Graded persistence diagrams and persistence landscapes

We introduce a refinement of the persistence diagram, the graded persistence diagram. It is the Mobius inversion of the graded rank function, which is obtained from the rank function using the unary numeral system. Both persistence diagrams and graded persistence diagrams are integer-valued functions on the Cartesian plane. Whereas the persistence diagram takes non-negative values, the graded persistence diagram takes values of 0, 1, or -1. The sum of the graded persistence diagrams is the persistence diagram. We show that the positive and negative points in the k-th graded persistence diagram correspond to the local maxima and minima, respectively, of the k-th persistence landscape. We prove a stability theorem for graded persistence diagrams: the 1-Wasserstein distance between k-th graded persistence diagrams is bounded by twice the 1-Wasserstein distance between the corresponding persistence diagrams, and this bound is attained. In the other direction, the 1-Wasserstein distance is a lower bound for the sum of the 1-Wasserstein distances between the k-th graded persistence diagrams. In fact, the 1-Wasserstein distance for graded persistence diagrams is more discriminative than the 1-Wasserstein distance for the corresponding persistence diagrams.Comment: accepted for publication in Discrete and Computational Geometr

This Rose Brings My Heart To You

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Pierrot and Pierrette : Valz Exquisite

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Sarah

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The Effects of a Problem Solving Model as an Alternative in the General Mathematics Curriculum

The purpose of this study was to evaluate an approved problem solving module as a model for use in the general mathematics curriculum, and assess its effectiveness in bringing about literacy in and a better attitude toward mathematics. Included in the module were topics on fundamental operations, fractions, decimals and percents, and problem solving. Specifically, the module focused on concerns of the National Assessment of Educational Progress (NAEF), that is the inability of students and young adults to use numbers skillfully enough to meet the demands of a modern society. The sample used in this study consisted of eight intact classes- -four control and four experimental groups. The groups were composed of students enrolled in ninth grade general mathematics from Logan City School District, Cache County School District, and Box Elder County School District. The three districts are comprised of urban, semi-rural, and rural developments in northern Utah. The total of 210 students included 131 males and 79 females. There were 117 students in the experimental groups and 93 students in the control groups. The diagrammatic problem solving module (composed of 20 lessons) was distributed among the teachers of the experimental groups for their perusal of content, lesson plans, and approach. When the teachers finished previewing the module, sessions were scheduled to accommodate each of their questions and concerns, and to cement the philosophy of the unit. Also, sessions were held with the control group teachers to delineate the kinds of experiences that would enhance equitable comparisons. To facilitate the use of the Solomon Four-Group Design, a least squares analysis of variance (ANOVA) technique with unequal numbers of subjects per treatment was used to analyze the null hypotheses at the .05 level of significance. The ANOVA was used for both the Algebra Readiness Test and the Mathematics Attitude Scale. On the basis of the findings of the five null hypotheses that were replicated in five different fashions, in regard to treatment effects, the following conclusions seem warranted. 1. There was a significantly more favorable attitude toward mathematics among students who were taught using the diagrammatic method. 2. There was a significantly more favorable performance among students on fundamental operations who were taught using the diagrammatic method. 3. There was a significantly more favorable performance among students on fractions who were taught using the diagrammatic method. 4. There was no difference in the performance among students on decimals and percents who were taught using the diagrammatic method and students who were taught using the traditional method. 5. There was a significantly favorable difference in the performance of students on problem solving who were taught using the diagrammatic method. 6. Sex differences were highly significant on one of the five replications, favoring the females. Inasmuch as the number of males far exceeded the number of females, a conclusion was not reached on sex differences. 7. Students who took both pretests and posttests, using the attitude scale, scored significantly higher than those who took only the posttest. Therefore it was concluded that pretesting served to raise scores on the attitude scale. 8. Students who took both pretests and posttests, using the readiness battery, did not perform significantly better than students who only took the posttest. Therefore, it was concluded that pretesting did not serve to enhance scores on the readiness battery

Isle D\u27Amour : Hesitation Waltz

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Isle D\u27Amour : Isle of Love

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While They Were Dancing Around

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Mister Butterfly

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My White Dove

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