The Skorokhod reflection problem for functions with discontinuities (contractive case)
|
|
- Elfrieda Walker
- 6 years ago
- Views:
Transcription
1 The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection problem for discontinuous functions (but rightcontinuous with left limits) are collected in these notes. In particular, we are concerned with existence and uniqueness in dimensions when the reflection matrix is of contractive type. Some proofs are new, while others are modifications and extensions of existing proofs for the more well-known reflection problem for continuous functions. The article has an overview character, in that the results and proofs are self-contained and are presented fully even in the simplest case of dimension one. Some variations in the proofs, resulting in short and intuitive arguments, reflect the author s own taste. We are also concerned with continuity and discontinuity properties of the reflection mapping under a variety of topologies. The reflection mapping has become a standard tool in numerous areas and the scope of this article is to provide a rigorous and accessible review of part of the subject. 1 Introduction: the physics of the problem The Skorokhod reflection problem is defined in a domain of, which is assumed to be a closed convex set with nontrivial interior. Associated to the boundary is a cone field, i.e., an assignment of a cone to each point of. This is all that is needed for the definition of the problem. Physically, what happens can be described as follows. We are given a free or unconstrained motion "!#!%$'&(, which is a function from ) &*+, into. The function is taken to be continuous or, more generally, right-continuous with left limits at all points of time, i.e. an element of the space - ) &*+,. The problem is to find a function./0.1"!##!2$3&(, of locally bounded variation, such that "!546.1"!# belongs to, for all times!, and, what is more important,. should be chosen so that it is minimal 1 with respect to the cone field. Historically, the problem was first formulated for,7) &*+, in [Skorokhod 1961] as a means of pathwise construction of a solution to a stochastic differential equation with reflection. The physical description of this section can be 8 takis@alea.ece.utexas.edu 1 in some sense to be made precise later 1
2 2 felt in the paper of [Tanaka 1979], where the cone field consists the normal cones at the boundary points of a convex region. If "! happens to be in for all!, then. is identically zero. Otherwise,. should increase only at those times! for which the constrained motion "! 4.1"!# belongs to the boundary. Now, this increase should be consistent with the cone field, i.e., if at time!, "!54.1"!#, then the increase of. at the same time should be contained in the cone " "!#146."!#. The above physical definition can be made precise. We are not going to that in its generality but only work with some special, yet important, cases. We refer to the above problem (i.e. the rigorously defined version of it) as the SRP in with cone field. In many cases, the more restricted version of it, where the free motions are supposed to be continuous, is of importance. By solution to the problem we mean finding the function. or the resulting motion 4.. There are many questions that arise. First, it is clear that not all cone fields make sense. For example, if the cone points toward the exterior of at some point, then it should be intuitively clear that there is no solution for a particularly chosen free process. On the other hand, even if the cone points toward the interior, we are still not sure whether we have 0, 1 or multiple solutions. To date, we do not know much about existence and uniqueness of the SRP in general. However, we know how to give precise existence conditions for the SRP in the positive orthant. Regarding uniqueness, not much is known, even in this special case. One canonical case which is well-studied is what we will call the contractive case. This refers to a SRP in, the closed positive orthant, with specially chosen cone field. Our purpose is to review this case in detail, starting from the trivial case of dimension 1, and provide proofs for free motions in - ) &*+,. These are extensions of existing proofs for continuous free motions. After presenting the proofs, we will deal with some continuity results and point out some discontinuities of the reflection mapping. 2 Dimension one A closed convex domain here is either a compact interval ), or an interval of the type ) +, or +. We shall consider only the case ) &*+,. Associated with the boundary point &( is a trivial cone that either points down or up. In the former case, it is easy to see that the SRP makes no sense. As before, - ) &*+, is the set of right-continuous with left limits functions with real values, while - ) &*+ is the set of functions in - ) &*+, which are increasing 2. Rigorously then, the problem can be cast as follows. - ). ) 4. & Definition 1 (Standard formulation of the SRP). Given a function &*+,, we say that - &*+, is a solution to the SRP iff (i) $,&, (ii) 1 "!#,&.1"!#. Let us note that, by convention, we set.1& &, and the integrator. in the above Lebesgue- Stieltjes integral assigns mass.& 7.1& at the point! &. That the problem has a unique 2 increasing is used in the sense of non-decreasing
3 3 solution is well-known. We repeat the steps that lead to the proof of this fact, as well as the standard formula. Theorem 1. The SRP as specified in Definition 1 has a unique solution. Proof. Let - ) &*+, be given. Note that we will not assume that & &. Any real value for & is possible. By convention, we let & &. Same convention applies to all elements of - ) &*+. Suppose that. is a solution (if any) of the SRP of Definition 1. Then, for any &!, Putting the above together we arrive at 3 :."! $.1 $ because $,&*."! $,&* since, by assumption,. is non-negative.."! $ #& (1) We thus proved that if there is a solution then (1) should hold. The right hand side of (1) is important so we give it a name: (2). "! ) & Note that we only used requirement (i) to derive this. Next we show that. is a solution itself. It is immediate that. is increasing and that 4. $,&. So (i) is satisfied. Let 4.. It remains to show (ii), i.e. that 1 "!,&. "! &. We claim that if "! &, then there is &, such that. "! 4. "!. The claim follows from right continuity: Given any &, we can find &, such that "!#, for all /)!#!4!. Since "!,&, we can choose &"#$", "!#. So we have that, for all )!#!14%, 6 "!# % "!# / "!#. "!. Hence, from (2), Again from (2), & ('. "!14# )+ (' & ) &,-. ) * & (3) & (' Due to (3), the last term can be dropped, and hence. "! 4/ ) &01.. But the latter quantity is, due to (2),. "!. Thus, for all! $ &, "! & implies that. "! 4. "! and so the integral in (ii) is zero. We have constructed a solution given by formula (2). For this solution we have "! ) "! "!2 The above proof is not the only one. One can devise other proofs of existence, perhaps more sophisticated and shorter. But the proof given is, at least, very mundane. 3 where 3 stands for maximum and 4 for minimum
