and Lemma 2.2, we find that for any \gamma > 0, g - \gamma is the convolution inverse of g \gamma .  ...  , exists on (0, T 1 -\epsilon 0 ) and | v \delta (t) -v(t)| < \epsilon . (51) Proof.  ... 
doi:10.1137/17m1160318 fatcat:cpbhqlboordorbpzjf2w5c4rrq

Semigroups of locally lipschitzian operators and applications [chapter]

Yoshikazu Kobayashi, Shinnosuke Oharu
1993 Lecture notes in mathematics  
A continuous function $u:[0, \tau]arrow X$ is said to be a mild solution of (DI) on $[0, \tau]$ , provided that for each $\epsilon>0$ there is an e-approximate solution $v^{\epsilon}$ of (DI) on $[0, \  ...  In order to advance a general theory of nonlinear semigroups, it is necessary to restrict the continuity of the operators $S(t)$ .  ... 
doi:10.1007/bfb0085480 fatcat:qqquit4crffjrlilahnr67g3o4

A QUASI-DECOMPOSABLE ABELIAN GROUP WITHOUT PROPER ISOMORPHIC QUOTIENT GROUPS AND PROPER ISOMRPHIC SUBGROUPS II

Takashi Ito, John M. Irwin
1969 Journal of Faculty of Science Hokkaido University Series I Mathematics  
Introduction This paper is a continuation of our paper [2] with the same title. In [2] one of theorems was stated without detailed proof.  ...  The purpose of this paper is just to give a complete proof of it. So we shall omit an origin of our problem and general background for it, see the first section of [2] and references in [2] .  ... 
doi:10.14492/hokmj/1530064871 fatcat:zyvabwy4andsdorx3p5jzbzbgy

Alpha Epsilon Delta (N1)

M. L. MOORE
1959 Science  
Severinghaus (associate Alpha Epsilon Delta (NI) SCIENCE, VOL. 129 NEW! Versatile laboratory aids that save time and money!  ...  Ideal for equilibrium dialysis, continuous-flow dialysis, and micro and thin-layer dialysis. Furnished with explosion-proof motor, if desired.  ... 
doi:10.1126/science.129.3347.502 pmid:17783995 fatcat:px33tnlp6zfytamla3vzt236my

Which Mathematical Logic is the Logic of Mathematics?

Jaakko Hintikka
2012 Logica Universalis  
History of Logic vs. History of Mathematics. Jaakko Hintikka. 032012 17 BIBLIOGRAPHY  ...  Acknowledgements This paper was written when Jaakko Hintkkka was a Distinguished Visiting Fellow of the Collegium for Advanced Studies of the University of Helsinki.  ...  It would have been an impressive proof of the significance of Frege's logic if he had pointed out how it enables us to formulate the epsilon-delta technique.  ... 
doi:10.1007/s11787-012-0065-6 fatcat:7egblszkrjg7lmnutjlz2igfqu

Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations

Takayasu Matsuo
2008 Journal of Computational and Applied Mathematics  
As examples of the application of the present method, it is shown that several dissipa-$tive/cooervative$ Galerkin schemes in the literature can be systematically derived.  ...  A new method is proposed for designing Galerkin schemes that retain the energy dissipation or conservation properties of nonlinear partial differential equations such as the Cahn-Hilliard equation or the  ...  The first class is that given by all real-valued PDEs of the form of equation (3) : $\frac{\partial u}{\partial t}=(-1)^{s+1}(\frac{\partial}{\partial x})^{2\epsilon}\frac{\delta G}{\delta u}$ $s=0,1,2  ... 
doi:10.1016/j.cam.2007.08.001 fatcat:74rtpioogbhs5nhmqiolylvms4

On a stability of heat kernel estimates under generalized non-local Feynman-Kac perturbations for stable-like processes [chapter]

