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Topology in Chemistry. 1st Edition. Discrete Mathematics of Molecules. 0 star rating Write a review. Authors: D H Rouvray R B King. Hardcover ISBN.
Table of contents
- Recommended for you
- Schultz Polynomials and Their Topological Indices of Jahangir Graphs J2,m
- Distance-based topological polynomials and indices of friendship graphs
Springer-Verlag, Berlin. Digest J Nanomater Bios 4 4 — Hedyari A Wiener and Schultz indices of V-naphtalenic nanotori. Optoelectron Adv Mater Rapid Commun 5 7 — Digest J Nanomater Bios 5 1 — Hosoya H On some counting polynomials in chemistry. Discrete Appl Math — Hua H Wiener and Schultz molecular topological indices of graphs with specified cut edges. Digest J Nanomater Bios 4 1 — Asian Acad Res J Multidiscip 3 1 — Chem Phys Lett — J Chem Inform Comput Sci Schultz HP Topological organic chemistry 1. Graph theory and topological indices of alkanes. Whole-molecule Schultz topo- logical indices of alkanes.
J Nanomater Trinajstic N Chemical graph theory. Wiener H Structural determination of paraffin boiling points. J Am Chem Soc — Wiener H Relations of the physical properties of the isomeric alkanes to molecular structure: surface tension, specific dispersion, and critical solution temperature in aniline. J Phys Chem — Zhou B Bounds for the Schultz molecular topological index.
Download references. WG and MRF proposed the idea for computing the distance-based topological indices of friendship graph which was implemented and computations were founds by MI and MRRK which were verified by all the authors. All authors read and approved the final manuscript. Correspondence to Muhammad Imran. Reprints and Permissions. Search all SpringerOpen articles Search.
Abstract Drugs and chemical compounds are often modeled as graphs in which the each vertex of the graph expresses an atom of molecule and covalent bounds between atoms are represented by the edges between their corresponding vertices. Background Recent years have witnessed the rapid development in nanomaterials and drugs, which keeps in pace with the development of pharmacopedia.
Full size image. The present multi-level persistent homology is able to describe any selected interactions of interest and delivers two benefits in characterizing biomolecules. Firstly, the pairwise non-covalent interactions can be reflected by the 0th dimensional barcodes. Secondly, such treatment generates more higher dimensional barcodes and the small structural fluctuation among different conformations of the same molecule can be captured.
The persistent barcode representation of the molecule can be significantly enriched to better distinguish between different molecular structures and isomers. A different conformation of the ligand is generated by using the Frog2 web server [ ]. The persistent barcodes generated using Rips complex with the distance matrices M are identical and only have 0th dimensional bars due to the simple structure.
In this case, the 0th dimensional bars only reflect the length of each bond and therefore fail to distinguish the two slightly different conformations of the same molecule. However, when the modified distance matrices are employed, the barcode representation is significantly enriched and is able to capture the tiny structural perturbation between the conformations. An illustration of the outcome from the modified distance matrix is shown in Fig 8. Illustration of representation ability of in reflecting structural perturbations among conformations of the same molecule.
Right: The persistence diagram showing the 1st and 2nd dimensional results generated using Rips complex with for two conformations. It is worth noticing that the barcodes generated using Rips complex with M are identical for the two conformations. In protein-ligand binding analysis and analysis involving interactions, we are interested in the change of topological invariants induced by interactions that are caused by binding or other processes. Similar to the idea of multi-level persistent homology, we can design a distance matrix to focus on the interactions of interest.
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In applications, A 1 and A 2 can be respectively a set of atoms of the protein and a set of atoms of the ligand in a protein-ligand complex. In this case, the characterization of interactions between ligand and protein is an important task. In the modeling of point mutation induced protein stability changes, A 1 could be the set of atoms at the mutation site and A 2 could be the set of atoms of surrounding residues close to the mutation site. Similar treatment can be used for protein-protein and protein-nucleic acid interactions.
