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Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis.

Research & information: general --- Mathematics & science --- integro–differential systems --- Cauchy matrix --- exponential stability --- distributed control --- delay differential equation --- ordinary differential equation --- asymptotic equivalence --- approximation --- eigenvalue --- oscillation --- variable delay --- deviating argument --- non-monotone argument --- slowly varying function --- Crank–Nicolson scheme --- Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- variable coefficients --- stability analysis --- Lane-Emden-Klein-Gordon-Fock system with central symmetry --- Noether symmetries --- conservation laws --- differential equations --- non-monotone delays --- fractional calculus --- stochastic heat equation --- additive noise --- chebyshev polynomials of sixth kind --- error estimate --- fractional difference equations --- delay --- impulses --- existence --- fractional Jaulent-Miodek (JM) system --- fractional logistic function method --- symmetry analysis --- lie point symmetry analysis --- approximate conservation laws --- approximate nonlinear self-adjointness --- perturbed fractional differential equations --- integro–differential systems --- Cauchy matrix --- exponential stability --- distributed control --- delay differential equation --- ordinary differential equation --- asymptotic equivalence --- approximation --- eigenvalue --- oscillation --- variable delay --- deviating argument --- non-monotone argument --- slowly varying function --- Crank–Nicolson scheme --- Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- variable coefficients --- stability analysis --- Lane-Emden-Klein-Gordon-Fock system with central symmetry --- Noether symmetries --- conservation laws --- differential equations --- non-monotone delays --- fractional calculus --- stochastic heat equation --- additive noise --- chebyshev polynomials of sixth kind --- error estimate --- fractional difference equations --- delay --- impulses --- existence --- fractional Jaulent-Miodek (JM) system --- fractional logistic function method --- symmetry analysis --- lie point symmetry analysis --- approximate conservation laws --- approximate nonlinear self-adjointness --- perturbed fractional differential equations

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“Symmetry Breaking in Cells and Tissues” presents a collection of seventeen reviews, opinions and original research papers contributed by theoreticians, physicists and mathematicians, as well as experimental biologists, united by a common interest in biological pattern formation and morphogenesis. The contributors discuss diverse manifestations of symmetry breaking in biology and showcase recent developments in experimental and theoretical approaches to biological morphogenesis and pattern formation on multiple scales.

Research & information: general --- Biology, life sciences --- actin waves --- curved proteins --- dynamic instability --- podosomes --- diffusion --- cell polarity --- Cdc42 --- stress --- cellular memory --- phase separation --- prions --- apoptotic extrusion --- oncogenic extrusion --- contractility --- actomyosin --- bottom-up synthetic biology --- motor proteins --- pattern formation --- self-organization --- cell motility --- signal transduction --- actin dynamics --- intracellular waves --- polarization --- direction sensing --- symmetry-breaking --- biphasic responses --- reaction-diffusion --- membrane and cortical tension --- cell fusion --- cortexillin --- cytokinesis --- Dictyostelium --- myosin --- symmetry breaking --- cytoplasmic flow --- phase-space analysis --- nonlinear waves --- actin polymerization --- bifurcation theory --- mass conservation --- spatial localization --- activator–inhibitor models --- developmental transitions --- cell polarization --- mathematical model --- fission yeast --- reaction–diffusion model --- small GTPases --- Cdc42 oscillations --- pseudopod --- Ras activation --- cytoskeleton --- chemotaxis --- neutrophils --- natural variation --- modelling --- activator-substrate mechanism --- mass-conserved models --- intracellular polarization --- partial differential equations --- sensitivity analysis --- GTPase activating protein (GAP) --- fission yeast Schizosaccharomyces pombe --- CRY2-CIBN --- optogenetics --- clustering --- positive feedback --- network evolution --- Saccharomyces cerevisiae --- polarity --- modularity --- neutrality --- n/a

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Partial differential equations (PDEs) have been used in theoretical ecology research for more than eighty years. Nowadays, along with a variety of different mathematical techniques, they remain as an efficient, widely used modelling framework; as a matter of fact, the range of PDE applications has even become broader. This volume presents a collection of case studies where applications range from bacterial systems to population dynamics of human riots.

cross diffusion --- Turing patterns --- non-constant positive solution --- animal movement --- correlated random walk --- movement ecology --- population dynamics --- taxis --- telegrapher’s equation --- invasive species --- linear determinacy --- population growth --- mutation --- spreading speeds --- travelling waves --- optimal control --- partial differential equation --- invasive species in a river --- continuum models --- partial differential equations --- individual based models --- plant populations --- phenotypic plasticity --- vegetation pattern formation --- desertification --- homoclinic snaking --- front instabilities --- Evolutionary dynamics --- G-function --- Quorum Sensing --- Public Goods --- semi-linear parabolic system of equations --- generalist predator --- pattern formation --- Turing instability --- Turing-Hopf bifurcation --- bistability --- regime shift --- carrying capacity --- spatial heterogeneity --- Pearl-Verhulst logistic model --- reaction-diffusion model --- energy constraints --- total realized asymptotic population abundance --- chemostat model --- social dynamics --- wave of protests --- long transients --- ghost attractor --- prey–predator --- diffusion --- nonlocal interaction --- spatiotemporal pattern --- Allen–Cahn model --- Cahn–Hilliard model --- spatial patterns --- spatial fluctuation --- dynamic behaviors --- reaction-diffusion --- spatial ecology --- stage structure --- dispersal

