共查询到20条相似文献,搜索用时 15 毫秒
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本文中,我们讨论了在定义模糊n-方体数值映射的某种类型的积分时要用到的序有界模糊n-方体数集的上确界与下确界的问题。我们证明了序有界模糊n-方体数集的上、下确界的存在性,并给出了它们的表示。 相似文献
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The length of a field is the smallest integer m such that any totally positive quadratic form of dimension m represents all sums of squares. We investigate this field invariant and compare it to others such as the u-invariant, the Pythagoras number, the Hasse number, and the Mordell function related to sums of squares of linear forms. 相似文献
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The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric
piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities.
With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give
a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise
algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions
for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition
for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros
in every n-cell in the n-complex.
This work was supported by National Natural Science Foundation of China (Grant Nos. 10271022, 60373093, 60533060), the Natural
Science Foundation of Zhejiang Province (Grant No. Y7080068) and the Foundation of Department of Education of Zhejiang Province
(Grant Nos. 20070628 and Y200802999) 相似文献
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P. F. Kurchanov A. D. Myshkis A. M. Filimonov 《Journal of Applied Mathematics and Mechanics》1991,55(6):870-876
The problem of the vibrations of a one-dimensional chain of n + 1 linearly connected material points, under the action of a constant force F applied suddenly to the end point, is considered and the exact solution is analysed. It is proved that the supremum of the connecting force between the end point and its predecessor is equal to (2Fn)/(n + 1) if and only if the number of points is either prime or a power of 2. The behaviour of the supremum if this condition is not satisfied is demonstrated in the special case of six points. A discussion of the relationship with vibrations of the corresponding continuous medium and with the convergence of the method of straight lines is presented. 相似文献
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Warren W. Esty 《Historia Mathematica》1979,6(4):437-441
Ancient portraits of mathematicians are very rare. Since portraiture of living persons on ancient coins was restricted to rulers and members of their families, there are no contemporary portraits of mathematicians on ancient coins. However, a posthumous portrait of Pythagoras, who died c. 497 B.C., is the first portrait of a man, as opposed to a god, on a coin. Portraits did not exist at the time of Pythagoras; thus this coin, minted c. 430 B.C., is an extremely early portrait of a mathematician. 相似文献
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There is a well-known way to generalize the Riemann-Roch operator for Kahler manifold to that for Hermitian manifold. In this paper we show a slightly different way to get a generalized Riemann-Roch operator, which is just the Dirac operator. The difference between the two operators is that the latter one enables the so-called Pythagoras equalities. 相似文献
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Eliza P. de Jager 《Topology and its Applications》2007,154(10):2117-2126
We show that the infimum of any family of proximally symmetric quasi-uniformities is proximally symmetric, while the supremum of two proximally symmetric quasi-uniformities need not be proximally symmetric. On the other hand, the supremum of any family of transitive quasi-uniformities is transitive, while there are transitive quasi-uniformities whose infimum with their conjugate quasi-uniformity is not transitive. Moreover we present two examples that show that neither the supremum topology nor the infimum topology of two transitive topologies need be transitive. Finally, we prove that several operations one can perform on and between quasi-uniformities preserve the property of having a complement. 相似文献
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Lokenath Debnath 《International Journal of Mathematical Education in Science & Technology》2016,47(3):329-355
‘Number rules the universe.’The Pythagoras 相似文献
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J. M. H. Peters 《International Journal of Mathematical Education in Science & Technology》2013,44(6):733-736
When Pythagoras investigated frequencies of vibrating strings under tension, he found that by dividing the string in half it sounds an octave higher, by dividing it into three he obtained a note called a perfect fifth with its frequency three times that of the original string. By taking J of the string he found it vibrated at a frequency of the perfect fourth with frequency ratio to the fundamental of 4:3. With simple ratios like these he was able to construct a musical scale which subsequently developed into the diatonic scale which has been with us ever since. However, if Pythagoras had listened very carefully he may have wondered whether he should keep the ratio 4 : 3 for the perfect fourth or substitute another with ratio 11 : 8 related to the eleventh harmonic of the vibrating string. The matter is investigated below. 相似文献
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We study the problem of discrepancy of finite point sets in the unit square with respect to convex polygons, when the directions of the edges are fixed, when the number of edges is bounded, as well as when no such restrictions are imposed. In all three cases, we obtain estimates for the supremum norm that are very close to best possible. 相似文献
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it is said that when Pythagoras discovered his famous theorem, in a right-angled triangle the squares of the smaller sides
sum up to the square of the hypoteneuse, he sacrificed a hundred oxen to thank the gods. 相似文献
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Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded. 相似文献
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Helena Malinowski 《Positivity》2018,22(4):1039-1063
In Archimedean vector lattices bands can be introduced via three different coinciding notions. First, they are order closed ideals. Second, they are precisely those ideals which equal their double disjoint complements. The third concept is that of an ideal which contains the supremum of any of its bounded subsets, provided the supremum exists in the vector lattice. We investigate these three notions and their relationships in the more general setting of Archimedean pre-Riesz spaces. We introduce the notion of a supremum closed ideal, which is related to the third aforementioned notion in vector lattices. We show that for a directed ideal I in a pervasive pre-Riesz space with the Riesz decomposition property these three concepts coincide, provided the double disjoint complement of I is directed. In pervasive pre-Riesz spaces every directed band is supremum closed and every supremum closed directed ideal I equals its double disjoint complement, provided the double disjoint complement of I is directed. In general, in Archimedean pre-Riesz spaces the three notions differ. For this we provide appropriate counterexamples. 相似文献
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In this paper, we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum. As an application, we obtain that the continuous fuzzy-valued function on a closed interval exists supremum and infimum and give the precise representation. We also show that the bounded fuzzy-valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of Riemann and Riemann–Stieltjes by the usual way. 相似文献
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Paul Poncet 《Linear algebra and its applications》2011,435(7):1672-1680
A maxitive measure is the analogue of a finitely additive measure or charge, in which the usual addition is replaced by the supremum operation. In contrast to charges, maxitive measures often have a density. We show that maxitive measures can be decomposed as the supremum of a maxitive measure with density, and a residual maxitive measure that is null on compact sets under specific conditions. 相似文献
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Ulrich Bauer Axel Munk Hannes Sieling Max Wardetzky 《Foundations of Computational Mathematics》2017,17(1):1-33
We investigate the problem of estimating the number of modes (i.e., local maxima)—a well-known question in statistical inference—and we show how to do so without presmoothing the data. To this end, we modify the ideas of persistence barcodes by first relating persistence values in dimension one to distances (with respect to the supremum norm) to the sets of functions with a given number of modes, and subsequently working with norms different from the supremum norm. As a particular case, we investigate the Kolmogorov norm. We argue that this modification has certain statistical advantages. We offer confidence bands for the attendant Kolmogorov signatures, thereby allowing for the selection of relevant signatures with a statistically controllable error. As a result of independent interest, we show that taut strings minimize the number of critical points for a very general class of functions. We illustrate our results by several numerical examples. 相似文献