For biomolecules, the interaction strength between pair of atoms usually does not align linearly to their Euclidean distances. For example, van der Waals interaction is often described by the Lennard-Jones potential. Therefore, kernel function filtration can be used to emphasize certain geometric scales. This filtration can be incorporated in the element specific persistent homology 7 Additionally, one can simultaneously use two or more correlation functions characterized by different scales to generate a multiscale representation of biomolecules [ ].
One form of the correlation function based filtration matrix is constructed by flexibility and rigidity index. Note that the rigidity index is given by [ ] 9 This expression is closely related to the rigidity density based volumetric filtration [ 90 ]. Electrostatic effects are some of the most important effects in biomolecular structure, function, and dynamics. The embedding of electrostatics in topological invariants is of particular interest and can be very useful in describing highly charged biomolecules such as nucleic acids and their complexes.
We introduce electrostatics interaction induced distance functions in Eq 10 to address the electrostatic interactions among charged atoms. The abstract distance between two charged particles are rescaled according to their charges and their geometric distance, and is modeled as 10 where d ij is the distance between the two atoms, q i and q j are the partial charges of the two atoms, and c is a nonzero tunable parameter.
The form of the function is adopted from sigmoid function which is widely used as an activation function in artificial neural networks. Such function regularizes the input signal to the [0, 1] interval. Other functions can be similarly used. This formulation can be extended to systems with dipole or higher order multipole approximations to electron density.
The weak interactions due to long distances or neutral charges result in correlation values close to 0. The parameter c is rather physical but chosen to effectively spread the computed values over the 0, 1 interval so that the results can be used by machine learning methods. Another simple choice of charge correlation functions is However, this choice will lead to a different filtration domain.
Eq 11 can be used for electrostatic filtration as well. In this case, the filtration parameter can be the charge density value and cubical complex based filtration can be used. Multicomponent persistent homology refers to the construction of multiple persistent homology components from a given object to describe its properties.
Obviously, element specific persistent homology leads to multi-component persistent homology. Nevertheless, in element specific persistent homology, the emphasis is given to the appropriate selection of important elements for describing certain biological properties or functions. For example, in biological context, electronegative atoms are selected for describing hydrogen bond interactions, polar atoms are selected for describing hydrophilic interactions, and carbon atoms are selected for describing hydrophobic interactions.
Note that in chemical context, an atom may have many sharply different chemical and physical properties, depending on its oxidation states. Whereas, in multicomponent persistent homology, the emphasis is placed on the systematic generation of topological invariants from different combinatorial possibilities and the construction of 2D or high-dimensional persistent maps for deep convolutional neural networks.
Barcode representation of topological invariants offers a visualization of persistent homology analysis. In machine learning analysis, we convert the barcode representation of topological invariants into structured feature arrays for machine learning. To this end, we introduce two methods, i. These methods are discussed below. Python code is given in S1 Code for the generation of features used in the final models in the Results section. Note that the above discussion should be applied to three topological dimensions, i. In general, this approach enables the description of bond lengths, including the length of non-covalent interactions, in biomolecules and was referred to as binned persistent homology in our earlier work [ 14 , 27 , 94 ].
Another method of feature vector generation from a set of barcodes is to extract important statistics of barcode collections such as maximum values and standard deviations. Three statistic feature vectors , , and can then be generated in the sense of the statistics of the collection of barcodes. The generation of is the same by examining the set Death. Statistics feature vectors are collected from barcodes of three topological dimensions, i.
A more thorough description of sets of barcodes is to first divide the sets into subsets and extract features analogously to the barcode statistics method. As shown in Fig 9 , a persistence diagram can be divided into slices in different directions.
