The bifunctional formalism: an alternative treatment of density functionals

The bifunctional formalism presents an alternative how to obtain the functional value from its functional derivative by exploiting homogeneous density scaling. In the bifunctional formalism the density dependence of the functional derivative is suppressed. Consequently, those derivatives have to be treated as formal functional derivatives. For a pointwise correspondence between the true and the formal functional derivative, the bifunctional expression yields the same value as the density functional. Within the bifunctional formalism the functional value can directly be obtained from its derivative (while the functional itself remains unknown). Since functional derivatives are up to a constant uniquely defined, this approach allows for a pointwise comparison between approximate potentials and reference potentials. This aspect is especially important in the field of orbital-free density functional theory, where the burden is to approximate the kinetic energy. Since in the bifunctional approach the potential is approximated directly, full control is given over the latter, and consequently over the final electron densities obtained from variational procedure. Besides the bifunctional formalism itself another concept is introduced, dividing the total non-interacting kinetic energy into a known functional part and a remainder, called Pauli kinetic energy. Only the remainder requires further approximations. For practical purposes sufficiently accurate Pauli potentials for application on atoms, molecular and solid-state systems are presented.

A new Wilson line-based action for gluodynamics

We perform a canonical transformation of fields that brings the Yang-Mills action in the light-cone gauge to a new classical action, which does not involve any triple-gluon vertices. The lowest order vertex is the four-point MHV vertex. Higher point vertices include the MHV and $$ \overline{\mathrm{MHV}} $$ MHV ¯ vertices, that reduce to the corresponding amplitudes in the on-shell limit. In general, any n-leg vertex has 2 ≤ m ≤ n − 2 negative helicity legs. The canonical transformation of fields can be compactly expressed in terms of path-ordered exponentials of fields and their functional derivative. We apply the new action to compute several tree-level amplitudes, up to 8-point NNMHV amplitude, and find agreement with the standard methods. The absence of triple-gluon vertices results in fewer diagrams required to compute amplitudes, when compared to the CSW method and, obviously, considerably fewer than in the standard Yang-Mills action.

Scalar-on-function local linear regression and beyond

It is common to want to regress a scalar response on a random function. This paper presents results that advocate local linear regression based on a projection as a nonparametric approach to this problem. Our asymptotic results demonstrate that functional local linear regression outperforms its functional local constant counterpart. Beyond the estimation of the regression operator itself, local linear regression is also a useful tool for predicting the functional derivative of the regression operator, a promising mathematical object on its own. The local linear estimator of the functional derivative is shown to be consistent. For both the estimator of the regression functional and the estimator of its derivative, theoretical properties are detailed. On simulated datasets we illustrate good finite sample properties of the proposed methods. On a real data example of a single-functional index model we indicate how the functional derivative of the regression operator provides an original, fast, and widely applicable estimation method.

Jets and differential linear logic

We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic, and hence a computational theory of non-linear partial differential equations and their solutions based on variational calculus.

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and ψ -Caputo fractional derivative.

JT gravity at finite cutoff

We compute the partition function of 2D2D Jackiw-Teitelboim (JT) gravity at finite cutoff in two ways: (i) via an exact evaluation of the Wheeler-DeWitt wavefunctional in radial quantization and (ii) through a direct computation of the Euclidean path integral. Both methods deal with Dirichlet boundary conditions for the metric and the dilaton. In the first approach, the radial wavefunctionals are found by reducing the constraint equations to two first order functional derivative equations that can be solved exactly, including factor ordering. In the second approach we perform the path integral exactly when summing over surfaces with disk topology, to all orders in perturbation theory in the cutoff. Both results precisely match the recently derived partition function in the Schwarzian theory deformed by an operator analogous to the T\bar TTT‾ deformation in 2D2D CFTs. This equality can be seen as concrete evidence for the proposed holographic interpretation of the T\bar TTT‾ deformation as the movement of the AdS boundary to a finite radial distance in the bulk.

