Smooth Graphs Research Articles

Interpretable generalized neural additive models for mortality prediction of COVID-19 hospitalized patients in Hamadan, Iran

BackgroundThe high number of COVID-19 deaths is a serious threat to the world. Demographic and clinical biomarkers are significantly associated with the mortality risk of this disease. This study aimed to implement Generalized Neural Additive Model (GNAM) as an interpretable machine learning method to predict the COVID-19 mortality of patients.MethodsThis cohort study included 2181 COVID-19 patients admitted from February 2020 to July 2021 in Sina and Besat hospitals in Hamadan, west of Iran. A total of 22 baseline features including patients' demographic information and clinical biomarkers were collected. Four strategies including removing missing values, mean, K-Nearest Neighbor (KNN), and Multivariate Imputation by Chained Equations (MICE) imputation methods were used to deal with missing data. Firstly, the important features for predicting binary outcome (1: death, 0: recovery) were selected using the Random Forest (RF) method. Also, synthetic minority over-sampling technique (SMOTE) method was used for handling imbalanced data. Next, considering the selected features, the predictive performance of GNAM for predicting mortality outcome was compared with logistic regression, RF, generalized additive model (GAMs), gradient boosting decision tree (GBDT), and deep neural networks (DNNs) classification models. Each model trained on fifty different subsets of a train-test dataset to ensure a model performance. The average accuracy, F1-score and area under the curve (AUC) evaluation indices were used for comparison of the predictive performance of the models.ResultsOut of the 2181 COVID-19 patients, 624 died during hospitalization and 1557 recovered. The missing rate was 3 percent for each patient. The mean age of dead patients (71.17 ± 14.44 years) was statistically significant higher than recovered patients (58.25 ± 16.52 years). Based on RF, 10 features with the highest relative importance were selected as the best influential features; including blood urea nitrogen (BUN), lymphocytes (Lym), age, blood sugar (BS), serum glutamic-oxaloacetic transaminase (SGOT), monocytes (Mono), blood creatinine (CR), neutrophils (NUT), alkaline phosphatase (ALP) and hematocrit (HCT). The results of predictive performance comparisons showed GNAM with the mean accuracy, F1-score, and mean AUC in the test dataset of 0.847, 0.691, and 0.774, respectively, had the best performance. The smooth function graphs learned from the GNAM were descending for the Lym and ascending for the other important features.ConclusionsInterpretable GNAM can perform well in predicting the mortality of COVID-19 patients. Therefore, the use of such a reliable model can help physicians to prioritize some important demographic and clinical biomarkers by identifying the effective features and the type of predictive trend in disease progression.

INFLUENCE OF DESIGN FEATURES OF RELATIONSHIPS ON THE STRESS STATE OF REINFORCED CONCRETE FLOOR SLABS

The article shows the influence of the design features of the bonds on the stress state of reinforced concrete floor slabs. Resolving equations for the calculation of plates are presented. A numerical calculation is made for several variants of fixing smooth and ribbed plates and graphs of changes in internal forces depending on the stiffness characteristics of the bonds are presented.

Local coloring problems on smooth graphs

We construct a smooth locally finite Borel graph $G$ and a local coloring problem $\Pi$ such that $G$ has a coloring $V(G) \to \mathbb{N}$ that solves $\Pi$, but no such coloring can be Borel.

Lipschitz graphs and currents in Heisenberg groups

AbstractThe main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension$k\leq n$in sub-Riemannian Heisenberg groups${\mathbb H}^{n}$. For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between${\mathbb H}$-rectifiability and ‘Lipschitz’${\mathbb H}$-rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.

TIME-OPTIMAL PATH PLANNING MODEL USING GENETIC ALGO- RITHM IN RRR ROBOT

The mathematical expression of the kinematic equations of each joint is utilized for the path planning using a quantic polynomial in joint space. In this study, a time optimization model for path planning using genetic algorithms with a vari- ety of crossover fraction and mutation rates is investigated. The optimization process is performed with MATLAB. Optimization using boundary conditions is performed with MATLAB. The result of the simulation, smooth speed graphs, angular position graphs, and the time when joint movements will complete the orbit as soon as possible are obtained. As a result of this study, a path planning model that can be applied to any robot is developed in joint space based on time optimization and can be used to shorten the task time, especially in task-based robots.

Regularity for Shape Optimizers: The Degenerate Case

AbstractWe consider minimizers of urn:x-wiley:00103640:media:cpa21810:cpa21810-math-0001 where F is a function nondecreasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often‐studied F=λN. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler‐Lagrange equation for the minimizers that is not otherwise available. © 2019 Wiley Periodicals, Inc.