Schultz Polynomials and Their Topological Indices of Jahangir Graphs J2,m
The barcodes that fall in each slice form a subset. Each subset is described in terms of feature vector by using the barcode statistics method. When the persistence diagram is sliced horizontally, members in each subset have similar death values and the barcode statistics feature vector is generated for the set of birth values. Similarly, members in each subset have similar birth values if the persistence diagram is sliced vertically, and the barcode statistics feature vector is generated for the set of death values.
The barcode statistics feature vectors are generate for both set of birth values and set of death values if the persistence diagram is sliced diagonally, where members in each subset have similar persistence. This type of feature vector generation describes the set of barcodes in more detail but will produce longer feature vectors. The barcodes are plotted as persistence diagrams with the horizontal axis being birth and the vertical axis being death. From left to right, the subsets are generated according to the slicing of death, birth, and persistence values.
The construction of multi-dimensional persistence is an interesting topic in persistent homology. In general, it is believed that multi-dimensional persistence has better representational power for complex systems described by multiple parameters [ 43 ]. Although multidimensional persistence is hard to compute, one can compute persistence for one parameter while fixing the rest of the parameters to a sequence of fixed values.
In the case where there are two parameters, a bifiltration can be done by taking turns to fix one parameter to a sequence of fixed values while computing persistence for the other parameter. For example, one can take a sequence of resolutions and compute persistence for distance with each fixed resolution. The sequence of outputs can be stacked to form a multidimensional representation [ ].
Computing persistence multiple times and stacking the results is especially useful when the parameters that are not chosen to be the filtration parameter are naturally discrete with underlying orders. For example, the multi-component or element specific persistent homology will result in many persistent homology computations over different selections of atoms.
These results can be ordered by the percentage of atoms used of the whole molecule or by their importance scores in classical machine learning methods. Also, multiple underlying dimensions exist in the element specific persistent homology characterization of molecules. This property enables 2D or 3D topological representation of molecules.
Based on the observation that the performance of the predictor degenerates when too many element combinations are used, we order the element combinations according to their individual performance on the task using methods of ensemble of trees. Combining the dimension of spatial scale and dimension of element combinations, a 2D topological representation is obtained.
Such representation is expected to work better in the case of complex geometry such as protein-ligand complexes. For 0th dimensional, since all bars start from zero, there is no need for. These eight 2D representations are regarded as eight channels of a 2D topological image. In protein-ligand binding analysis, 2D topological features are generated for the barcodes of a protein-ligand complex and for the differences between barcodes of the protein-ligand complex and those of the protein.
Therefore, we have a total of 16 channels in a 2D image for the protein-ligand complex. This channel image can be fed into the training or the prediction of convolutional neural networks. In the characterization of protein-ligand complexes using alpha complexes, 2D features are generated from the alpha complex based on persistent homology computations of protein and protein-ligand complex. A total of element combinations are considered. Fig 10 illustrates 16 channels of sample 1wkm in PDBBind database. These images are directly used in deep convolutional neural networks for training and prediction.
For each map, the horizontal axis is the dimension of spatial scale and the vertical axis is element combinations ordered by their importance. When there are fewer element combinations considered which can hardly form another axis, the axis of element combinations can be added into the original channels to form 1D representations that can be used in 1D CNN.
Three machine learning algorithms, including k-nearest neighbors KNN regression, gradient boosting trees and deep convolutional neural networks, are integrated with our topological representations to construct topological learning algorithms. One of the simplest machine learning algorithms is k-nearest neighbors KNN for classification or for regression. In KNN regression, for a given object, its property values is obtained by the average or the weighted average of the values of its k nearest neighbors induced by a given metric of similarity.
Then, the problem becomes how to construct a metric on the dataset. In the present work, instead of computing similarities from constructed feature vectors, the similarity between biomolecules can simply be derived from distances between barcodes generated from different biomolecules. Popular barcode space metrics include the bottleneck distance [ ] and more generally, the Wasserstein metrics [ 95 , 96 ]. The definition of the two metrics is summarized as follows.