4 4 What about uniqueness? Again there are many ways to prove that the starred solution above is the only one. We adopt the most straightforward one. We first argue that any solution must start from 4.1& & 0 This actually follows easily from (ii). According to our convention (stated just after Definition 1), & &.1& " & 46.& #.1& 2 If & & then.& & and so.1& &. If & &, then & 4.& &, and there is no choice other than.& &. The only case left is & "3&. But we know that.1& $'. & 2$ &. So.1&,&. Thus the first term of the display above becomes null:.1& 4 & &. Suppose then that. is another solution, different from.. From what we showed earlier,. can only be larger than or equal to., while.1&. &. Hence our assumption that. is distinct from. can only mean that. is strictly larger than. at some point. Since. and. start from the same value, we must have that.. for some & and for all sufficiently small. (4) But then.1. $. So.1 4 '&. From (ii) of Definition 1 we deduce the existence of some (small) positive for which.1 4%.1 and this contradicts (4). Hence. is unique. There is another formulation of the SRP which is equally useful. It is the following: Definition 2 (Monotonic formulation of the SRP). Given - ) &*+,, consider the set. ) &*+ %46. $&(, equipped with the componentwise partial ordering. A solution to the SRP - is defined as a minimal element of this set. Our previous discussion showed the following: Theorem 2. The SRP as specified in Definition 2 has a unique solution that coincides with.. This definition is more general than the previous one, because it does not require that have any structure. Indeed, we need not specify that is an element of - ) &*+,. Any 1) &*+ can be used. Similarly, the topology of the domain ) &*+, is irrelevant: only the order structure is needed. Under such generalizations, the problem of Definition 2 makes sense and Theorem 2 holds. 3 Continuity properties We now view the SRP as defining a mapping from - ) &*+, into itself, associating to an -) &*+, the function. of (2): "!# ) & 4 here stands for ; we also use for (5)
5 5 We also let "!# "!54 "!# ) ) "!# "!2 (6) The point of this section is to show that and are continuous under any reasonable topology on - ) &*+. Two topologies are considered in this section: the -topology, or topology of uniform convergence on compact sets, and the -topology introduced in [Skorokhod 1961]. A third topology will be considered in a later section. To start, note that (5), (6) imply that if is continuous then, are also continuous functions. The trivial inequality % where, are two collections of real numbers indexed by the same set, is responsible for the inequalities & "!# "!# & "! "! & "!# "!# (7) & "!# "! & "!# "!# # - ) for any,&. Thus, if we equip &*+, with the -topology, which is induced by the norm we have implying that Recall now that the is continuous under this topology. & (8) topology (the so-called usual Skorokhod topology): Let be the set of strict time changes of ) &*+,, i.e. the set of continuous strictly increasing functions ) &*+ ) &*+,, with & 3& and +, 7+. We say that such that -converges to the identity (id), and -converges to. It is clear that " #! id -converges to iff there is a sequence (9) is a metric for the topology. To see that and are -continuous (i.e. when both the domain and codomain are given the topology) observe the following simple lemma: Lemma 1. For any, and any,- ) &*+,, we have ", and " 1. Proof. Direct application of formula (5) for and (6) for. Theorem 3. Both and are -continuous.
6 6 Proof. Suppose that -converges to. Then there is, -converging to id, such that -converges to. Then, for the same sequence, we have ". But -converges to (composition is -continuous), and is -continuous, so -converges to. Similar argument holds for. The reason that we discussed the and topologies now, is that we are going to use them to prove existence and uniqueness in many dimensions. One more point should be stressed. The space -) &*+ is not complete under the metric of (9). The reason is that measuring the deviation of from id in a uniform fashion is too crude. To make the space complete we need to measure the deviation of from id by measuring the deviations of its slopes, in a sense; see [Billingsley 1968, pg. 112]. So we introduce the norm on by and then set It can be shown that that " #.&. "!#! %, generate the same topology, and that " # This obvious inequality will be useful in the next section. 4 Multidimensional case 2 (10) is complete. Furthermore, notice (11) We consider a very canonical instance of the SRP in. The boundary is the union of the -dimensional faces &( The relative interior (with respect to ) of face is the set 5 &*#,& for all # Given vectors (or, rather, directions) # each associate the positive cone spanned by those directions for which in, we defined the cone field as follows: To we associate the positive cone (half line) generated by. To any other, we. Formally then, "1 1" & $,& Thus, to the origin &, we associate the cone spanned by all vectors. The vectors also called reflection directions. The reflection matrix is a matrix. # # are whose columns are the vectors