Daehong Kim, Kazuhiro Kuwae
2014 Festschrift Masatoshi Fukushima  
{f}$ with $\tilde{f}(\partial)=0$ (the proof is similar to the proof of [6, Theorem 2.1.3] if X is transient, for general case, it can be proved by way of time change method).  ...  $f|_{F_{k}\cup\{\partial\}}$ ) is continuous for each $k\in \mathbb{N}$ . Since X is conservative, any $\mathcal{E}$ -nest $\{F_{k}\}$ of closed sets is automatically a strict $\mathcal{E}$ -nest.  ... 
doi:10.1142/9789814596534_0016 fatcat:m53fkzgfsjdx3l4hrynntykwoi

Value Functions and Transversality Conditions for Infinite Horizon Optimal Control Problems [chapter]

Nobusumi Sagara
2008 Communications in Computer and Information Science  
In particular, the role of the transversality conditions at infinity is clarified.  ...  This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary  ...  Note also that the conditions (i) to (iii) and (v) of the hypothesis imply Hypothesis A.2 and, thus, the convexity of the value function.  ... 
doi:10.1007/978-3-540-87477-5_30 fatcat:ro6egqjjgbdtxdcpqotkenmxk4

Value Functions and Transversality Conditions for Infinite-Horizon Optimal Control Problems

Nobusumi Sagara
2010 Set-Valued and Variational Analysis  
In particular, the role of the transversality conditions at infinity is clarified.  ...  This paper investigates a relationship between the maximum principle with an infinite horizon and dynamic programming and sheds new light upon the role of the transversality condition at infinity as necessary  ...  Note also that the conditions (i) to (iii) and (v) of the hypothesis imply Hypothesis A.2 and, thus, the convexity of the value function.  ... 
doi:10.1007/s11228-009-0132-1 fatcat:e3jogx4qubduvjf4xbsxocssle

Large deviations for random matrices [chapter]

Fumio Hiai, Dénes Petz
2006 Mathematical Surveys and Monographs  
Although it was proved in [BG] , our proof is more transparent in a bit more general setting.  ...  }\subset X$ such that $\lim_{\epsilonarrow}\sup_{0}\epsilon\log P\epsilon(K_{\delta}^{c})<-1/\delta$ .  ... 
doi:10.1090/surv/077/06 fatcat:347c4nqxybgcra64btzdign4vu

A UNIFIED THEOREM WITH SOME APPLICATIONS TO GENERALIZATIONS OF G. REEB'S THEOREM

Toshiyuki MAEBASHI
1963 Journal of Faculty of Science Hokkaido University Series I Mathematics  
_{*}(X)\Vert\leqq\epsilon/\delta$ .  ...  Hence we see $f(z)\in\sigma_{2\cdot f(y)}$ , namely, $f(\sigma_{1\delta})\subset\sigma_{2\epsilon}$ . This completes the proof.  ... 
doi:10.14492/hokmj/1530691416 fatcat:hmziuac23nf3ngdzqlr2drtdee

Experiences with a method for enclosing solutions of systems of equations

Gerhard Heindl
1995 Journal of Computational and Applied Mathematics  
In a recent paper the author has demonstrated how to derive inclusions of solutions of systems of equations from hypernorm estimates and a procedure for computing upper bounds for vectors of the type (  ...  Two algorithms for computing componentwise inclusions, one for solutions of nonlinear problems and one for solutions of linear problems are presented.  ...  (infl(HS, epsilon) denotes the epsilon inflation of HS as produced, e.g., by the operator inflated in [12, p. 178].  ... 
doi:10.1016/0377-0427(94)00084-e fatcat:36ve272wxjdwdigatmavtc4pxe

Linear Evolution Equations in Banach Spaces

Naoki Tanaka
1991 Proceedings of the London Mathematical Society  
PROOF: By Theorem 2.1 there exists a unique pair $(\{V_{1}(t, s)\}, \{V_{2}(t,s)\})$ of strongly continuous families of bounded linear operators defined on the triangle $\triangle=\{(t,s);0\leq s\leq t  ...  PROOF: By virtue of Lemma 2.3 we can show (2.2) in the same way as in the proof of [2,Theorem 2.1]. To prove (2.3), let $0\leq r\leq s\leq T$ and let $\lambda>0$ be such that $\lambda\omega_{3}<1$ .  ... 
doi:10.1112/plms/s3-63.3.657 fatcat:7x5tllda45hajkpljd75xd4nae
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