The bottleneck distance is defined as , where the minimum is taken over all possible bijections from subsets of A to subsets of B. The Wasserstein metric, a L p generalized analog to the bottleneck distance can be defined with the penalty [ 96 ] 15 and the corresponding distance. It approaches the bottleneck distance by setting p goes to infinity. Wasserstein metric measures the closeness of barcodes generated from different biomolecules.
It will be interesting to consider other distances for metric spaces, such as Hausdorff distance, Gromov-Hausdorff distance [ ], and Yau-Hausdorff distance [ ] for biomolecular analysis. However, an exhaustive study of this issue is beyond the scope of the present work. The barcode space metrics can be directly used to assess the representation power of various persistent homology methods on biomolecules without being affected by the choice of machine learning models and hyperparameters.
We show in the section of results that the barcode space metrics induced similarity measurement is significantly correlated to molecule functions. Wasserstein metric measures from biomolecules can also be directly implemented in a kernel based method such as nonlinear support vector machine algorithm for classification and regression tasks. However, this aspect is not explored in the present work. Gradient boosting trees is an ensemble method which ensembles individual decision trees to achieve the capability of learning complex feature target maps and can effectively prevent overfitting by using shrinkage technique.
The gradient boosting trees method is realized using the GradientBoostingRegressor module in scikit-learn software package [ ] version 0. A set of parameters found to be efficient in our previous study on the protein-ligand binding affinity prediction [ 27 ] is used uniformly unless specified. The deep convolutional neural networks in this work are implemented using Keras [ ] version 1.
For TopBP-DL Complex , a widely used convolutional neural network architecture is employed beginning with convolution layers followed by dense layers. Due to the limited computation resources, parameter optimization is not performed, while most parameters are adopted from our earlier work [ 94 ]. Reasonable parameters are assigned manually. The detailed architecture is shown in Fig The Adam optimizer with learning rate 0.
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The loss function is the mean squared error function. The network is trained with a batch size of 16 and epochs. The training data is shuffled for each epoch. The numbers in convolution layers mean the number of filters and filter size from left to right. The dense layers are drawn with number of neurons and activation function. The pooling size of the pooling layers and dropout rate of the dropout layers are listed.
The Adam optimizer with learning rate set to 0. The loss function is binary cross-entropy. The network is trained with a batch size of and 10 epochs. The batch size is larger than that used in TopBP-DL due to the much larger training set in this problem. Because of the same reason, the training process converges to a small loss very fast with only a few training steps.
The 1D image-like layers are shown in sharp-corner rectangles. Code for generating the features used in the final models in the Results section. It takes PDB files for proteins and Mol2 files for ligands as inputs. Abstract This work introduces a number of algebraic topology approaches, including multi-component persistent homology, multi-level persistent homology, and electrostatic persistence for the representation, characterization, and description of small molecules and biomolecular complexes.
Author summary Conventional persistent homology neglects chemical and biological information during the topological abstraction and thus has limited representational power for complex chemical and biological systems. Introduction Arguably, machine learning has become one of the most important developments in data science and artificial intelligence.
Results Rational drug design and discovery have rapidly evolved into some of the most important and exciting research fields in medicine and biology. Download: PPT. Fig 1. An illustration of the topology based machine learning algorithms used in scoring and virtual screening. Ligand based protein-ligand binding affinity prediction In this section, we address the representation of small molecules by element specific persistent homology, especially the proposed multi-level persistent homology designed for small molecules.
Data set. Models and performance. Table 1.
Distance-based topological polynomials and indices of friendship graphs
Complex based protein-ligand binding affinity prediction In this section, we develop topological representations of protein-ligand complexes. Data sets.
Model and performance. Table 3. Structure-based virtual screening In this section, we examine the performance of the proposed method for the main application in this paper, which is structure-based virtual screening which involves protein-compound complexes obtained by attempting to dock the candidates to the target proteins. DUD data set. Data processing. Topology based machine learning models. Topology based deep learning model. The final model. Discussion Ligand based protein-ligand binding affinity prediction We conduct several experiments on ligand based protein-ligand binding affinity prediction in this section which leads to the final models.