7 7 To say that a (locally bounded variation) function "!#!%$3&(, with values in increases, at time!, in a direction contained in the cone spanned by #, is to say that there are nonnegative numbers. "! #. "!, not all zero, such that "!. "! "!#. This simple observation, along with the informal definition of the SRP given in the Introduction, motivates the following: Definition 3 (Standard formulation in ). Given a matrix and - ) &*+, we say that. - ) &*+, is a solution to the SRP( ) iff (i) 4. $,&, (ii) 1 "!,&.#"!# &,. We will use the notation SRP( ) for this problem. As in dimension 1, we want to consider conditions that ensure existence and uniqueness. A word of caution: when we say uniqueness we may mean uniqueness of the reflector. or uniqueness of its representation.. In more general domains and/or for more general cone fields, this can be an issue. However, for the case considered here, the matrix will turn out to be invertible, so there is no ambiguity. Our assumption then is: Assumption C: The matrix ( =identity) is non-negative with spectral radius ". We refer to this as the contractive case for obvious reasons. We then have the following theorem: Theorem 4. Given a matrix satisfying assumption C, the SRP on with reflection matrix has a unique solution. The idea of [Harrison and Reiman 1981] and [Reiman 1984] is to consider an equivalent problem. The proof presented here differs slightly from their proof in that may be a discontinuous function. Also, existence and uniqueness are proved without reference to the alternative formulation of the SRP (see Definition 4) below. Proof. First consider a trivial instance of the problem, namely the SRP( ) (i.e. take the reflection matrix to be identity). Then the problem is a collection of 1-dimensional problems, and hence existence and uniqueness is immediate coordinate by coordinate. Let. denote this solution, when the free process is. The index refers to the identity reflection matrix. We also set 34. The results of Section 3 show that and are continuous both in the and topologies. Now consider satisfying Assumption C and set. Our equation 4. (see (i) of Definition 3) can be written as and since (ii) should hold, it follows that " ".. (12) and this is a fixed point equation. Since ", the inverse of is non-negative matrix. With denoting the all ones column, let, and is a
8 Notice that has strictly positive components. Instead of the usual norm on, we introduce the -norm defined by: This induces a norm on the matrices, which is easily seen to be: With this norm, we have " &, i.e. To see this, observe that & for all, or, $" for all. Thus, is a contractive linear function on ". (It was to be expected that is contractive under some norm, because ", but what we show here is that it is so under the - norm, and this is of relevance.) We also introduce the corresponding uniform norm on - ) &*+,, as in (8), and the metric, as in (10): " #! & # - ) &*+,2 The first one generates the -topology. The second one generates the complete and " #. It now follows, from (7) and the definition of, that: "! "!# "! "! Fix now a free motion matrix norm yield: Thus ) ) # - ) &*+,. The above inequalities and the fact that 2". 2". ". 8 topology. They are both is an induced ". ". ". 2".... is a contraction under the norm. Let be the operator applied times to a fixed / ( is thus Cauchy under the norm. Hence it metric. But this metric is complete and generates the topology. Hence -converges to, say,.. Now is -continuous and hence -continuous. Thus ) &*+. The sequence is Cauchy under the converges to.. But 2 and this of course converges to.. Thus. 2". and this proves that the limit point. solves the SRP( ). If there are two solutions, then the previous
9 9 inequality shows that they must be identical. So there is a unique solution. As in dimension one, we have another problem: Definition 4 (Monotonic formulation of the SRP). Given a matrix and, - ) &*+,, consider the set. - ) &*+, 4. $,&(, equipped with the componentwise partial ordering. A solution to the SRP( ) is a minimal element of this set. The equivalence of the two definitions needs a proof. Theorem 5. Assume that condition C holds. Then the SRP( ) of Definition 4 has a unique solution which coincides with the solution of the SRP( ) of Definition 3. Proof. We must show that any. - ) &*+,, such that 24. $ &, satisfies.$., where. is the unique solution of the SRP( ) in its standard formulation. Recall the definition of from (12) through which is is componentwise defined as in (5). It is then clear that is increasing in the natural partial order of functions: For all. #. - ) &*+,.6... Fix. - ) &*+,, such that 24.$,&. Thus ". 14. $,&. From Theorem 2 we obtain and so, by induction,. $ "... $. for all $ We now show that we can pass to the limit in this inequality. Recall from the proof of Theorem 4 that. -converges to.. Thus, there is a sequence such that -converges to id, and. -converges to. ; in particular, the convergences hold pointwise; the inequality in the display above also holds pointwise. Hence. "!##,.1 "!## for all! and. Taking limits we get. "!# #.1 "!#. But, given any,&, we can find such that "!! 4! for all $. Thus, for all $ we have.1 "!##,."! 4. Hence #.1 "!##,."!, by the right-continuity of.. Thus, in the end, we get This concludes the proof.. "!6.1"!# for all! $,&. 5 Behavior under weaker topologies Even though the the notion of convergence. We introduce the - ) &*+,, let topology is standard in the space - ) &*+,, it is often necessary to weaken " "!# "!##! $,&( topology, defined in [Skorokhod 1956]. Given
10 10 be the graph of, and let "1 "! 5 "! 4 "!##!$&( (13) be the linear extension of the graph of. Both sets are subsets of ) &*+,. Definition 5. An where the first coordinate -parametrization of an element of - ) &*+, is a bijection &, while the second coordinate is simply continuous, i.e. ) &*+,. ) &*+ is continuous, increasing 5, with Definition 6. We say that a sequence - ) &*+ exist -parametrizations of, (, and -converges to. " & -converges to an,6- ) &*+, if there -parametrization of, such that Note that this is a well-defined notion of convergence on - ) &*+, that leads to the usual definition of closed sets. The topology can be seen to be strictly weaker than : for instance, if we consider "! 1"! $ and "!# "! &, 6 (, we see that there is no way for to -converge to, but that does, indeed, -converge to ; see Figure 1. 1 x x n 1 1+1/n t Figure 1: converges in to, but not in 5.1 -continuity in dimension one We specialize to dimension first. The following propositions and lemma are responsible for the -continuity of the reflection mapping. Note that for an increasing function we let "! $,&!! $,&* be its generalized (right-continuous) inverse function. We also let "! "! $,& $!! $,& 5 but by no means strictly increasing in general
