This article shows that continuous functions on compact intervals may be approximated uniformly with scattered translates of the Poisson kernel (alpha(2) +x(2))(-1), where a > 0 is a fixed real parameter.
This article shows that continuous functions on compact intervals may be approximated uniformly with scattered translates of the Poisson kernel (alpha(2) +x(2))(-1), where a > 0 is a fixed real parameter.
The extension of lattice linear space of continuous bounded functions on a completely regular space, generated by mu-Riemann-integrable functions on this space, is considered in the paper. To characterize this mu-Riem...
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The extension of lattice linear space of continuous bounded functions on a completely regular space, generated by mu-Riemann-integrable functions on this space, is considered in the paper. To characterize this mu-Riemann extension some new functionally-analytical category of c-latlineals with refinements (equivalent to cr-latlineals) is used. On its base the notion of cr-completion of some definite type is introduced. A functionally-analytical characterization of the mu-Riemann extension as some cr(mu)-completion of some definite type of the cr(mu)-latlineal of continuous bounded functions is given.
In this paper we extend the techniques and the basic results of the classical Galois theory of the fields extension [InlineMediaObject not available: see fulltext.] is an algebraic closure of Q, to the algebras extens...
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Solving optimisation problems is a promising near-term application of quantum computers. Quantum variational algorithms (QVAs) leverage quantum superposition and entanglement to optimise over exponentially large solut...
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Solving optimisation problems is a promising near-term application of quantum computers. Quantum variational algorithms (QVAs) leverage quantum superposition and entanglement to optimise over exponentially large solution spaces using an alternating sequence of classically tunable unitaries. However, prior work has primarily addressed discrete optimisation problems. In addition, these algorithms have been designed generally under the assumption of an unstructured solution space, which constrains their speedup to the theoretical limits for the unstructured Grover's quantum search algorithm. In this paper, we show that QVAs can efficiently optimise continuous multivariable functions by exploiting general structural properties of a discretised continuous solution space with a convergence that exceeds the limits of an unstructured quantum search. We present the quantum multivariable optimisation algorithm and demonstrate its advantage over pre-existing methods, particularly when optimising high-dimensional and oscillatory functions.
In this paper we give the definition of [lambda, rho]-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of [lambda, rho]-continuous functi...
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In this paper we give the definition of [lambda, rho]-continuity of real-valued functions defined on an open interval, which is an example of path continuity. We give some properties of [lambda, rho]-continuous functions. The aim of the paper is to find the maximal additive class and the maximal multiplicative class for the family of [lambda, rho]-continuous functions.
This paper is devoted to relationships among various classes of I-a.e. continuous functions (i.e., of functions whose sets of discontinuity points belong to certain σ-ideals I consisting of boundary sets). For instan...
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This paper is devoted to relationships among various classes of I-a.e. continuous functions (i.e., of functions whose sets of discontinuity points belong to certain σ-ideals I consisting of boundary sets). For instance, if K is the σ-ideal of first category sets and I denotes the σ-ideal of all sets that are: of Lebesque measure zero, σ-porous, or countable, then the set of I-a.e. continuous functions is uniformly porous in the space of all K-a.e. continuous Darboux functions from ℝ² into ℝ² equipped with the metric of uniform convergence. As a tool in the proofs, symmetric Cantor sets in ℝ² are used.
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