Fig 2. Statistics of ligands in 7 protein clusters in S dataset. Feature vectors for gradient boosting trees. Barcode space metrics for k-nearest neighbor regression. Fig 3. An illustration of similarities between ligands measured by their barcode space Wasserstein distances. Table 7. Experiments for ligand-based protein-ligand binding affinity prediction of 7 protein clusters and protein-ligand complexes.
Performance of multi-component persistent homology. Table 8. Performance of different approaches on the S dataset. Robustness of topological learning models. Fig 4. Plot of performance against number of element combinations used. Complex based protein-ligand binding affinity prediction Having demonstrated the representational power of the present topological learning method for characterizing small molecules, we further examine the method on the task of characterizing protein-ligand complex.
Groups of topological features and their performance in association with GBT. Table 9. Experiments for protein-ligand-complex-based protein-ligand binding affinity prediction for the PDBBind datasets. Robustness of GBT algorithm against redundant element combination features and potential overfitting. Usefulness of more than 2 element types for interactive 0th dimensional barcodes. Table Performance of different protein-ligand complex based approaches on the PDBBind datasets. Importance of atomic charge in electrostatic persistence.
Relevance of elements that are rare with respect to the data sets. Fig 6. Assessment of performance of the model on samples with elements that are rare in the data sets. Structure-based virtual screening In our final model TopVS reported in Table 6 , we use topological descriptors of both protein-compound interactions and only the compounds i. Conclusion Persistent homology is a relatively new branch of algebraic topology and is one of the main tools in topological data analysis.
Methods Persistent homology The concept of persistent homology is built on the mathematical concept of homology, which associates a sequence of algebraic objects, such as abelian groups, to topological spaces. Simplicial complex. Persistent homology. Simplicial complexes and filtration.
Biological considerations The development of persistent homology was motivated by its potential in the dimensionality reduction, abstraction and simplification of biomolcular complexity [ 36 ]. Covalent bonds. Non-covalent interactions. Chirality, cis effect and trans effect. Multi-leveled protein structures.
Protein-ligand, protein-protein, and protein-nucleic acid complexes. Element specific persistent homology One important issue is how to protect chemical and biological information during the topological simplification. Distance matrix induced persistent homology Biomolecular systems are not only complex in geometry, but also in chemistry and biology. Multi-level persistent homology.
Fig 8. Multi-level persistent homology on simple small molecules. Interactive persistent homology. Correlation function based persistent homology. Flexibility and rigidity index based filtration matrix. Electrostatic persistence. Multi-component persistent homology. Feature generation from topological invariants Barcode representation of topological invariants offers a visualization of persistent homology analysis. Counts in bins. Barcode statistics. Persistence diagram slice and statistics. Fig The 2D topological maps of the 16 channels of sample 1wkm.
Machine learning algorithms Three machine learning algorithms, including k-nearest neighbors KNN regression, gradient boosting trees and deep convolutional neural networks, are integrated with our topological representations to construct topological learning algorithms. K-nearest neighbors algorithm via barcode space metrics. Gradient boosting trees. Deep convolutional neural networks. Supporting information. S1 Text. Extra results and records.
S1 Code. Feature generation. References 1. Imagenet classification with deep convolutional neural networks. In: Advances in neural information processing systems; Simonyan K, Zisserman A. Very deep convolutional networks for large-scale image recognition. Deep learning. Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups.
View Article Google Scholar 5. Schmidhuber J. Deep learning in neural networks: An overview. Neural Networks. Multimodal deep learning. Modeling epoxidation of drug-like molecules with a deep machine learning network. ACS Central Science.