11 11 In the sequel we fix - ) &*+ and consider an notation, we set It is helpful to recall that "! # "!# ranges in in " as ranges over ) &*+. -parametrization 5 2 Proposition 1. Consider - ) &*+ and let 5 be an Proof. Recall that 3 1 2) &*+,. To ease the " as! ranges over ) &*+,, while # ranges -parametrization. Then (14) (15) "1 is a bijection. Hence any point in "1 has a unique pre-image in ) &*+,. In particular, consider the point ## for an arbitrary $&. This ## actually belongs to "1 which is contained in ". Hence it has a unique pre-image under. Call this pre-image. (See Figure 2.) Thus, * ##. x x trajectory x ( λ (u)) p(u) 0 t 0 λ(u) t u λ 1 ( λ(u) ) = v λ trajectory u Figure 2: Behavior of the -component of an function parametrization around a typical jump of a typical
12 12 But * *#, and so * # ##2 (16) Equating the first components of (16) we have, and it is easily seen that # #. Equating the second components of (16) we have * #. Substituting for we have # ## # and this holds for an arbitrary. But id, because is continuous. Hence, setting "! in the above display, we get "!# "! and this proves (14). To prove (15), i.e. that "! #% "!, for any!, we observe that # for some. So we set in place of in the last display, take limits, and argue similarly. Proposition 2. Under the assumptions and notation above, we have: Proof. Owing to proposition 1, we have to prove or that (see eqrefname2) (17) (18) # " for all! $&. But this follows from the fact that is strictly increasing. The second equality is proved similarly. Lemma 2. Consider - ) &*+, and let 1 be an 5 is an -parametrization for. -parametrization for. Then Proof. We only need to prove that. ' 5 1 is a bijection from ) &*+, into the linear extension of the graph of. By the assumption that 5 is an -parametrization for, we have that is a bijection from ) &*+, into ". So it suffices to show that there is a bijection "1 1 such that.2, or that the following diagram commutes: ) &*+ ) &*+ " 1 The function is defined in the obvious way: "! 5 "! 4 "!# "! "! 4 1 "!##! $ &* & % (19)
13 To show that it is one-to-one, take two distinct points in 13 " and show that the images are distinct. We only need consider distinct points of the form "! "! 4 "!## and "! "! 4 "!## for. If their -images were equal we would have 1 "! 4 "!# "! 4 1 "!#, and this means that 1 "! 1 "!# which means that "! "! (by the very definition of ). But then "! "! 4 "!# "!# "!## "! "! 4 "!##, and this contradicts the assumption that the points are distinct. It is equally clear that covers 1. So is a bijection. We now check that the range of To check that this holds we use the definition (19) of and verify that the second coordinates are equal by applying (17) and (18). Theorem 6 ( -continuity in one dimension). 1- ) &*+ - ) &*+, is continuous when both domain and codomain are given the Proof. Suppose that is an -topology. convergent sequence with limit. Thus there are -parametrizations is -continuous, of, and 5 of, such that -converges to 5. Since we have that -converges to 1. On the other hand, due to Lemma 2, is an -limit the function. -parametrization of and 1 is an Now notice that, unfortunately, - ) &*+, is not a topological vector space [Whitt 1980] be- continuous, as the following little example shows; see Figure 3. Let convergent to cause addition is not -parametrization of. Hence has as "!# 1! &, "! "!, (. Both sequences are 1"!$,& and 1"! $,&, respectively. We would expect that their sum converges to the sum of their limits which is zero. But their sum is not -convergent. So continuity of cannot be deduced from the continuity of. It must be considered separately. It is easily verified that Propositions 1, 2 and Lemma 2 remain true with in place of. Thus Theorem 6 is also true with in place of, showing that is also continuous in dimension one. 5.2 Dimension two and higher In this section we show, by examples, that not everything is fine with the $. topology in dimension The following observation was made by [Lin 1996]. There is something unnatural about the way that jumps are connected in the topology. Namely, it is required that an -parametrization of a function ' - ) &*+, connect "! and "! by a straight line; see (13) and Definition 6. Unfortunately though, the mapping does not necessarily map straight segments into straight segments. This geometrical observation is the reason behind the fact that is not -continuous for $. To put it in another way, look at the natural map of (19) (with in place of ). The image of "1 under is not in general. For example, consider the SRP( ) in dimension with the trivial reflection matrix & &
14 !!!!! 14 1 x n 0 1/n t 1 x n 1 x n + x n 0 1/n 2/n t Figure 3: The sum of the and consider the function "! " where & "! "! "!# "!# limits of, is zero; however, 4 does not converge in #"!*!!! #"!!!!! $ &* (20) ( until time!, " ; see Figure 4. This function remains at the point moves linearly from to 77&* between! and!, and remains at thereafter. Since is identity, we obtain its reflection by reflecting each component individually: The straight segment joining straight segments ) and "! " "!# "! 2 (21) by is bent by the reflection and becomes the union of the and ) &, where 37 &. How this specifically affects the continuity of at the point can be exhibited by choosing! and! so close to a fixed time, say, so that "! -approximates a function that jumps from to at time. Specifically, let!, that "!#, let "! be the function given by (20), and "! its reflection as in (21). It is clear -converges to the function "!# 1"! " 4 1"! $. The reflection of is 1"! ". However, does not converge in at all. It is easy then to conjecture that is not the right topology for. A weaker topology in its codomain or a stronger topology in its domain, that takes into account the special nature of, is required. Under such a modification one hopes that will become continuous. The problem is to find a modification that also has a number of other interesting properties (preserves continuity of certain
