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Real and complex analysis
... You can explain it to average calculus students, and even lead them to conjecture it on their own; the only thing that's hard is convincing them that it's nontrivial! The Cut Property hasn't been entirely forgotten ( [1], [16, p. 53], and [20]) and it's well-known among people who study the axiomatization of Euclidean geometry [10] or the theory of partially ordered sets and lattices [28]. But it deserves to be better known among the mathematical community at large. ...
... The Ratio Test (16) implies completeness by way of the Archimedean Property (2) and the Cut Property (3): Note that the Ratio Test implies that 1 2 + 1 4 + 1 8 + . . . converges, implying that R is Archimedean (the sequence of partial sums 1 2 , 3 4 , 7 8 , . . . isn't even a Cauchy sequence if there exists an ǫ > 0 that is less than 1/n for all n). ...
Many of the theorems of real analysis, against the background of the ordered
field axioms, are equivalent to Dedekind completeness, and hence can serve as
completeness axioms for the reals. In the course of demonstrating this, the
article offers a tour of some less-familiar ordered fields, provides some of
the relevant history, and considers pedagogical implications.
In this paper we consider the problem of differential inclusion in time scales whose vector field is a multifunction, that is, a function that maps points to sets. It is provided conditions of existence without requiring compactness of the vector field; it is required that the vector field is closed, convex, and lower semicontinuous. In previous work in literature, it is required that the field is either scalar or compact, convex, and has closed graph.
In the classical risk model with initial capital u, let
τ(u) be the time of ruin, X+(u) be the risk reserve just
before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu
(1998) defined the function mδ(u) =E[e-δ
τ(u) w(X+(u), Y+(u)) 1 (τ(u) < ∞) ], where
δ ≥ 0 can be interpreted as a force of interest and
w(r,s) as a penalty function, meaning that mδ(u) is the
expected discounted penalty payable at ruin. This function is
known to satisfy a defective renewal equation, but easy explicit
formulae for mδ(u) are only available for certain special
cases for the claim size distribution. Approximations thus arise
by approximating the desired mδ(u) by that associated with
one of these special cases. In this paper a functional approach is
taken, giving rise to first-order correction terms for the above
approximations.
We extend the well-known fictitious play (FP) algorithm to compute pure-strategy Bayesian-Nash equilibria in private-value games of incomplete information with finite actions and continuous types (G-FACTs). We prove that, if the frequency distribution of actions (fictitious play beliefs) converges, then there exists a pure-strategy equilibrium strategy that is consistent with it. We furthermore develop an algorithm to convert the converged distribution of actions into an equilibrium strategy for a wide class of games where utility functions are linear in type. This algorithm can also be used to compute pure ϵ-Nash equilibria when distributions are not fully converged. We then apply our algorithm to find equilibria in an important and previously unsolved game: simultaneous sealed-bid, second-price auctions where various types of items (e.g., substitutes or complements) are sold. Finally, we provide an analytical characterisation of equilibria in games with linear utilities. Specifically, we show how equilibria can be found by solving a system of polynomial equations. For a special case of simultaneous auctions, we also solve the equations confirming the results obtained numerically.
Consider a matrix $\Sigma_n$ with random independent entries, each
non-centered with a separable variance profile. In this article, we study the
limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$
and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to
$\Sigma_n \Sigma_n^*$ as the dimensions of matrix $\Sigma_n$ go to infinity at
the same pace. Such quantities arise in the study of functionals of $\Sigma_n
\Sigma_n^*$ which do not only depend on the eigenvalues of $\Sigma_n
\Sigma_n^*$, and are pivotal in the study of problems related to non-centered
Gram matrices such as central limit theorems, individual entries of the
resolvent, and eigenvalue separation.
Let (Formula Presented), is a real valued function which is in Lip α, 0 < α < 1, on the unit (k — 1)-sphere S in k-dimensional Euclidean space, Ek, k ≧ 2 with the additional property that (FORMULA PRESENTED) where σ is the natural surface measure for S. (K(x) is usually called a Calderón-Zygmund kernel in Lip α.) Let μ be a Borei measure of finite total variation on Ek and set (Formula Presented). Also designate the principal-valued Fourier transform of K by K(y) and the principal-valued convolution of K with μ by (Formula Presented). Define (Formula Presented). Then if k is an even integer or if k — 3, the following result is established: lim(Formula Presented) almost everywhere.
Proofs are given of set-valued analogues of Rouche’s theorem, the argument principle, and the Picard theorems. This is achieved by developing a rudimentary theory of upper semicontinuous multivalued lifts.
We consider quantum-mechanical potentials giving rise to minimal (or maximal) eigenvalue gaps subject to LP constraints in n-dimensions. We prove existence and characterization theorems for optimizing potentials. The tunneling effect through a single barrier is shown always to be the cause of minimal gaps, and in some cases the gap minimizers are shown to be specific double-well potentials.
We prove a generalization of Livsic's Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie groups. Combining this with Topological Superrigidity, we find a Holder geometric structure for multiplicity free Anosov actions.
Let f be a probability density on the real line, let n be any positive integer, and assume the condition (R) that log f is locally integrable with respect to Lebesgue measure. Then either log f is almost everywhere equal to a polynomial of degree less than n, or the order statistic of n independent and identically distributed observations from the location-scale parameter model generated by f is minimal sufficient. It follows, subject to (R) and n 3, that a complete sufficient statistic exists in the normal case only. Also, for f with (R) infinitely divisible but not normal, the order statistic is always minimal sufficient for the corresponding location-scale parameter model. The proof of the main result uses a theorem on the harmonic analysis of translation and dilation invariant function spaces, attributable to K.O. Leland (1968) and L. Schwartz (1947). 1 Introduction and main results 1.1 Aim. Perhaps the most natural first step in the analysis of a statistical model consists in de...
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