Heat Kernels, Stochastic Processes and Functional Inequalities

The general topic of the 2016 workshop Heat kernels, stochastic processes and functional inequalities was the study of linear and non-linear diffusions in geometric environments including smooth manifolds, fractals and graphs, metric spaces and in random environments. The workshop brought together leading researchers from analysis, geometry and probability, and provided an excellent opportunity for interactions between scientists from these areas at different stages of their career. The unifying themes were heat kernel analysis, mass transportation problems and functional inequalities while the program straddled across a great variety of subjects and across the divide that exists between discrete and continuous mathematics. Other unifying concepts such as the notions of metric measure space, Otto Calculus and Lott-Sturm-Villani synthetic Ricci curvature bounds played an important part in the discussions. Novel directions including the study of Liouville quantum gravity were included. The workshop provided participants with an opportunity to discuss how these ideas and techniques can be used to approach problems regarding optimal transport, Riemannian and sub-Riemannian geometry, and analysis and stochastic processes in random media.

Re-creating Hottel’s emissivity charts for water vapor and extending them to 40 bar pressure using HITEMP-2010 data base

Hottel’s water vapor emissivity chart and the pressure correction chart have been re-created using the most recent HITEMP-2010 data base to provide more accurate values. Hottel’s standard emissivity has been redefined in order to produce smooth and accurate graphs and correlations. The new charts allow for calculating the water vapor emissivities at homogeneous pathways up to 200 bar equivalent pressure, in the 450–3000 K temperature range and in the 0.01–3000 bar cm pressure path length range; a departure from the original Hottel’s emissivity values can be larger than 300% at high pressures. Departures from Leckner’s (1972) correlations are typically in the ± 40% range. Besides the new paper graphs, showing the standard emissivity as well as the pressure correction factor at different temperatures, the calculated emissivities have been tabulated (see Supplementary material) at equivalent pressures ranging from 0.05 bar up to 250 bar. Using appropriate interpolation techniques, these charts can be used as look-up tables to calculate accurate total emissivity values without using pressure correction factors. To this end, linear interpolation for temperature scaling and logarithmic interpolation for pressure and pressure path length scaling are included in the Supplementary material.

Regularity in the amino acid sequences of proteins

T transit of proteins from linear to tertiary states should be exactly tractable because the tertiary states have unambiguous crystalline forms. Since the amino acids interact in the water environment, the solution for the folding problem should give due consideration to the hydrophobic / hydrophilic indices of the amino acids, as mentioned in many studies. Representing amino acids by these indices, we find two different kinds of regularity in amino acid sequence of cytosolic and membrane proteins. If we take the mean hydrophobicity for every few percent of amino acids in proteins and plot them in a cumulative fashion from the N to the C termini, we get two different kind of smooth graphs for cytosolic and membrane proteins.. More than 100 cytosolic proteins and 30 membrane proteins of widely varying number of amino acids and structures could be represented in two such smooth graphs with small standard errors. What we know so far, for cytosolic proteins, is that this regularity comes from a simple working principle which can be based on our observation on the kinetics of folding of a number of proteins starting from the N-termini and extending to the C-termini through a sequential interaction with the same set of nucleotides in the peptidyl transferase center (PTC) of the ribosome.

Weak Liouville-Arnol′d Theorems and Their Implications

This paper studies the existence of invariant smooth Lagrangian graphs for Tonelli Hamiltonian systems with symmetries. In particular, we consider Tonelli Hamiltonians with n independent but not necessarily involutive constants of motion and obtain two theorems reminiscent of the Liouville-Arnold theorem. Moreover, we also obtain results on the structure of the configuration spaces of such systems that are reminiscent of results on the configuration space of completely integrable Tonelli Hamiltonians.

Enumeration of labeled connected bicyclic and tricyclic graphs without bridges

We consider undirected simple connected graphs. A bridge in a connected graph is an edge whose deletion makes the graph disconnected. A smooth graph is a graph without endpoints. Obviously, the set of graphs without bridges is a subset of the set of smooth graphs. Smooth graphs were enumerated by Wright [1]. Hanlon and Robinson [2] enumerated both labeled and unlabeled graphs without bridges. They obtained a functional equation and a nonlinear partial differential equation for the generating function of labeled graphs. However, their formulas are awkward and have not been reduced to a form convenient for computations. Apparently, that is why Sloane’s famous “On-Line Encyclopedia of Integer Sequences” [3] only provides data on the number of unlabeled graphs without bridges. The aim of the present paper is to obtain closed-form expressions for the number of labeled biand tricyclic graphs without bridges as well as the corresponding asymptotics for the number of such graphs as the number of vertices tends to infinity. Recall that a cutpoint of a connected graph is a vertex whose deletion (the edges incident to the vertex are deleted as well) makes the graph disconnected. A block is a connected graph without cutpoints or a maximal nontrivial connected subgraph without cutpoints [4, p. 200]. A graph is said to be uni-, bi-, or tricyclic if its cyclomatic number is 1, 2, or 3, respectively. For k ≥ 0, we denote the number of labeled blocks with n vertices and n+ k edges, the number of labeled connected graphs without bridges with n vertices and n+ k edges, and the number of labeled smooth graphs with n vertices and n+ k edges by u(n, n+ k), l(n, n+ k), and v(n, n + k), respectively. The cyclomatic number of any of these graphs is k + 1. Let Uk(w) and Lk(w) be the exponential generating functions,

Maximum independent sets in 3- and 4-regular Hamiltonian graphs

Smooth 4-regular Hamiltonian graphs are generalizations of cycle-plus-triangles graphs. While the latter have been shown to be 3-choosable, 3-colorability of the former is NP-complete. In this paper we first show that the independent set problem for 3-regular Hamiltonian planar graphs is NP-complete, and using this result we show that this problem is also NP-complete for smooth 4-regular Hamiltonian graphs. We also show that this problem remains NP-complete if we restrict the problem to the existence of large independent sets (i.e., independent sets whose size is at least one third of the order of the graphs).