15 15 x 2 r 1 A = (2,1) graph of z r 2 0 C= (1,0) x 1 (0, 1) = B graph of x Figure 4: Straight-segment graphs can be bent when reflected in 2 or more dimensions popular functionals, for example). While this article was being written, [Whitt 1999b] proposed an interesting modification which makes the mapping continuous. Loosely speaking, a weaker topology is obtained by allowing jumps to be connected, not necessarily by straight segments, but continuous segments constrained in a box; for details we refer to the preprint [Whitt 1999a]. 5.3 A very weak topology Finally, we discuss the topology. Again this was introduced by [Skorokhod 1956], where its exact definition can be found. is considerably weaker than because it ignores the! -dependence of functions and only looks at their graphs. Loosely speaking, two functions # are -close if their graphs are uniformly close. It was observed in [Lin 1996] that if we endow both domain and codomain of with, then is discontinuous even in dimension one. This is shown by the following example; see Figure 5: Let be defined by linearly interpolating between the points & &,, &,, and "! for all! $. Then - converges to "!# 1"!%$. Now, with,, we have: is obtained by linearly interpolating between the points & &,, &, and "!# & for all!$ ; on the other hand, "! &. It is clear that the graph of is not close to the graph of. Hence does not -converge to. References [Billingsley 1968] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
16 16 1 z n 0 1 3/n 1 t 1 x n 1/n 1/n 1/n Figure 5: converges in to 1"!$, but does not converge in [Harrison and Reiman 1981] Harrison, J.M. and Reiman, M.I. (1981). Reflected Brownian motion on an orthant. Ann. Probab. 9, [Lin 1996] Lin, S.-J. (1996). Lévy and fractional Brownian motions for modeling and performance analysis in high-speed communication networks. Ph.D. dissertation, Univ. of Texas at Austin. [Reiman 1984] Reiman, M.I. (1984). Open queueing networks in heavy traffic. Math. Oper. Res. 9, [Skorokhod 1961] Skorokhod, A.D. (1961). Stochastic equations for diffusions in a bounded region. Theory Probab. Appl. 6, [Skorokhod 1956] Skorokhod (1956). Limit theorems for stochastic processes. Theory Probab. Appl. 1, [Tanaka 1979] Tanaka, H. (1979). Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9, [Whitt 1980] Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res 5, [Whitt 1999a] Whitt, W. (1999a). On the Skorohod topologies. Preprint. [Whitt 1999b] Whitt, W. (1999b). The reflection map is Lipschitz with appropriate Skorohod metrics. Preprint.
THE SKOROKHOD OBLIQUE REFLECTION PROBLEM IN A CONVEX POLYHEDRON
GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 2, 1996, 153-176 THE SKOROKHOD OBLIQUE REFLECTION PROBLEM IN A CONVEX POLYHEDRON M. SHASHIASHVILI Abstract. The Skorokhod oblique reflection problem is studied
More informationTheorems. Theorem 1.11: Greatest-Lower-Bound Property. Theorem 1.20: The Archimedean property of. Theorem 1.21: -th Root of Real Numbers
Page 1 Theorems Wednesday, May 9, 2018 12:53 AM Theorem 1.11: Greatest-Lower-Bound Property Suppose is an ordered set with the least-upper-bound property Suppose, and is bounded below be the set of lower
More informationare Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication
7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =
More informationA Criterion for the Stochasticity of Matrices with Specified Order Relations
Rend. Istit. Mat. Univ. Trieste Vol. XL, 55 64 (2009) A Criterion for the Stochasticity of Matrices with Specified Order Relations Luca Bortolussi and Andrea Sgarro Abstract. We tackle the following problem:
More informationGENERALIZED CONVEXITY AND OPTIMALITY CONDITIONS IN SCALAR AND VECTOR OPTIMIZATION
Chapter 4 GENERALIZED CONVEXITY AND OPTIMALITY CONDITIONS IN SCALAR AND VECTOR OPTIMIZATION Alberto Cambini Department of Statistics and Applied Mathematics University of Pisa, Via Cosmo Ridolfi 10 56124
More informationCHAPTER 2: CONVEX SETS AND CONCAVE FUNCTIONS. W. Erwin Diewert January 31, 2008.
1 ECONOMICS 594: LECTURE NOTES CHAPTER 2: CONVEX SETS AND CONCAVE FUNCTIONS W. Erwin Diewert January 31, 2008. 1. Introduction Many economic problems have the following structure: (i) a linear function
More informationINDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43
INDEX Abel s identity, 131 Abel s test, 131 132 Abel s theorem, 463 464 absolute convergence, 113 114 implication of conditional convergence, 114 absolute value, 7 reverse triangle inequality, 9 triangle
More informationMATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES
MATH 4200 HW: PROBLEM SET FOUR: METRIC SPACES PETE L. CLARK 4. Metric Spaces (no more lulz) Directions: This week, please solve any seven problems. Next week, please solve seven more. Starred parts of
More informationLecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University
Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................
More information1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3
Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationON THE UNIQUENESS PROPERTY FOR PRODUCTS OF SYMMETRIC INVARIANT PROBABILITY MEASURES
Georgian Mathematical Journal Volume 9 (2002), Number 1, 75 82 ON THE UNIQUENESS PROPERTY FOR PRODUCTS OF SYMMETRIC INVARIANT PROBABILITY MEASURES A. KHARAZISHVILI Abstract. Two symmetric invariant probability
More informationOptimization and Optimal Control in Banach Spaces
Optimization and Optimal Control in Banach Spaces Bernhard Schmitzer October 19, 2017 1 Convex non-smooth optimization with proximal operators Remark 1.1 (Motivation). Convex optimization: easier to solve,
More informationPart V. 17 Introduction: What are measures and why measurable sets. Lebesgue Integration Theory
Part V 7 Introduction: What are measures and why measurable sets Lebesgue Integration Theory Definition 7. (Preliminary). A measure on a set is a function :2 [ ] such that. () = 2. If { } = is a finite
More informationON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS
Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)
More informationPart III. 10 Topological Space Basics. Topological Spaces
Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.
