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Path integrals on finite sets

Erik G. F. Thomas
1996 Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications  
The path integral coresponding to a "two particle system without interaction" is the direct product of the corresponding path integrals.  ...  The propagator for a "two particle system with interaction" can be obtained by repeated integration.  ... 
doi:10.1007/bf00047924 fatcat:zcqcsotvb5f7dkyx52dgivcukq

On traces of Fourier integral operators localized at a finite set of points [article]

P. A. Sipailo
2016 arXiv   pre-print
(Φ) on X (in the sense of relative theory). We discuss the situation when i^!(Φ) has the form of a Fourier--Mellin operator and, in particular, is localized at a finite set of points.  ...  Given a smooth embedding of manifolds i: X M and a Fourier integral operator Φ acting on M, obtained by quantization of a canonical transformation, consider its trace i^!  ...  Note that Fourier-Mellin operators can be associated with a finite set of points in an obvious way.  ... 
arXiv:1612.06231v1 fatcat:ajduaxia5rflridcujwlir3l44

Singular integral operators on Nakano spaces with weights having finite sets of discontinuities [article]

Alexei Yu. Karlovich
2010 arXiv   pre-print
We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves.  ...  to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves.  ...  By P C(Γ) we denote the set of all a ∈ L ∞ (Γ) for which the one-sided limits a(t ± 0) := lim τ →t±0 a(τ ) exist and are finite at each point t ∈ Γ; here τ → t−0 means that τ approaches t following the  ... 
arXiv:1002.4813v1 fatcat:7aetrv35r5hstoomy3eolufnv4

Singular integral operators on Nakano spaces with weights having finite sets of discontinuities

Alexei Yu. Karlovich
2011 Banach Center Publications  
We extend this result to the case of Nakano spaces (also known as variable Lebesgue spaces) with certain weights having finite sets of discontinuities on arbitrary Carleson curves. where Γ(t, R) := {τ  ...  to the Cauchy singular integral operator, acting on Lebesgue spaces over Lyapunov curves.  ...  By P C(Γ) we denote the set of all a ∈ L ∞ (Γ) for which the one-sided limits a(t ± 0) := lim τ →t±0 a(τ ) exist and are finite at each point t ∈ Γ; here τ → t−0 means that τ approaches t following the  ... 
doi:10.4064/bc92-0-10 fatcat:ysbqz4m7ujberp2qxuaeghmx7q

A note on product of measures [article]

Grzegorz Andrzejczak
2018 arXiv   pre-print
The operation applies to arbitrary (not necessarily σ-finite) measures and is consistent with the Fubini--Tonelli theorem.  ...  Corollary 2 . 9 . 29 The Lebesgue integral with respect to an arbitrary measure µ equals dµ σ i.e. integrability as well as the integral depend on the σ−finite component µ σ only.Given any measurable space  ...  [S] ) assures that the Daniell-Stone integral d(μ ⊗ν) is equal to the Lebesgue integral with respect to a measure, say λ, and the corresponding σ−finite sets form the σ−ring S σ µ ⊗ T σ ν .  ... 
arXiv:1810.11485v1 fatcat:upev2obcj5abrneijhz2gprd4i

The arithmetic puncturing problem and integral points [article]

David McKinnon, Yi Zhu
2018 arXiv   pre-print
In particular, if an algebraic variety V has a dense set of rational points, they ask whether or not the set of D-integral points is potentially dense, where D is a set of codimension at least two.  ...  We also discuss some stronger notions of integrality of points, and give some positive answers to some cases of the analogous question in the stronger context.  ...  One can generalize this notion to that of an S-integral point, by permitting intersections over a finite set S of places.  ... 
arXiv:1806.03180v1 fatcat:etedoqobdnfbzg5rwcxflhwpnq

Separability of stochastic processes

S. H. Coleman
1963 Proceedings of the American Mathematical Society  
A consistent family of integrals on continuous functions of a finite number of variables is extended by the procedure (developed by McShane and Bourbaki) outlined in §1 to an integral on functions defined  ...  The integral. The integration theory used here can be found in detail in [l; 4; 6]. A set of elementary functions on a set 0 is a vector lattice 8 of realvalued functions on Q.  ...  Suppose that, for each finite subset A of T, I a. is an integral on the continuous functions based on A with 7¿(1) = 1.  ... 
doi:10.1090/s0002-9939-1963-0143248-3 fatcat:dna5kz4so5hyvidoktfypir6du

