| 1. |
- Kostykin, Vadim, et al.
(författare)
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On the Existence of Unstable Bumps in Neural Networks
- 2013
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Ingår i: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 75:4, s. 445-458
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Tidskriftsartikel (refereegranskat)abstract
- We study the neuronal field equation, a nonlinear integro-differential equation of Hammerstein type. By means of the Amann three fixed point theorem we prove the existence of bump solutions to this equation. Using the Krein-Rutman theorem we show their Lyapunov instability.
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| 2. |
- Leśnik, Karol, et al.
(författare)
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Abstract Cesàro Spaces. Optimal Range
- 2014
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Ingår i: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 81:2, s. 227-235
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Tidskriftsartikel (refereegranskat)abstract
- Abstract Cesàro spaces are investigated from the optimal domain and optimal range point of view. There is a big difference between the cases on [0, ∞) and on [0, 1], as we can see in Theorem 1. Moreover, we present an improvement of Hardy’s inequality on [0, 1] which plays an important role in these considerations.
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| 3. |
- Maad Sasane, Sara, et al.
(författare)
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Generators for rings of compactly supported distributions
- 2011
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Ingår i: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 69:1, s. 63-71
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Tidskriftsartikel (refereegranskat)abstract
- Let CUnknown control sequence '\tt' denote a closed convex cone in \mathbb RdRd with apex at 0. We denote by E¢(C)Unknown control sequence '\tt' the set of distributions on \mathbb RdRd having compact support contained in CUnknown control sequence '\tt'. Then E¢(C)Unknown control sequence '\tt' is a ring with the usual addition and with convolution. We give a necessary and sufficient analytic condition on [^(f)]1,..., [^(f)]nf1fn for f1,... ,fn Î E¢(C)Unknown control sequence '\tt' to generate the ring E¢(C)Unknown control sequence '\tt'. (Here [^( · )] denotes Fourier-Laplace transformation.) This result is an application of a general result on rings of analytic functions of several variables by Lars Hörmander. En route we answer an open question posed by Yutaka Yamamoto.
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| 4. |
- Shulman, Victor, et al.
(författare)
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Closable Multipliers
- 2011
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Ingår i: Integral Equations and Operator Theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 69:1, s. 29-62
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Tidskriftsartikel (refereegranskat)abstract
- Let (X, mu) and (Y, nu) be standard measure spaces. A function. phi is an element of L-infinity(XxY, mu x nu) is called a (measurable) Schur multiplier if the map S phi, defined on the space of Hilbert-Schmidt operators from L-2(X, mu) to L2(Y, nu) by multiplying their integral kernels by phi, is bounded in the operator norm. The paper studies measurable functions phi for which S phi is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if phi is of Toeplitz type, that is, if phi(x, y) = f(x - y), x, y is an element of G, where G is a locally compact abelian group, then the closability of phi is related to the local inclusion of f in the Fourier algebra A(G) of G. If phi is a divided difference, that is, a function of the form (f(x) - f(y))/(x - y), then its closability is related to the "operator smoothness" of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.
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