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Discrete Comput Geom. Sumners DW. Knot theory and DNA. In: Proceedings of Symposia in Applied Mathematics. Darcy IK, Vazquez M. Determining the topology of stable protein-DNA complexes. Biochemical Society Transactions. Heitsch C, Poznanovic S. Discrete and Topological Models in Molecular Biology. DasGupta B, Liang J. Models and Algorithms for Biomolecules and Molecular Networks. Shi X, Koehl P. Geometry and topology for modeling biomolecular surfaces. Far East J Applied Math. Topological persistence and simplification. Bendich P, Harer J. Persistent Intersection Homology. Stability of Persistence Diagrams.
Foundations of Computational Mathematics. Persistent Homology for Kernels, Images, and Cokernels. SODA 09; Proximity of persistence modules and their diagrams. In: Proc. Persistence-based clustering in riemannian manifolds. Carlsson G, Zomorodian A. The theory of multidimensional persistence. Discrete Computational Geometry. Zigzag persistent homology and real-valued functions.
ACM Sympos. Persistent cohomology and circular coordinates. Discrete and Comput Geom. Carlsson G, De Silva V. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel.
Siegel modular varieties are the most basic examples of Shimura varieties. Construction The Siegel modular variety A can be constructed as the complex analytic spaces constructed as the quotient of the Siegel upper half-space of degree g by the action of a symplectic group. Graphical representation of a rotaxane Structure of a rotaxane that has a cyclobis paraquat-p-phenylene macrocycle. The name is derived from the Latin for wheel rota and axle axis. The two components of a rotaxane are kinetically trapped since the ends of the dumbbell often called stoppers are larger than the internal diameter of the ring and prevent dissociation unthreading of the components since this would require significant distortion of the covalent bonds.
Much of the research concerning rotaxanes and other mechanically interlocked molecular architectures, such as catenanes, has been focused on their efficient synthesis or their utilization as artificial molecular machines. However, examples of rotaxane substructure have been found in naturally occurring peptides, including: cystine knot peptides, cyclotides. Olympiadane is a mechanically-interlocked molecule composed of five interlocking macrocycles that resembles the Olympic rings. The molecule is a linear pentacatenane or a catenane.
It was synthesized and named by Fraser Stoddart and coworkers in See also Olympicene References Amabilino, D. Browne, M. The New York Times. Retrieved 3 January An artificial chemistry is a chemical-like system that usually consists of objects, called molecules, that interact according to rules resembling chemical reaction rules.
Artificial chemistries are created and studied in order to understand fundamental properties of chemical systems, including prebiotic evolution, as well as for developing chemical computing systems. Artificial chemistry is a field within computer science wherein chemical reactions—often biochemical ones—are computer-simulated, yielding insights on evolution, self-assembly, and other biochemical phenomena. The field does not use actual chemicals, and should not be confused with either synthetic chemistry or computational chemistry. Rather, bits of information are used to represent the starting molecules, and the end products are examined along with the processes that led to them.
The field originated in artificial life but has shown to be a versatile method with applications in many fields such as chemistry, economics, sociology and. A collection of circles and the corresponding unit disk graph Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth.
The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object. Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology. History Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late A triangle immersed in a saddle-shape plane a hyperbolic paraboloid , as well as two diverging ultraparallel lines.
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
enter The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations. The differential geometry of surfaces captures many of the key ideas and techniques endemic to this field.
History of development Differential geometry arose and developed as a result of and in connection to the mathematical a. Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities.
In these contexts, the capital letters and the small letters represent distinct and unrelated entities. In mathematical finance, the Greeks are the variables denoted by Greek letters used to descri. In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges one in each direction. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century, in the work of Ramon Llull.
Topoisomerases are enzymes that participate in the overwinding or underwinding of DNA. The winding problem of DNA arises due to the intertwined nature of its double-helical structure. In order to prevent and correct these types of topological problems caused by the double helix, topoisomerases bind to DNA and cut the phosphate backbone of either one or both the DNA strands.