More informationNotions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy
Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.
More informationThe Lebesgue Integral
The Lebesgue Integral Brent Nelson In these notes we give an introduction to the Lebesgue integral, assuming only a knowledge of metric spaces and the iemann integral. For more details see [1, Chapters
More informationTopological properties
CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological
More informationSome Properties of the Augmented Lagrangian in Cone Constrained Optimization
MATHEMATICS OF OPERATIONS RESEARCH Vol. 29, No. 3, August 2004, pp. 479 491 issn 0364-765X eissn 1526-5471 04 2903 0479 informs doi 10.1287/moor.1040.0103 2004 INFORMS Some Properties of the Augmented
More informationMATHS 730 FC Lecture Notes March 5, Introduction
1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists
More informationSome SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen
Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text
More informationA Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries
A Discrete Duality Between Nonmonotonic Consequence Relations and Convex Geometries Johannes Marti and Riccardo Pinosio Draft from April 5, 2018 Abstract In this paper we present a duality between nonmonotonic
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More information3 Integration and Expectation
3 Integration and Expectation 3.1 Construction of the Lebesgue Integral Let (, F, µ) be a measure space (not necessarily a probability space). Our objective will be to define the Lebesgue integral R fdµ
More informationg 2 (x) (1/3)M 1 = (1/3)(2/3)M.
COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is
More informationDS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.
DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1
More information(x, y) = d(x, y) = x y.
1 Euclidean geometry 1.1 Euclidean space Our story begins with a geometry which will be familiar to all readers, namely the geometry of Euclidean space. In this first chapter we study the Euclidean distance
More informationSet, functions and Euclidean space. Seungjin Han
Set, functions and Euclidean space Seungjin Han September, 2018 1 Some Basics LOGIC A is necessary for B : If B holds, then A holds. B A A B is the contraposition of B A. A is sufficient for B: If A holds,
More informationMaths 212: Homework Solutions
Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then
More information1 Directional Derivatives and Differentiability
Wednesday, January 18, 2012 1 Directional Derivatives and Differentiability Let E R N, let f : E R and let x 0 E. Given a direction v R N, let L be the line through x 0 in the direction v, that is, L :=
More informationIntroduction to Real Analysis Alternative Chapter 1
Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces
More informationChapter 2. Metric Spaces. 2.1 Metric Spaces
Chapter 2 Metric Spaces ddddddddddddddddddddddddd ddddddd dd ddd A metric space is a mathematical object in which the distance between two points is meaningful. Metric spaces constitute an important class
More informationLines, parabolas, distances and inequalities an enrichment class
Lines, parabolas, distances and inequalities an enrichment class Finbarr Holland 1. Lines in the plane A line is a particular kind of subset of the plane R 2 = R R, and can be described as the set of ordered
More informationarxiv: v1 [math.fa] 1 Nov 2017
NON-EXPANSIVE BIJECTIONS TO THE UNIT BALL OF l 1 -SUM OF STRICTLY CONVEX BANACH SPACES V. KADETS AND O. ZAVARZINA arxiv:1711.00262v1 [math.fa] 1 Nov 2017 Abstract. Extending recent results by Cascales,
More informationProof. We indicate by α, β (finite or not) the end-points of I and call
C.6 Continuous functions Pag. 111 Proof of Corollary 4.25 Corollary 4.25 Let f be continuous on the interval I and suppose it admits non-zero its (finite or infinite) that are different in sign for x tending
More informationMath 341: Convex Geometry. Xi Chen
Math 341: Convex Geometry Xi Chen 479 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA E-mail address: xichen@math.ualberta.ca CHAPTER 1 Basics 1. Euclidean Geometry
More informationOn the mean connected induced subgraph order of cographs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 71(1) (018), Pages 161 183 On the mean connected induced subgraph order of cographs Matthew E Kroeker Lucas Mol Ortrud R Oellermann University of Winnipeg Winnipeg,
More informationThe small ball property in Banach spaces (quantitative results)
The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence
More informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........
More informationChapter 2 Linear Transformations
Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more
More informationChapter 2 Metric Spaces
Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the so-called prime and maximal ideals. Let
More informationOn Reflecting Brownian Motion with Drift
Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability
More informationat time t, in dimension d. The index i varies in a countable set I. We call configuration the family, denoted generically by Φ: U (x i (t) x j (t))
Notations In this chapter we investigate infinite systems of interacting particles subject to Newtonian dynamics Each particle is characterized by its position an velocity x i t, v i t R d R d at time
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationProblem 3. Give an example of a sequence of continuous functions on a compact domain converging pointwise but not uniformly to a continuous function
Problem 3. Give an example of a sequence of continuous functions on a compact domain converging pointwise but not uniformly to a continuous function Solution. If we does not need the pointwise limit of
More informationLinear Algebra. Preliminary Lecture Notes
Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date May 9, 29 2 Contents 1 Motivation for the course 5 2 Euclidean n dimensional Space 7 2.1 Definition of n Dimensional Euclidean Space...........