Separability of Stochastic Processes

S. H. Coleman
1963 Proceedings of the American Mathematical Society  
A consistent family of integrals on continuous functions of a finite number of variables is extended by the procedure (developed by McShane and Bourbaki) outlined in §1 to an integral on functions defined  ...  The integral. The integration theory used here can be found in detail in [l; 4; 6]. A set of elementary functions on a set 0 is a vector lattice 8 of realvalued functions on Q.  ...  Suppose that, for each finite subset A of T, I a. is an integral on the continuous functions based on A with 7¿(1) = 1.  ... 
doi:10.2307/2033981 fatcat:dxfmcpyvrngf5a4whwdc7t6yxq

The radon-nikodym theorem. I

A.C. Zaanen
1961 Indagationes Mathematicae (Proceedings)  
finite, and XAn the characteristic function of the set An E r of finite ,u-measure) is a linear vector lattice Ls, and J(s) = .2f cn,u(An) is an elementary integral on Ls.  ...  Similarly, by extending the elementary step function integral with respect to v on the ring .1h of all sets of finite v-measure, we obtain ff.  ... 
doi:10.1016/s1385-7258(61)50016-9 fatcat:cu2yts4nw5fkhbaqnrwrntnqna

On the theory of improper definite integrals

Eliakim Hastings Moore
1901 Transactions of the American Mathematical Society  
Now we take as type X"° of the set 3*°° = {±oo } on finite intervals the (extended) type of proper definite integrals defined in § 1 16°.  ...  f depends upon the type f of proper definite integrals, except that now the set 30 is of arbitrary content 7(30) and does not necessarily lie on a finite interval.  ... 
doi:10.1090/s0002-9947-1901-1500580-0 fatcat:hxzvhmhw35ajzc4kqgawh7tkpq

Page 255 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 97, Issue 2 [page]

1960 American Mathematical Society. Transactions of the American Mathematical Society  
If either fyxt+ or fyx- defines a o-finite measure on Gin W’, then x is said to be a-integrable on © in W’. If x is o-integrable on § in W, we will just say that x is o-integrable.  ...  In the following (W, §, u) will be a fixed g-finite measure space, where W is an abstract set with points w, § is a s-algebra of subsets of W, yu is a o-finite complete measure on W.  ... 

Probability on Finite Set and Real-Valued Random Variables

Hiroyuki Okazaki, Yasunari Shidama
2009 Formalized Mathematics  
Probability on Finite Set and Real-Valued Random Variables In the various branches of science, probability and randomness provide us with useful theoretical frameworks.  ...  If dom f = ∅ and M (dom f ) < +∞, then f is integrable on M . (8) Let O be a non empty finite set and f be a partial function from O to R.  ...  Then X is integrable on P .(31) Let O be a non empty finite set, P be a probability on the trivial σ-field of O, X be a real-valued random variable of the trivial σ-field of O, F be a finite sequence of  ... 
doi:10.2478/v10037-009-0014-x fatcat:wf7qobx2gbgh5ldx4tebrof2la

Integral Function Bases [article]

Raymond Hemmecke, Robert Weismantel
2004 arXiv   pre-print
Integral bases, a minimal set of solutions to Ax≤ b, x∈^n that generate any other solution to Ax≤ b, x∈^n, as a nonnegative integer linear combination, are always finite and are at the core of the Integral  ...  In this paper we present one generalization of the notion of integral bases to the nonlinear situation with the intention of creating an integral basis method also for nonlinear integer programming.  ...  (b) If a finite integral generating set of S exists, then there is a unique integral basis of S.  ... 
arXiv:math/0410225v1 fatcat:nyjue5bekzf5zmwaobedf6ktje

Page 468 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 2, Issue 4 [page]

1901 American Mathematical Society. Transactions of the American Mathematical Society  
and does not necessarily lie on a finite interval.  ...  (a) With respect to any interval a) and point-set 2 or interval-set J the no- tation =, or J, denotes the point-set of = or interval-set lying on J which lies ah on ab. * The extremities + 1 are the transforms  ... 

Page 411 of Mathematical Reviews Vol. 43, Issue 2 [page]

1972 Mathematical Reviews  
The paper’s approach to vector integration is now the | following. The integral of an X-valued function x(-) with | respect to p is to be a (finitely additive) set function y(:)=(e—f, x du) on &.  ...  For ¢ the identity function, one gets the weak and total variations and the spaces V¥ and W of set functions of weak finite variation and totally finite variation, respec- _ tively.  ... 
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