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Dataset-Learning Duality and Emergent Criticality. [PDF]
Entropy (Basel)Kukleva E, Vanchurin V.
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Solitary and soliton solutions of the nonlinear fractional Chen Lee Liu model with beta derivative. [PDF]
Sci RepHussain A +5 more
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Journal of Mathematical Physics, 1966
The integral equation P ∫ cK(ζ′,ζ)ζ′−ζφ(ζ′) dζ′=h(ζ)φ(ζ)+f(ζ)is shown to have simple solutions obtained by standard and elementary methods if h and K have appropriate analytic properties.
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The integral equation P ∫ cK(ζ′,ζ)ζ′−ζφ(ζ′) dζ′=h(ζ)φ(ζ)+f(ζ)is shown to have simple solutions obtained by standard and elementary methods if h and K have appropriate analytic properties.
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Singularities of Solutions of Singular Integral Equations
Ukrainian Mathematical Journal, 2002This paper deals with a singular integral equation \[ Sq+Tq=f,\tag{1} \] where \(q(x)\) is an unknown function, \[ Sq(x):=aq(x)+\frac{1}{\pi }\text{v.p.} \int_{-1}^{1} \frac{q(\tau)}{\tau -x} d\tau,\;Tq(x):=\int_{-1}^{1}K(x,\tau)q(\tau) d\tau. \] It is assumed that the functions \(f\) and \(K\) smoothly depend on additional parameters.
Kapustyan, V. E., Il'man, V. M.
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2003
In this chapter we discuss some recent results for Fredholm and Volterra integral equations, which deal with the existence of positive (and possibly multiple) solutions of certain classes of these equations. In Section 3.2 we provide some existence results for the nonsingular Fredholm integral equations.
Ravi P. Agarwal, Donal O’Regan
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In this chapter we discuss some recent results for Fredholm and Volterra integral equations, which deal with the existence of positive (and possibly multiple) solutions of certain classes of these equations. In Section 3.2 we provide some existence results for the nonsingular Fredholm integral equations.
Ravi P. Agarwal, Donal O’Regan
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Approximate Solution of a Singular Integral Equation Using the Sobolev Method
Lobachevskii Journal of Mathematics, 2022K. Shadimetov, D. Akhmedov
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2002
The celebrated Cauchy singular integral operator on a Jordan curve, or more precisely, its 1-periodic counterpart is perhaps the most important brick in the theory of periodic integral and pseudodifferential operators. In this chapter, we first treat the Cauchy singular operators in the Holder spaces C α (Γ) and after that we extend the results to L 2 ...
Jukka Saranen, Gennadi Vainikko
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The celebrated Cauchy singular integral operator on a Jordan curve, or more precisely, its 1-periodic counterpart is perhaps the most important brick in the theory of periodic integral and pseudodifferential operators. In this chapter, we first treat the Cauchy singular operators in the Holder spaces C α (Γ) and after that we extend the results to L 2 ...
Jukka Saranen, Gennadi Vainikko
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1995
Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by $$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left(
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Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by $$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left(
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