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Register a new userSubdifferential of the supremum function: Moving back and forth between continuous and non-continuous settings
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In this paper we develop general formulas for the subdifferential of the pointwise supremum of convex functions, which cover and unify both the compact continuous and the non-compact non-continuous settings. From the non-continuous to the continuous setting, we proceed by a compactification-based approach which leads us to problems having compact index sets and upper semi-continuously indexed mappings, giving rise to new characterizations of the subdifferential of the supremum by means of upper semicontinuous regularized functions and an enlarged compact index set. In the opposite sense, we rewrite the subdifferential of these new regularized functions by using the original data, also leading us to new results on the subdifferential of the supremum. We give two applications in the last section, the first one concerning the nonconvex Fenchel duality, and the second one establishing Fritz-John and KKT conditions in convex semi-infinite programming.
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In this paper, we are interested in the estimation of Particle Size Distributions (PSDs) during a batch crystallization process in which particles of two different shapes coexist and evolve simultaneously. The PSDs are estimated thanks to a measurement of an apparent Chord Length Distribution (CLD), a measure that we model for crystals of spheroidal shape. Our main result is to prove the approximate observability of the infinite-dimensional system in any positive time. Under this observability condition, we are able to apply a Back and Forth Nudging (BFN) algorithm to reconstruct the PSD.
Meirowitz [17] showed existence of continuous behavioural function equilibria for Bayesian games with non-finite type and action spaces. A key condition for the proof of the existence result is equi-continuity of behavioural functions which, according to Meirowitz [17, page 215], is likely to fail or difficult to verify. In this paper, we advance the research by presenting some verifiable conditions for the required equi-continuity, namely some growth conditions of the expected utility functions of each player at equilibria. In the case when the growth is of second order, we demonstrate that the condition is guaranteed by strong concavity of the utility function. Moreover, by using recent research on polynomial decision rules and optimal discretization approaches in stochastic and robust optimization, we propose some approximation schemes for the Bayesian equilibrium problem: first, by restricting the behavioral functions to polynomial functions of certain order over the space of types, we demonstrate that solving a Bayesian polynomial behavioural function equilibrium is down to solving a finite dimensional stochastic equilibrium problem; second, we apply the optimal quantization method due to Pflug and Pichler [18] to develop an effective discretization scheme for solving the latter. Error bounds are derived for the respective approximation schemes under moderate conditions and both aca- demic examples and numerical results are presented to explain the Bayesian equilibrium problem and their approximation schemes.
In this paper, we consider the back and forth nudging algorithm that has been introduced for data assimilation purposes. It consists of iteratively and alternately solving forward and backward in time the model equation, with a feedback term to the observations. We consider the case of 1-dimensional transport equations, either viscous or inviscid, linear or not (Burgers equation). Our aim is to prove some theoretical results on the convergence, and convergence properties, of this algorithm. We show that for non viscous equations (both linear transport and Burgers), the convergence of the algorithm holds under observability conditions. Convergence can also be proven for viscous linear transport equations under some strong hypothesis, but not for viscous Burgers equation. Moreover, the convergence rate is always exponential in time. We also notice that the forward and backward system of equations is well posed when no nudging term is considered.
As proved in [16], for a Tychonoff space $X$, a locally convex space $C_{p}(X)$ is distinguished if and only if $X$ is a $Delta$-space. If there exists a linear continuous surjective mapping $T:C_p(X) to C_p(Y)$ and $C_p(X)$ is distinguished, then $C_p(Y)$ also is distinguished [17]. Firstly, in this paper we explore the following question: Under which conditions the operator $T:C_p(X) to C_p(Y)$ above is open? Secondly, we devote a special attention to concrete distinguished spaces $C_p([1,alpha])$, where $alpha$ is a countable ordinal number. A complete characterization of all $Y$ which admit a linear continuous surjective mapping $T:C_p([1,alpha]) to C_p(Y)$ is given. We also observe that for every countable ordinal $alpha$ all closed linear subspaces of $C_p([1,alpha])$ are distinguished, thereby answering an open question posed in [17]. Using some properties of $Delta$-spaces we prove that a linear continuous surjection $T:C_p(X) to C_k(X)_w$, where $C_k(X)_w$ denotes the Banach space $C(X)$ endowed with its weak topology, does not exist for every infinite metrizable compact $C$-space $X$ (in particular, for every infinite compact $X subset mathbb{R}^n$).
The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $mathbb R^m$ this notion is near to the separate continuity for which it is required only the continuity on the straight lines which are parallel to coordinate axes. The classical Lebesgue theorem states that every separately continuous function $f:mathbb R^mtomathbb R$ is of the $(m-1)$-th Baire class. In this paper we prove that every linearly continuous function $f:mathbb R^mtomathbb R$ is of the first Baire class. Moreover, we obtain the following result. If $X$ is a Baire cosmic topological vector space, $Y$ is a Tychonoff topological space and $f:Xto Y$ is a Borel-measurable (even BP-measurable) linearly continuous function, then $f$ is $F_sigma$-measurable. Using this theorem we characterize the discontinuity point set of an arbitrary linearly continuous function on $mathbb R^m$. In the final part of the article we prove that any $F_sigma$-measurable function $f:partial Uto mathbb R$ defined on the boundary of a strictly convex open set $Usubsetmathbb R^m$ can be extended to a linearly continuous function $bar f:Xto mathbb R$. This fact shows that in the ``descriptive sense the linear continuity is not better than the $F_sigma$-measurability.
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