Borel-amenable reducibilities for sets of reals

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

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Exhaustive operators and vector measures

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500015571 ◽

1975 ◽

Vol 19 (3) ◽

pp. 291-300 ◽

Cited By ~ 12

Author(s):

N. J. Kalton

Keyword(s):

Linear Operator ◽

Vector Space ◽

Compact Hausdorff Space ◽

Hausdorff Space ◽

Continuous Functions ◽

Vector Measures ◽

Borel Functions ◽

Image Position ◽

Real Topological Vector Space ◽

Sequence Of Functions

Let S be a compact Hausdorff space and let Φ: C(S)→E be a linear operator defined on the space of real-valued continuous functions on S and taking values in a (real) topological vector space E. Then Φ is called exhaustive (7) if given any sequence of functions fn ∈ C(S) such that fn ≧ 0 andthen Φ(fn)→0 If E is complete then it was shown in (7) that exhaustive maps are precisely those which possess regular integral extensions to the space of bounded Borel functions on S; this is equivalent to possessing a representationwhere μ is a regular countably additive E-valued measure defined on the σ-algebra of Borel subsets of S.

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Oscillation criteria for differential equations with piecewise constant argument

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.32.2001.344 ◽

2001 ◽

Vol 32 (4) ◽

pp. 293-304

Author(s):

Zhiguo Luo ◽

Jianhua Shen

Keyword(s):

Differential Equation ◽

Continuous Functions ◽

Variable Coefficients ◽

Piecewise Constant ◽

Piecewise Constant Argument ◽

Greatest Integer Function ◽

Special Case ◽

Oscillation And Nonoscillation Criteria ◽

Oscillation And Nonoscillation ◽

Constant Argument

We obtain some new oscillation and nonoscillation criteria for the differential equation with piecewise constant argument $$ x'(t) + a(t)x(t) + b(x) x([t-k]) = 0, $$ where $ a(t) $ and $ b(t) $ are continuous functions on $ [-k, \infty) $, $ b(t) \ge 0 $, $ k $ is a positive integer and $ [ \cdot ] $ denotes the greatest integer function. The method used is based on the treatment of certain difference equation with variable coefficients. Our results extend theorems in [15]. As a special case, our results also improve the conclusions obtained by Aftabizadeh, Wiener and Xu [3].

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VI. On the general theory integration

Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character ◽

10.1098/rsta.1905.0006 ◽

1905 ◽

Vol 204 (372-386) ◽

pp. 221-252 ◽

Cited By ~ 14

Keyword(s):

Arbitrary Point ◽

Continuous Functions ◽

Common Value ◽

Theory Integration ◽

Positive Quantity ◽

The Common ◽

Definition Of ◽

Special Case ◽

Ordinary Integral ◽

Definite Limit

Riemann was the first to consider the theory of integration of non-continuous functions. As is well known, his definition of the integral of a function between the limits a and b is as follows:— Divide the segment ( a, b ) into any finite number of intervals, each less, say, than a positive quantity, or norm d ; take the product of each such interval by the value of the function at any point of that interval, and form the sum of all these products; if this sum has a limit, when d is indefinitely diminished which is independent of the mode of division into intervals, and of the choice of the points in those intervals at which the values of the function are considered, this limit is called the integral of the function from a to b . The most convenient mode, however, of defining a Riemann (that is an ordinary) integral of a function, is due to Darboux; it is based on the introduction of upper and lower integrals (intégrale par excès, par défaut: oberes, unteres Integral). The definitions of these are as follows:— It may be shown that, if the interval ( a, b ) be divided as before, and the sum of the products taken as before, but with this difference, that instead of the value of the function at an arbitrary point of the part, the upper (lower) limit of the values of the function in the part be taken and multiplied by the length of the corresponding part, these summations have, whatever he the type of function, each of them a definite limit, independent of the mode of division and the mode in which d approaches the value zero. This limit is called the upper (lower) integral of the function. In the special case in which these two limits agree, the common value is called the integral the function .

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On certain non-archimedean functions analogous to complex analytic functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700024837 ◽

1976 ◽

Vol 14 (1) ◽

pp. 23-36 ◽

Cited By ~ 1

Author(s):

Kurt Mahler

Keyword(s):

Power Series ◽

Analytic Functions ◽

Convergent Series ◽

Continuous Functions ◽

Algebraic Extension ◽

Finite Degree ◽

Uniformly Convergent ◽

Image Position ◽

Derivatives Of ◽

Special Case

Let Qp and K be the rational p–adic field and an algebraic extension of Qp of finite degree, respectively, and let I and Ik be the subsets of Qp and of K consisting of the p–adic integers of these fields.It is known that the continuous functions f: I → Qp can be written aswhere this series converges uniformly on I, and that such continuous functions need not be differentiable at any point. We here study continuous functions F: IK → K which for all X on IK are the sum of a uniformly convergent seriesIt is proved that such functions F(X) have at every point of IK derivatives of all orders. In the special case when K is totally ramified, they cannot in general be developed into power series that converge everywhere on IK, but this is possible when K is not totally ramified.