PENGARUH MODEL PEMBELAJARAN NUMBERED HEAD TOGETHER DAN REALISTICS MATHEMATIC EDUCATION TERHADAP KEMAMPUAN BERPIKIR KRITIS DAN KEMAMPUAN PEMECAHAN MASALAH MATEMATIS SISWA DI SMA NEGERI 11 MEDAN

<p class="AfiliasiCxSpFirst" align="left"><strong>Abstrak:</strong></p><p class="AfiliasiCxSpMiddle">Penelitian ini bertujuan untuk mengetahui pengaruh kemampuan berpikir kritis dan kemampuan pemecahan masalah matematis siswa yang diajar dengan model pembelajaran <em>Numbered Head Together </em>dan model pembelajaran <em>Realistics Mathematic Education </em>di kelas XI SMA Negeri 11 Medan. Penelitian ini merupakan peneltian kuantitatif dengan jenis penelitian kuasi eksperimen. Populasi penelitian ini adalah seluruh siswa kelas XI SMA Negeri 11 Medan yang terdiri dari 2 kelas dan berjumlah 60 siswa, yang juga dijadikan sampel pada penelitian ini yakni sebagai sampel jenuh. Instrumen tes yang digunakan adalah dengan tes kemampuan berpikir kritis dan tes kemampuan pemecahan masalah berbentuk uraian. Analisis data dilakukan dengan analisis vaian (ANAVA). Hasil temuan ini menunjukkan: 1) Kemampuan berpikir kritis matematika siswa yang diajar dengan menggunakan model pembelajaran <em>Numbered Head Together</em> lebih baik dari pada siswa yang diajar dengan model pembelajaran <em>Realistics Mathematic Education </em>pada materi turunan fungsi; 2)Kemampuan pemecahan masalah matematika siswa yang diajar dengan menggunakan model pembelajaran <em>Numbered Head Together</em> tidak lebih baik dari pada siswa yang diajar dengan model pembelajaran <em>Realistics Mathematic Education </em>pada materi turunan fungsi; 3) Kemampuan berpikir kritis dan pemecahan masalah matematika siswa yang diajar dengan menggunakan model pembelajaran <em>Numbered Head Together</em> lebih baik dari pada siswa yang diajar dengan model pembelajaran <em>Realistics Mathematic Education </em>pada materi turunan fungsi; 4) Terdapat interaksi yang signifikan antara model pembelajaran yang digunakan terhadap kemampuan berpikir kritis dan pemecahan masalah matematika siswa pada materi turunan fungsi. Simpulan dalam penelitian ini menjelaskan bahwa kemampuan berpikir kritis dan pemecahan masalah matematika siswa lebih sesuai diajarkan dengan Model Pembelajaran <em>Numbered Head Together</em> dari pada Model Pembelajaran <em>Realistics Mathematic Education </em>pada materi turunan fungsi<em>.</em></p><p class="AfiliasiCxSpMiddle" align="left"><strong> </strong></p><p class="AfiliasiCxSpLast" align="left"><strong>Kata Kunci</strong>:</p><p>Kemampuan Berpikir Kritis, Kemampuan Pemecahan Masalah Matematis, Model Pembelajaran <em>Numbered Head Together</em>, Model Pembelajaran <em>Realistics Mathematic Education</em></p><p> </p><p class="AfiliasiCxSpFirst" align="left"><strong><em>Abstract:</em></strong></p><p class="AfiliasiCxSpMiddle"><em>This study aims to determine the effect of critical thinking skills and mathematical problem solving abilities of students who are taught with the Numbered Head Together learning model and the Realistic Mathematic Education learning model in class XI of SMA Negeri 11 Medan. This research is a quantitative research with the type of research that is quasi experiment. The population of this study was all students of class XI of SMA Negeri 11 Medan consisting of 2 classes and totaling 60 students, who were also sampled in this study as saturated samples. The test instrument used was a critical thinking ability test and a problem solving ability test in the form of a description. Data analysis was performed by analysis of variants (ANAVA). These findings show: 1) The ability to think critically the mathematics of students taught using the Numbered Head Together learning model is better than students taught with the Realistic Mathematic Education learning model on functional derivative material; 2) The ability to solve mathematical problems of students who are taught using the Numbered Head Together learning model is no better than students who are taught with Realistic Mathematic Education learning models on functional derivative material; 3) The ability to think critically and solve mathematical problems of students taught using the Numbered Head Together learning model is better than students taught with Realistic Mathematic Education learning models on functional derivative materials; 4) There is a significant interaction between the learning models used in critical thinking skills and students' mathematical problem solving on functional derivative material. The conclusion in this study explains that the ability to think critically and solve students' mathematical problems is more suitable to be taught with the Numbered Head Together Learning Model than the Realistic Mathematic Education Learning Model on functional derivative material.</em></p><p class="AfiliasiCxSpMiddle"><em> </em></p><p class="AfiliasiCxSpLast" align="left"><strong><em>Keywords</em></strong><em>:</em></p><p><em>Critical Thinking Ability, Mathematical Problem Solving Ability, Numbered Head Together Learning Model, Realistic Mathematic Education Learning Model</em></p>