Smooth Graphs for Visual Exploration of Higher-Order State Transitions

In this paper, we present a new visual way of exploring state sequences in large observational time-series. A key advantage of our method is that it can directly visualize higher-order state transitions. A standard first order state transition is a sequence of two states that are linked by a transition. A higher-order state transition is a sequence of three or more states where the sequence of participating states are linked together by consecutive first order state transitions. Our method extends the current state-graph exploration methods by employing a two dimensional graph, in which higher-order state transitions are visualized as curved lines. All transitions are bundled into thick splines, so that the thickness of an edge represents the frequency of instances. The bundling between two states takes into account the state transitions before and after the transition. This is done in such a way that it forms a continuous representation in which any subsequence of the time series is represented by a continuous smooth line. The edge bundles in these graphs can be explored interactively through our incremental selection algorithm.We demonstrate our method with an application in exploring labeled time-series data from a biological survey, where a clustering has assigned a single label to the data at each time-point. In these sequences, a large number of cyclic patterns occur, which in turn are linked to specific activities. We demonstrate how our method helps to find these cycles, and how the interactive selection process helps to find and investigate activities.

Parameterized Algorithms in Smooth 4-Regular Hamiltonian Graphs

Smooth 4-regular hamiltonian graphs are generalizations of cycle plus triangles graphs. It has been shown that both the independent set and 3-colorability problems are NP-Complete in this class of graphs. In this paper we show that these problems are fixed parameter tractable if we choose the number of inner cycles as parameter.

The Dirichlet energy of mappings from ${\bf B^3}$ into a manifold: density results and gap phenomenon

Weak limits of graphs of smooth maps \(u_k: B^n\to \mathcal{Y}\) with equibounded Dirichlet integral give rise to elements of the space \(\mathrm{cart}^{2,1}(B^n\times \mathcal{Y})\). We assume that the 2-homology group of \(\mathcal{Y}\) has no torsion and that the Hurewicz homomorphism \(\pi_2(\mathcal{Y})\to H_2(\mathcal{Y},{\mathbb{Q}})\) is injective. Then, in dimension n = 3, we prove that every element T in \( \mathrm{cart} ^{2,1}(B^3\times \mathcal{Y})\), which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {uk} with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.

A characterization of graphs which can be approximated in area by smooth graphs

For vector valued maps, convergence in W1,1 and of all minors of the Jacobian matrix in L1 is equivalent to convergence weakly in the sense of currents and in area for graphs. We show that maps defined on domains of dimension n S 3 can be approximated strongly in this sense by smooth maps if and only if the same property holds for the restriction to a.e. 2-dimensional plane intersecting the domain.

New procedure for presenting data by integrated or differentiated smooth graphs

A new algorithm for presenting data by integrated or differentiated smooth graphs was found while looking for a fast and flexible procedure for presenting experimental data by smooth graphs based on different line shape functions. The algorithm to be presented is applicable at data points that are both widely separated and scattered within the same set, as well as data of a higher dimensional frame.

Graphs of Finite Mass Which Cannot Be Approximated by Smooth Graphs with Equibounded Area

We show that there exist graphs of Cartesian maps, of finite mass, which cannot be approximated weakly as currents by graphs of smooth maps with equibounded area.

Graphs of finite mass which cannot be approximated in area by smooth graphs

In this note we shall show that there are obstructions of homotopic and not just of homological nature to the approximability in area of graphs of finite mass by graphs of smooth maps. We shall present a class of closed graphs with finite mass which cannot be approximated weakly and in area by smooth graphs.

Enumeration of smooth labelled graphs

SynopsisAn (n, q) graph is a graph on n labelled points and q lines without loops or multiple lines. We write ν(n, q) for the number of smooth (n, q) graphs, i.e. connected graphs without end points, and ν = V(Z, Y) = ∑n,q ν(n,q)ZnYq /n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form for V (not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied by V. We interpret this equation in purely combinatorial terms. We write Vk = ∑ n ν(n, n + k)Xn/n! and find a recurrence formula for Vk for successive k. We use these and other results to find an asymptotic expansion for ν(n,q) as n→∞ when (q/n) − log n − log log n→ + ∞ and an asymptotic approximation to ν(n,n + k) when 0 < k = o and to log ν(n, n + k) when k < (1−ε).