This intermediate break allows the DNA to be untangled or unwound, and, at the end of these processes, the DNA backbone is resealed again. Since the overall chemical composition and connectivity of the DNA do not change, the DNA substrate and product are chemical isomers, differing only in their global topology, resulting in the name for these enzymes. Topoisomerases are isomerase enzymes that act. Dynamical systems theory is an area of mathematics used to describe the behavior of the complex dynamical systems, usually by employing differential equations or difference equations.
When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler—Lagrange equations of a least action principle.
When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations. This theory deals with the long-term qualitative behavior of dynamical systems, and studie.
David van Dantzig September 23, — July 22, was a Dutch mathematician, well known for the construction in topology of the dyadic solenoid. He was a member of the Significs Group. At the University of Amsterdam he was succeeded by Jan Hemelrijk. Originally working on topics in differential geometry and topology, after World War I. A smart battery pack must be charged by a smart battery charger. Functions Safety circuit for 4 cell LiFePO4 batteries with balancer Monitor A BMS may monitor the state of the battery as represented by various items, such as: Voltage: total voltage, voltages of individual cells, minimum and maximum cell voltage or voltage of periodic taps Temperature: average temperature, coolant intake temperature, coolant output temperature, or temperatures of individual cells State of charge SOC or depth of discharge DOD , to indicate the cha.
In polymer chemistry, chain walking or chain running is a mechanism that operates during some alkene polymerization reactions. This reaction gives rise to branched and hyperbranched hydrocarbon polymers. This process is also characterized by accurate control of polymer architecture and topology. The potential applications of polymers formed by this reaction are diverse, from drug delivery to phase transfer agents, nanomaterials, and catalysis. A palladium catalyst that promotes cha. Topology chemistry topic In chemistry, topology provides a convenient way of describing and predicting the molecular structure within the constraints of three-dimensional 3-D space.
The analysis of these topologies, when combined with simple electrostatic theory and a few empirical observatio Folders related to Topology chemistry : Mathematical chemistry Revolvy Brain revolvybrain Cheminformatics Revolvy Brain revolvybrain Supramolecular chemistry Revolvy Brain revolvybrain. Computational topology topic Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.
It has exponential run-time in the number of tetrah Folders related to Computational topology: Computational fields of study Revolvy Brain revolvybrain Computational topology Revolvy Brain revolvybrain Computational complexity theory Revolvy Brain revolvybrain. Base topology topic In mathematics, a base or basis B for a topological space X with topology T is a collection of sets in X such that every open set in X can be written as a union of elements of B. Mathematical chemistry topic Mathematical chemistry is the area of research engaged in novel applications of mathematics to chemistry; it concerns itself principally with the mathematical modeling of chemical phenomena.
Outline of academic disciplines topic Collage of images representing different academic disciplines The following outline is provided as an overview of and topical guide to academic disciplines: An academic discipline or field of study is a branch of knowledge, taught and researched as part of higher education. General topology topic The Topologist's sine curve, a useful example in point-set topology. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected Folders related to General topology: General topology Revolvy Brain revolvybrain.
Molecular geometry topic Geometry of the water molecule Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. X-ray crystallography, neutron diffraction and electron diffraction can give m Folders related to Molecular geometry: Molecular geometry Revolvy Brain revolvybrain.
Theoretical chemistry topic J. It uses mathematical and physical methods to explain the structures and dynamics of chemical systems and to Folders related to Theoretical chemistry: Theoretical chemistry Revolvy Brain revolvybrain Chemistry Revolvy Brain revolvybrain Stuff Zijun Chen ZijunChen. Circuit topology topic The circuit topology of a linear polymer refers to arrangement of its intra-molecular contacts. For a chain with n contacts, the topology can be described by an n by n matrix in which each element illustrates the relation between a pair of contacts and may take one of the t Folders related to Circuit topology: Mathematical chemistry Revolvy Brain revolvybrain Molecular geometry Revolvy Brain revolvybrain Supramolecular chemistry Revolvy Brain revolvybrain.