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Discontinuities of the pressure for piecewise monotonic interval maps

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385701001122 ◽

2001 ◽

Vol 21 (1) ◽

pp. 197-232 ◽

Cited By ~ 1

Author(s):

PETER RAITH

Keyword(s):

Continuous Function ◽

Continuous Functions ◽

Finite Union ◽

Topological Pressure ◽

Stability Conditions ◽

Small Perturbations ◽

Interval Maps ◽

Closed Intervals ◽

Piecewise Monotonic ◽

Special Case

For a piecewise monotonic map T:X\to{\Bbb R}, where X is a finite union of closed intervals, define R(T)= \bigcap_{n=0}^{\infty}\overline{T^{-n}X}. The influence of small perturbations of T on the dynamical system (R(T),T) is investigated. If P is a finite and T-invariant subset of R(T), and if f_0:P\to{\Bbb R} is a non-negative continuous function, then it is proved that the infimum of the topological pressure p(R(T),T,f) over all non-negative continuous functions f:X\to{\Bbb R} with f|_P=f_0 equals the maximum of h_{\text{\rm top}}(R(T),T) and p(P,T,f_0). This result is used to obtain stability conditions, which are equivalent to the upper semi-continuity of the topological pressure for every continuous function f:X\to{\Bbb R}. In the case of a continuous piecewise monotonic map T:X\to{\Bbb R} one of these stability conditions is: there exists no endpoint of an interval of monotonicity of T which is periodic and contained in the interior of X. Furthermore, these results are applied to monotonic mod one transformations, another special case of piecewise monotonic maps.

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Symmetric spectral synthesis

Aequationes Mathematicae ◽

10.1007/s00010-020-00764-9 ◽

2020 ◽

Author(s):

Eszter Gselmann ◽

László Székelyhidi

Keyword(s):

Symmetric Group ◽

Spectral Synthesis ◽

Higher Dimensions ◽

Acta Math ◽

Continuous Functions ◽

Gelfand Pairs ◽

Space Of Continuous Functions ◽

Translation Invariant ◽

Special Case ◽

Straightforward Extension

AbstractAccording to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d.

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Disconjugacy and the limit-point condition for fourth-order operators

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500019508 ◽

1976 ◽

Vol 76 (1) ◽

pp. 77-80

Author(s):

Robert M. Kauffman

Keyword(s):

Differential Operator ◽

Fourth Order ◽

Limit Point ◽

Continuous Functions ◽

Oscillatory Solution ◽

Linearly Independent ◽

Point Condition ◽

General Class Of Functions ◽

Special Case ◽

Class Of Functions

SynopsisLet L denote the ordinary differential operator given by Lf = (pf″)″ + (qf′)′ + rf, with p″, q′ and r continuous functions on [0,∞), and with p>0, q ≦ 0, and r ≧ 0. It is proved that if the equation Lg = 0 possesses a non-oscillatory solution, then any non-trivial solution f to Lf = 0 such that f(0) = f′(0) = 0 is eventually bounded away from zero.This theorem is used to prove that, for a general class of functions q and r containing the polynomials as a very special case, the equation Lg = 0 has at most two linearly independent square integrable solutions, when p is identically one, q ≦ 0 and r ≧ 0.Finally, the main theorem is applied to show that certain sixth-order operators are limit-3.

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On the Structure of Finite Level and ω-Decomposable Borel Functions

Journal of Symbolic Logic ◽

10.2178/jsl.7804150 ◽

2013 ◽

Vol 78 (4) ◽

pp. 1257-1287 ◽

Cited By ~ 7

Author(s):

Luca Motto Ros

Keyword(s):

Banach Space ◽

Elementary Proof ◽

Borel Function ◽

Continuous Functions ◽

Full Description ◽

Countable Union ◽

Space Theory ◽

Measurable Functions ◽

Borel Functions

AbstractWe give a full description of the structure under inclusion of all finite level Borel classes of functions, and provide an elementary proof of the well-known fact that not every Borel function can be written as a countable union of Σα0-measurable functions (for every fixed 1 ≤ α < ω1). Moreover, we present some results concerning those Borel functions which are ω-decomposable into continuous functions (also called countably continuous functions in the literature): such results should be viewed as a contribution towards the goal of generalizing a remarkable theorem of Jayne and Rogers to all finite levels, and in fact they allow us to prove some restricted forms of such generalizations. We also analyze finite level Borel functions in terms of composition of simpler functions, and we finally present an application to Banach space theory.

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Neural Decision Tree Towards Fully Functional Neural Graph

Unmanned Systems ◽

10.1142/s2301385020500132 ◽

2020 ◽

Vol 08 (03) ◽

pp. 203-210

Author(s):

Han Xiao ◽

Ge Xu

Keyword(s):

Neural Networks ◽

Theoretical Analysis ◽

Decision Tree ◽

Back Propagation ◽

Continuous Functions ◽

Decision Function ◽

Propagation Process ◽

Neural Architecture ◽

Special Case

Though the traditional algorithms could be embedded into neural architectures with the proposed principle of [H. Xiao, Hungarian layer: Logics empowered neural architecture, arXiv: 1712.02555], the variables that only occur in the condition of branch could not be updated as a special case. To tackle this issue, we multiply the conditioned branches with Dirac symbol (i.e., [Formula: see text]), then approximate Dirac symbol with the continuous functions (e.g., [Formula: see text]). In this way, the gradients of condition-specific variables could be worked out in the back-propagation process, approximately, making a fully functional neural graph. Within our novel principle, we propose the neural decision tree (NDT), which takes simplified neural networks as decision function in each branch and employs complex neural networks to generate the output in each leaf. Extensive experiments verify our theoretical analysis and demonstrate the effectiveness of our model.

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