On the difference of inverse coefficients of convex functions

Let f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${\mathcal {S}}$$ S be the subclass of normalised univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . Let F be the inverse function of f defined in some set $$|\omega |\le r_{0}(f)$$ | ω | ≤ r 0 ( f ) , and be given by $$F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$$ F ( ω ) = ω + ∑ n = 2 ∞ A n ω n . We prove the sharp inequalities $$-1/3 \le |A_4|-|A_3| \le 1/4$$ - 1 / 3 ≤ | A 4 | - | A 3 | ≤ 1 / 4 for the class $${\mathcal {K}}\subset {\mathcal {S}}$$ K ⊂ S of convex functions, thus providing an analogue to the known sharp inequalities $$-1/3 \le |a_4|-|a_3| \le 1/4$$ - 1 / 3 ≤ | a 4 | - | a 3 | ≤ 1 / 4 , and giving another example of an invariance property amongst coefficient functionals of convex functions.

Parameter Estimation in the Age of Degeneracy and Unidentifiability

Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.

Compression and Decompression which Use Deep Neural Network

<p>an auto-encoder which can be split into two parts is designed. The two parts can work well separately. The top half is an abstract network which is trained by supervised learning and can be used to classify and regress. The bottom half is a concrete network which is accomplished by inverse function and trained by self-supervised learning. It can generate the input of abstract network from concept or label. It is tested by tensorflow and mnist dataset. The abstract network is like LeNet-5. The concrete network is the inverse of the abstract network.Lossy compression can achieved by the test. The large compression ratio which is 19.6 is achieved. The decompression performance is ok through regression which treats classification as regression.</p>

Radius of Star-Likeness for Certain Subclasses of Analytic Functions

In this paper, we investigate a normalized analytic (symmetric under rotation) function, f, in an open unit disk that satisfies the condition ℜfzgz>0, for some analytic function, g, with ℜz+1−2nzgz>0,∀n∈N. We calculate the radius constants for different classes of analytic functions, including, for example, for the class of star-like functions connected with the exponential functions, i.e., the lemniscate of Bernoulli, the sine function, cardioid functions, the sine hyperbolic inverse function, the Nephroid function, cosine function and parabolic star-like functions. The results obtained are sharp.

PENGARUH PENGGUNAAN POWERPOINT DENGAN VISUAL BASIC APPLICATION TERHADAP HASIL BELAJAR SISWA PADA MATERI FUNGSI INVERS DI SMK WIKRAMA BOGOR

The purpose of this study was to prove the effect of the use of powerpoint with Visual Basic Application (VBA) on student learning outcomes in the inverse function material. This VBA powerpoint is applied to students of SMK class XI Mathematics subject in Inverse Function at SMK Wikrama Bogor. The research method used is quantitative with a simple experimental design (Posttest Only Control Group Design). This study involved 2 groups, namely the control class and the experimental class. The experimental class was given treatment using powerpoint learning media with Visual Basic Application (VBA), while the control class was conventional learning. Each was given a post test. Based on the results of data processing, the results of the calculation of the average post-test value of each experimental class were 76.13 while the control class was 65.85. Based on these data, it shows that there are differences between the two groups, (1) The average value of the experimental class is above the minimum completeness criteria, which is 75, while the control class is below the minimum completeness criteria. (2) The experimental class gets more value for powerpoint learning media with Visual Basic Application (VBA) is able to improve student learning outcomes. ABSTRAKTujuan dari penelitian ini untuk membuktikan adanya pengaruh media pembelajaran powerpoint dengan Visual Basic Application (VBA) terhadap hasil belajar siswa pada materi fungsi invers. VBA powerpoint ini diterapkan pada siswa SMK kelas XI mata pelajaran Matematika materi Fungsi Invers di SMK Wikrama Bogor. Metode penelitian yang digunakan yaitu kuantitatif dengan desain eksperimen sederhana (Posttest Only Control Group Design). Penelitian ini melibatkan 2 kelompok yaitu kelas kontrol dan kelas eksperimen. Kelas ekperimen diberikan perlakuan menggunakan media pembelajaran powerpoint dengan Visual Basic Application (VBA), sedangkan kelas kontrol pembelajaran konvensional. Masing masing diberikan post test. Berdasarkan hasil pengolahan data menunjukkan hasil perhitungan nilai rata-rata post test dari masing masing kelas eksperimen 76,13 sedangkan kelas kontrol 65,85. Berdasarkan data tersebut menunjukkan adanya perbedaan dari kedua kelompok, yaitu (1) Nilai rata rata kelas eksperimen diatas kriteria ketuntasan minimal yaitu 75, sedangkan kelas kontrol dibawah kriteria ketuntasan minimal.(2) Kelas eksperimen memperoleh nilai lebih media pembelajaran powerpoint dengan Visual Basic Application (VBA) mampu meningkatkan hasil belajar siswa.

Parameter estimation in the age of degeneracy and unidentifiability

Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one's ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (\Ldaomega) models. These are an extension of the classical lambda-omega (\ldaomega) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square-wave to spike-like. We show that the \Ldaomega\, models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase-shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the \Ldaomega\, models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of \Ldaomega\, models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data.