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This Special Issue is devoted to some serious problems that the Fractional Calculus (FC) is currently confronted with and aims at providing some answers to the questions like “What are the fractional integrals and derivatives?”, “What are their decisive mathematical properties?”, “What fractional operators make sense in applications and why?’’, etc. In particular, the “new fractional derivatives and integrals” and the models with these fractional order operators are critically addressed. The Special Issue contains both the surveys and the research contributions. A part of the articles deals with foundations of FC that are considered from the viewpoints of the pure and applied mathematics, and the system theory. Another part of the Special issue addresses the applications of the FC operators and the fractional differential equations. Several articles devoted to the numerical treatment of the FC operators and the fractional differential equations complete the Special Issue.

fractional derivatives --- fractional integrals --- fractional calculus --- fractional anti-derivatives --- fractional operators --- integral transforms --- convergent series --- fractional integral --- fractional derivative --- numerical approximation --- translation operator --- distributed lag --- time delay --- scaling --- dilation --- memory --- depreciation --- probability distribution --- fractional models --- fractional differentiation --- distributed time delay systems --- Volterra equation --- adsorption --- fractional differential equations --- numerical methods --- smoothness assumptions --- persistent memory --- initial values --- existence --- uniqueness --- Crank–Nicolson scheme --- weighted Shifted Grünwald–Letnikov approximation --- space fractional convection-diffusion model --- stability analysis --- convergence order --- Caputo–Fabrizio operator --- Atangana–Baleanu operator --- fractional falculus --- general fractional derivative --- general fractional integral --- Sonine condition --- fractional relaxation equation --- fractional diffusion equation --- Cauchy problem --- initial-boundary-value problem --- inverse problem --- fractional calculus operators --- special functions --- generalized hypergeometric functions --- integral transforms of special functions

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“Symmetry Breaking in Cells and Tissues” presents a collection of seventeen reviews, opinions and original research papers contributed by theoreticians, physicists and mathematicians, as well as experimental biologists, united by a common interest in biological pattern formation and morphogenesis. The contributors discuss diverse manifestations of symmetry breaking in biology and showcase recent developments in experimental and theoretical approaches to biological morphogenesis and pattern formation on multiple scales.

Research & information: general --- Biology, life sciences --- actin waves --- curved proteins --- dynamic instability --- podosomes --- diffusion --- cell polarity --- Cdc42 --- stress --- cellular memory --- phase separation --- prions --- apoptotic extrusion --- oncogenic extrusion --- contractility --- actomyosin --- bottom-up synthetic biology --- motor proteins --- pattern formation --- self-organization --- cell motility --- signal transduction --- actin dynamics --- intracellular waves --- polarization --- direction sensing --- symmetry-breaking --- biphasic responses --- reaction-diffusion --- membrane and cortical tension --- cell fusion --- cortexillin --- cytokinesis --- Dictyostelium --- myosin --- symmetry breaking --- cytoplasmic flow --- phase-space analysis --- nonlinear waves --- actin polymerization --- bifurcation theory --- mass conservation --- spatial localization --- activator–inhibitor models --- developmental transitions --- cell polarization --- mathematical model --- fission yeast --- reaction–diffusion model --- small GTPases --- Cdc42 oscillations --- pseudopod --- Ras activation --- cytoskeleton --- chemotaxis --- neutrophils --- natural variation --- modelling --- activator-substrate mechanism --- mass-conserved models --- intracellular polarization --- partial differential equations --- sensitivity analysis --- GTPase activating protein (GAP) --- fission yeast Schizosaccharomyces pombe --- CRY2-CIBN --- optogenetics --- clustering --- positive feedback --- network evolution --- Saccharomyces cerevisiae --- polarity --- modularity --- neutrality --- actin waves --- curved proteins --- dynamic instability --- podosomes --- diffusion --- cell polarity --- Cdc42 --- stress --- cellular memory --- phase separation --- prions --- apoptotic extrusion --- oncogenic extrusion --- contractility --- actomyosin --- bottom-up synthetic biology --- motor proteins --- pattern formation --- self-organization --- cell motility --- signal transduction --- actin dynamics --- intracellular waves --- polarization --- direction sensing --- symmetry-breaking --- biphasic responses --- reaction-diffusion --- membrane and cortical tension --- cell fusion --- cortexillin --- cytokinesis --- Dictyostelium --- myosin --- symmetry breaking --- cytoplasmic flow --- phase-space analysis --- nonlinear waves --- actin polymerization --- bifurcation theory --- mass conservation --- spatial localization --- activator–inhibitor models --- developmental transitions --- cell polarization --- mathematical model --- fission yeast --- reaction–diffusion model --- small GTPases --- Cdc42 oscillations --- pseudopod --- Ras activation --- cytoskeleton --- chemotaxis --- neutrophils --- natural variation --- modelling --- activator-substrate mechanism --- mass-conserved models --- intracellular polarization --- partial differential equations --- sensitivity analysis --- GTPase activating protein (GAP) --- fission yeast Schizosaccharomyces pombe --- CRY2-CIBN --- optogenetics --- clustering --- positive feedback --- network evolution --- Saccharomyces cerevisiae --- polarity --- modularity --- neutrality