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Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a long time, ‘variational ’ problems have been identified mostly with the ‘calculus of variations’. In that venerable subject, built around the minimization of integral functionals, constraints were relatively simple and much of the focus was on infinitedimensional function spaces. A major theme was the exploration of variations around a point, within the bounds imposed by the constraints, in order to help characterize solutions and portray them in terms of ‘variational principles’. Notions of perturbation, approximation and even generalized differentiability were extensively investigated. Variational theory progressed also to the study of socalled stationary points, critical points, and other indications of singularity that a point might have relative to its neighbors, especially in association with existence theorems for differential equations.
Trichotomies for ideals of compact sets
 J. SYMBOLIC LOGIC
"... We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal. ..."
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Cited by 5 (3 self)
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We prove several trichotomy results for ideals of compact sets. Typically, we show that a “sufficiently rich” universally Baire ideal is either Π 0 3hard, or Σ 0 3hard, or else a σideal.
On some new ideals on the Cantor and Baire spaces
 Proc. Am. Math. Soc
, 1998
"... Abstract. We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire spac ..."
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Cited by 8 (7 self)
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Abstract. We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire
The Variable Discharge of Cortical Neurons: Implications for Connectivity, Computation, and Information Coding
 J. Neurosci
, 1998
"... this paper we propose that the irregular ISI arises as a consequence of a specific problem that cortical neurons must solve: the problem of dynamic range or gain control. Cortical neurons receive 300010,000 synaptic contacts, 85% of which are asymmetric and hence presumably excitatory (Peters, 198 ..."
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Cited by 341 (3 self)
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this paper we propose that the irregular ISI arises as a consequence of a specific problem that cortical neurons must solve: the problem of dynamic range or gain control. Cortical neurons receive 300010,000 synaptic contacts, 85% of which are asymmetric and hence presumably excitatory (Peters, 1987; Braitenberg and Schuz, 1991). More than half of these contacts are thought to arise from neurons within a 100200 #m radius of the target cell, reflecting the stereotypical columnar organization of neocortex. Because neurons within a cortical column typically share similar physiological properties, the conditions that excite one neuron are likely to excite a considerable fraction of its afferent input as well (Mountcastle, 1978; Peters and Sethares, 1991), creating a scenario in which saturation of the neuron's firing rate could easily occur. This problem is exacerbated by the fact that EPSPs from individual axons appear to exert substantial impact on the membrane potential (Mason et al., 1991; Otmakhov Received Sept. 15, 1997; revised Feb. 25, 1998; accepted March 3, 1998.
BAIRE CATEGORY FOR MONOTONE SETS
, 1994
"... Abstract. We study Baire category for downwardclosed subsets of 2 ω, showing that it behaves better in this context than for general subsets of 2 ω. We show that, in the downwardclosed context, the ideal of meager sets is prime and bcomplete, while the complementary filter is gcomplete. We also ..."
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Abstract. We study Baire category for downwardclosed subsets of 2 ω, showing that it behaves better in this context than for general subsets of 2 ω. We show that, in the downwardclosed context, the ideal of meager sets is prime and bcomplete, while the complementary filter is gcomplete. We also
BOOLEAN RINGS THAT ARE BAIRE SPACES
, 2001
"... Weak completeness properties of Boolean rings are related to the property of being a Baire space (when suitably topologised) and to renorming properties of the Banach spaces of continuous functions on the corresponding Stone spaces. ..."
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Cited by 2 (0 self)
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Weak completeness properties of Boolean rings are related to the property of being a Baire space (when suitably topologised) and to renorming properties of the Banach spaces of continuous functions on the corresponding Stone spaces.
COMPACT SUBSETS OF THE FIRST BAIRE CLASS
, 1999
"... Perhaps the earliest results about pointwise compact sets of Baire class1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire cla ..."
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Cited by 22 (3 self)
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Perhaps the earliest results about pointwise compact sets of Baire class1 functions are the two selection theorems of E. Helly found in most of the standard texts on real variable (see, e.g., [Lo], [N]). These two theorems are really theorems about a particular example of a compact set of Baire
Results 1  10
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