More informationCOMPLEX ANALYSIS Spring 2014
COMPLEX ANALYSIS Spring 2014 1 Preliminaries Homotopical topics Our textbook slides over a little problem when discussing homotopy. The standard definition of homotopy is for not necessarily piecewise
More informationContents Ordered Fields... 2 Ordered sets and fields... 2 Construction of the Reals 1: Dedekind Cuts... 2 Metric Spaces... 3
Analysis Math Notes Study Guide Real Analysis Contents Ordered Fields 2 Ordered sets and fields 2 Construction of the Reals 1: Dedekind Cuts 2 Metric Spaces 3 Metric Spaces 3 Definitions 4 Separability
More informationSPACES ENDOWED WITH A GRAPH AND APPLICATIONS. Mina Dinarvand. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 69, 1 (2017), 23 38 March 2017 research paper originalni nauqni rad FIXED POINT RESULTS FOR (ϕ, ψ)-contractions IN METRIC SPACES ENDOWED WITH A GRAPH AND APPLICATIONS
More informationDef. A topological space X is disconnected if it admits a non-trivial splitting: (We ll abbreviate disjoint union of two subsets A and B meaning A B =
CONNECTEDNESS-Notes Def. A topological space X is disconnected if it admits a non-trivial splitting: X = A B, A B =, A, B open in X, and non-empty. (We ll abbreviate disjoint union of two subsets A and
More informationAsymptotic Irrelevance of Initial Conditions for Skorohod Reflection Mapping on the Nonnegative Orthant
Published online ahead of print March 2, 212 MATHEMATICS OF OPERATIONS RESEARCH Articles in Advance, pp. 1 12 ISSN 364-765X print) ISSN 1526-5471 online) http://dx.doi.org/1.1287/moor.112.538 212 INFORMS
More informationMath 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008
Math 541 Fall 2008 Connectivity Transition from Math 453/503 to Math 541 Ross E. Staffeldt-August 2008 Closed sets We have been operating at a fundamental level at which a topological space is a set together
More informationElements of Convex Optimization Theory
Elements of Convex Optimization Theory Costis Skiadas August 2015 This is a revised and extended version of Appendix A of Skiadas (2009), providing a self-contained overview of elements of convex optimization
More informationIn English, this means that if we travel on a straight line between any two points in C, then we never leave C.
Convex sets In this section, we will be introduced to some of the mathematical fundamentals of convex sets. In order to motivate some of the definitions, we will look at the closest point problem from
More informationMAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9
MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended
More informationFinite-Dimensional Cones 1
John Nachbar Washington University March 28, 2018 1 Basic Definitions. Finite-Dimensional Cones 1 Definition 1. A set A R N is a cone iff it is not empty and for any a A and any γ 0, γa A. Definition 2.
More informationConvergence in shape of Steiner symmetrized line segments. Arthur Korneychuk
Convergence in shape of Steiner symmetrized line segments by Arthur Korneychuk A thesis submitted in conformity with the requirements for the degree of Master of Science Graduate Department of Mathematics
More informationStability Analysis and Synthesis for Scalar Linear Systems With a Quantized Feedback
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003 1569 Stability Analysis and Synthesis for Scalar Linear Systems With a Quantized Feedback Fabio Fagnani and Sandro Zampieri Abstract
More informationFixed point theorems of nondecreasing order-ćirić-lipschitz mappings in normed vector spaces without normalities of cones
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 18 26 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa Fixed point theorems of nondecreasing
More informationWeek 3: Faces of convex sets
Week 3: Faces of convex sets Conic Optimisation MATH515 Semester 018 Vera Roshchina School of Mathematics and Statistics, UNSW August 9, 018 Contents 1. Faces of convex sets 1. Minkowski theorem 3 3. Minimal
More informationAN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES
AN ELEMENTARY PROOF OF THE SPECTRAL RADIUS FORMULA FOR MATRICES JOEL A. TROPP Abstract. We present an elementary proof that the spectral radius of a matrix A may be obtained using the formula ρ(a) lim
More informationCommutative Banach algebras 79
8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationThe Arzelà-Ascoli Theorem
John Nachbar Washington University March 27, 2016 The Arzelà-Ascoli Theorem The Arzelà-Ascoli Theorem gives sufficient conditions for compactness in certain function spaces. Among other things, it helps
More information1 Differentiable manifolds and smooth maps
1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set
More informationConvex Analysis and Economic Theory AY Elementary properties of convex functions
Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory AY 2018 2019 Topic 6: Convex functions I 6.1 Elementary properties of convex functions We may occasionally
More informationA FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS
Fixed Point Theory, (0), No., 4-46 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html A FIXED POINT THEOREM FOR GENERALIZED NONEXPANSIVE MULTIVALUED MAPPINGS A. ABKAR AND M. ESLAMIAN Department of Mathematics,
More informationEconomics 204 Fall 2011 Problem Set 2 Suggested Solutions
Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit
More informationTopology, Math 581, Fall 2017 last updated: November 24, Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski
Topology, Math 581, Fall 2017 last updated: November 24, 2017 1 Topology 1, Math 581, Fall 2017: Notes and homework Krzysztof Chris Ciesielski Class of August 17: Course and syllabus overview. Topology
More informationMore Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order Restriction
Sankhyā : The Indian Journal of Statistics 2007, Volume 69, Part 4, pp. 700-716 c 2007, Indian Statistical Institute More Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order
More informationINVERSE FUNCTION THEOREM and SURFACES IN R n
INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that
More informationMetric spaces and metrizability
1 Motivation Metric spaces and metrizability By this point in the course, this section should not need much in the way of motivation. From the very beginning, we have talked about R n usual and how relatively
More informationBehaviour of Lipschitz functions on negligible sets. Non-differentiability in R. Outline
Behaviour of Lipschitz functions on negligible sets G. Alberti 1 M. Csörnyei 2 D. Preiss 3 1 Università di Pisa 2 University College London 3 University of Warwick Lars Ahlfors Centennial Celebration Helsinki,
More informationSeparation in General Normed Vector Spaces 1
John Nachbar Washington University March 12, 2016 Separation in General Normed Vector Spaces 1 1 Introduction Recall the Basic Separation Theorem for convex sets in R N. Theorem 1. Let A R N be non-empty,
More informationA NICE PROOF OF FARKAS LEMMA
A NICE PROOF OF FARKAS LEMMA DANIEL VICTOR TAUSK Abstract. The goal of this short note is to present a nice proof of Farkas Lemma which states that if C is the convex cone spanned by a finite set and if
More informationNotes taken by Graham Taylor. January 22, 2005
CSC4 - Linear Programming and Combinatorial Optimization Lecture : Different forms of LP. The algebraic objects behind LP. Basic Feasible Solutions Notes taken by Graham Taylor January, 5 Summary: We first
More informationGreen s Theorem in the Plane
hapter 6 Green s Theorem in the Plane Introduction Recall the following special case of a general fact proved in the previous chapter. Let be a piecewise 1 plane curve, i.e., a curve in R defined by a
More informationWhere is matrix multiplication locally open?
Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?
More informationGeometry and topology of continuous best and near best approximations
Journal of Approximation Theory 105: 252 262, Geometry and topology of continuous best and near best approximations Paul C. Kainen Dept. of Mathematics Georgetown University Washington, D.C. 20057 Věra
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationA VERY BRIEF REVIEW OF MEASURE THEORY
A VERY BRIEF REVIEW OF MEASURE THEORY A brief philosophical discussion. Measure theory, as much as any branch of mathematics, is an area where it is important to be acquainted with the basic notions and
More informationApplied Mathematics Letters
Applied Mathematics Letters 25 (2012) 974 979 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml On dual vector equilibrium problems
More informationDYNAMICAL CUBES AND A CRITERIA FOR SYSTEMS HAVING PRODUCT EXTENSIONS
DYNAMICAL CUBES AND A CRITERIA FOR SYSTEMS HAVING PRODUCT EXTENSIONS SEBASTIÁN DONOSO AND WENBO SUN Abstract. For minimal Z 2 -topological dynamical systems, we introduce a cube structure and a variation
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationCombinatorics in Banach space theory Lecture 12
Combinatorics in Banach space theory Lecture The next lemma considerably strengthens the assertion of Lemma.6(b). Lemma.9. For every Banach space X and any n N, either all the numbers n b n (X), c n (X)
More informationSTABILITY AND STRUCTURAL PROPERTIES OF STOCHASTIC STORAGE NETWORKS 1
STABILITY AND STRUCTURAL PROPERTIES OF STOCHASTIC STORAGE NETWORKS 1 by Offer Kella 2 and Ward Whitt 3 November 10, 1994 Revision: July 5, 1995 Journal of Applied Probability 33 (1996) 1169 1180 Abstract
More informationCourse 212: Academic Year Section 1: Metric Spaces
Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........
More informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More informationINSTITUTE of MATHEMATICS. ACADEMY of SCIENCES of the CZECH REPUBLIC. A universal operator on the Gurariĭ space
INSTITUTE of MATHEMATICS Academy of Sciences Czech Republic INSTITUTE of MATHEMATICS ACADEMY of SCIENCES of the CZECH REPUBLIC A universal operator on the Gurariĭ space Joanna Garbulińska-Wȩgrzyn Wiesław
More informationAvailable online at ISSN (Print): , ISSN (Online): , ISSN (CD-ROM):
American International Journal of Research in Formal, Applied & Natural Sciences Available online at http://www.iasir.net ISSN (Print): 2328-3777, ISSN (Online): 2328-3785, ISSN (CD-ROM): 2328-3793 AIJRFANS
More informationContents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping.
Minimization Contents: 1. Minimization. 2. The theorem of Lions-Stampacchia for variational inequalities. 3. Γ -Convergence. 4. Duality mapping. 1 Minimization A Topological Result. Let S be a topological
More informationMATH 426, TOPOLOGY. p 1.
MATH 426, TOPOLOGY THE p-norms In this document we assume an extended real line, where is an element greater than all real numbers; the interval notation [1, ] will be used to mean [1, ) { }. 1. THE p
More informationSTAT 7032 Probability Spring Wlodek Bryc
STAT 7032 Probability Spring 2018 Wlodek Bryc Created: Friday, Jan 2, 2014 Revised for Spring 2018 Printed: January 9, 2018 File: Grad-Prob-2018.TEX Department of Mathematical Sciences, University of Cincinnati,
More informationTechnical Results on Regular Preferences and Demand
Division of the Humanities and Social Sciences Technical Results on Regular Preferences and Demand KC Border Revised Fall 2011; Winter 2017 Preferences For the purposes of this note, a preference relation
More informationThe Skorokhod problem in a time-dependent interval
The Skorokhod problem in a time-dependent interval Krzysztof Burdzy, Weining Kang and Kavita Ramanan University of Washington and Carnegie Mellon University Abstract: We consider the Skorokhod problem
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets
FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be
More informationLebesgue-Stieltjes measures and the play operator
Lebesgue-Stieltjes measures and the play operator Vincenzo Recupero Politecnico di Torino, Dipartimento di Matematica Corso Duca degli Abruzzi, 24, 10129 Torino - Italy E-mail: vincenzo.recupero@polito.it
More information1. Continuous Functions between Euclidean spaces
Math 441 Topology Fall 2012 Metric Spaces by John M. Lee This handout should be read between Chapters 1 and 2 of the text. It incorporates material from notes originally prepared by Steve Mitchell and
More information