Product topic Look up product in Wiktionary, the free dictionary. Areas of mathematics topic Mathematics encompasses a growing variety and depth of subjects over history, and comprehension requires a system to categorize and organize the many subjects into more general areas of mathematics. Uniformity topic Look up uniformity in Wiktionary, the free dictionary. Finite topological space topic In mathematics, a finite topological space is a topological space for which the underlying point set is finite. This leads to a simpler description of topolog Folders related to Finite topological space: Topological spaces Revolvy Brain revolvybrain Combinatorics Revolvy Brain revolvybrain.
Discrete mathematics topic Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithms. However, there is no exact definition of the term "discrete mathemat Folders related to Discrete mathematics: Discrete mathematics Revolvy Brain revolvybrain.
Supramolecular chemistry topic Supramolecular chemistry is the domain of chemistry concerning chemical systems composed of a discrete number of molecules. Mechanically interlocked molecular architectures topic Mechanically interlocked molecular architectures MIMAs are molecules that are connected as a consequence of their topology. Residual topology topic Residual topology  is a descriptive stereochemical term to classify a number of intertwined and interlocked molecules, which cannot be disentangled in an experiment without breaking of covalent bonds, while the strict rules of mathematical topology allow such a disentanglement.
Computational chemistry topic Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. Subbase topic In topology, a subbase or subbasis for a topological space X with topology T is a subcollection B of T that generates T, in the sense that T is the smallest topology containing B. TCA topic Look up tca in Wiktionary, the free dictionary. The research required to solve mathematical problem Folders related to Mathematics: Main topic articles Revolvy Brain revolvybrain Main topic classifications Revolvy Brain revolvybrain Mathematics Revolvy Brain revolvybrain.
List of scientific journals topic The following is a partial list of scientific journals. Order theory topic Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. The idea of being greater than or less than another number is one of the basic intuitions of number systems compare with numeral systems in general although one usually is also interested in the actual difference of two numbers, which is not gi Folders related to Order theory: Order theory Revolvy Brain revolvybrain.
Molecular Borromean rings topic Molecular Borromean rings are an example of a mechanically-interlocked molecular architecture in which three macrocycles are interlocked in such a way that breaking any macrocycle allows the others to disassociate. Trifluoroacetic acid TFA is added to catalyse the imine bond-forming reaction Folders related to Molecular Borromean rings: Supramolecular chemistry Revolvy Brain revolvybrain Molecular topology Revolvy Brain revolvybrain.
Catenane topic Crystal structure of a catenane with a cyclobis paraquat-p-phenylene macrocycle reported by Stoddart and coworkers. Catenanes have been synthesised in two different ways: st Folders related to Catenane: Cyclophanes Revolvy Brain revolvybrain Macrocycles Revolvy Brain revolvybrain Organic semiconductors Revolvy Brain revolvybrain.
Differential equation topic Visualization of heat transfer in a pump casing, created by solving the heat equation. Topological index topic In the fields of chemical graph theory, molecular topology, and mathematical chemistry, a topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound.
Hexadentate ligand topic A hexadentate ligand in coordination chemistry is a ligand that combines with a central metal atom with six bonds.
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Magnetic skyrmion topic Fig. Most descriptions include the notion of topology - a categorization of shapes and the way in which an object is laid out in space - using a continuous-field approximation as defined in micromagne Folders related to Magnetic skyrmion: Magnetism Revolvy Brain revolvybrain Quasiparticles Revolvy Brain revolvybrain. Coarctate reaction topic Examples of linear, pericyclic, and coarctate transition states In the classification of organic reactions by transition state topology, a coarctate reaction from L.
Thus, the topology of the transition state of a c Folders related to Coarctate reaction: Organic reactions Revolvy Brain revolvybrain.