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Search results for: CALCULUS
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Unifying Aspects of Generalized Calculus
PublicationNon-Newtonian calculus naturally unifies various ideas that have occurred over the years in the field of generalized thermostatistics, or in the borderland between classical and quantum information theory. The formalism, being very general, is as simple as the calculus we know from undergraduate courses of mathematics. Its theoretical potential is huge, and yet it remains unknown or unappreciated.
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Some applications of fractional order calculus
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Simple Fractal Calculus from Fractal Arithmetic
PublicationNon-Newtonian calculus that starts with elementary non-Diophantine arithmetic operations of a Burgin type is applicable to all fractals whose cardinality is continuum. The resulting definitions of derivatives and integrals are simpler from what one finds in the more traditional literature of the subject, and they often work in the cases where the standard methods fail. As an illustration, we perform a Fourier transform of a real-valued...
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The evidential value of dental calculus in the identification process
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Advances in Calculus of Variations
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Fractional Differential Calculus
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Bell-Type Inequalities from the Perspective of Non-Newtonian Calculus
PublicationA class of quantum probabilities is reformulated in terms of non-Newtonian calculus and projective arithmetic. The model generalizes spin-1/2 singlet state probabilities discussed in Czachor (Acta Physica Polonica:139 70–83, 2021) to arbitrary spins s. For s → ∞ the formalism reduces to ordinary arithmetic and calculus. Accordingly, the limit “non-Newtonian to Newtonian” becomes analogous to the classical limit of a quantum theory
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Modelling heat transfer in heterogeneous media using fractional calculus
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Modeling Heat Transfer in Heterogeneous Media Using Fractional Calculus
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Fourier transforms on Cantor sets: A study in non-Diophantine arithmetic and calculus
PublicationFractals equipped with intrinsic arithmetic lead to a natural definition of differentiation, integration, and complex structure. Applying the formalism to the problem of a Fourier transform on fractals we show that the resulting transform has all the required basic properties. As an example we discuss a sawtooth signal on the ternary middle-third Cantor set. The formalism works also for fractals that are not self-similar.
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Neural Approximators for Variable-Order Fractional Calculus Operators (VO-FC)
PublicationThe paper presents research on the approximation of variable-order fractional operators by recurrent neural networks. The research focuses on two basic variable-order fractional operators, i.e., integrator and differentiator. The study includes variations of the order of each fractional operator. The recurrent neural network architecture based on GRU (Gated Recurrent Unit) cells functioned as a neural approximation for selected...
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Fractional Calculus and Applied Analysis
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Reliability Analysis of Data Storage Using Survival Signature and Logic Differential Calculus
PublicationThe Data storage system is an important part of any information system. All the necessary data that must be available for the successful operation of the information system are stored here. Therefore, it is advisable to think about the reliability of such a data storage system. As part of reliability engineering, it is possible to perform a reliability analysis of any system. Therefore, the data storage system can be analyzed as...
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Non-classical operational calculus applied to certain linear discrete time-system
PublicationW pracy zastosowano nieklasyczny rachunek operatorów do wyznaczania odpowiedzi pewnego dyskretnego układu dynamicznego. Pokazano metodę szczególnie przydatną do wyznaczania odpowiedzi pewnych dyskretnych niestacjonarnych układów dynamicznych, przy których zastosowanie przekształcenia Z sprawia duże trudności.
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Application of non-classical operational calculus to indicate hazards in numerical solutions of engineering problems
PublicationThe article addresses the application of non- classical operational calculus to approximative solutions of engineering problems. The engineering-sound examples show that a continuous–discrete problem transformation from differential unequivocal problem to a differential wildcard problem, triggering a change in solution quality. A number of approximative methods are capable to alter both quantitative and qualitative...
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Fractional Calculus Evaluation of Hyaluronic Acid Crosslinking in a Nanoscopic Part of Articular Cartilage Model System
PublicationThis work presents a study of the mechanism of physical crosslinking of hyaluronic acid in the presence of common phospholipids in synovial joint organ systems. Molecular dynamic simulations have been executed to understand the formation of hyaluronan networks at various phospholipid concentrations. The results of the simulations suggest that the mechanisms exhibit subdiffusion characteristics. Transportation quantities derive...
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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
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CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
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Calculus of gallbladder with other cholecystitis - Female, 53 - Tissue image [3110630005636871]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Calculus of gallbladder with other cholecystitis - Female, 75 - Tissue image [5060730012998381]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Calculus of gallbladder with other cholecystitis - Female, 53 - Tissue image [3110630005635611]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Calculus of gallbladder with other cholecystitis - Female, 75 - Tissue image [5060730012993001]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Calculus of gallbladder with other cholecystitis - Female, 75 - Tissue image [506073001299141]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Calculus of gallbladder with other cholecystitis - Female, 69 - Tissue image [6290730011495251]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: VS200 Olympus slide scanner (20x magnification) and saved to DICOM format.
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Variable Order Differential Models of Bone Remodelling * *This work was supported by FCT, through IDMEC, under LAETA, projects UID/EMS/50022/2013, BoneSys, joint Polish-Portuguese project Modelling and controlling cancer evolution using fractional calculus, PERSEIDS (PTDC/EMS-SIS/0642/2014) and IF/00653/2012
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Functional delay fractional equations
PublicationIn this paper, we discuss functional delay fractional equations. A Banach fixed point theorem is applied to obtain the existence (uniqueness) theorem. We also discuss such problems when a delay argument has a form α(t) = αt, 0 < α < 1, by Rusing the method of successive approximations. Some existence results are also formulated in this case. An example illustrates the main result.
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Diffusion equations with spatially dependent coefficients and fractal Cauer-type networks
PublicationIn this article, we formulate and solve the representation problem for diffusion equations: giving a discretization of the Laplace transform of a diffusion equation under a space discretization over a space scale determined by an increment h > 0, can we construct a continuous in h family of Cauer ladder networks whose constitutive equations match for all h > 0 the discretization. It is proved that for a finite differences discretization...
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About the Noether’s theorem for fractional Lagrangian systems and a generalization of the classical Jost method of proof
PublicationRecently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in [10] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [7]) using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular ([9],Theorem 32). However, the counterexample does not explain why and where the proof given in [10] does not work. In this paper, we make a detailed analysis...
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Systems of Nonlinear Fractional Differential Equations
PublicationUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives D(T)(q)x and D(T)(q)y. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.
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On the Fenchel–Moreau conjugate of G-function and the second derivative of the modular in anisotropic Orlicz spaces
PublicationIn this paper, we investigate the properties of the Fenchel–Moreau conjugate of G-function with respect to the coupling function c(x, A) = |A[x]2 |. We provide conditions that guarantee that the conjugate is also a G-function. We also show that if a G-function G is twice differentiable and its second derivative belongs to the Orlicz space generated by the Fenchel–Moreau conjugate of G then the modular generated by G is twice differentiable...
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Homoclinics for singular strong force Lagrangian systems in R^N
PublicationWe will be concerned with the existence of homoclinics for second order Hamiltonian systems in R^N (N>2) given by Hamiltonians of the form H(t,q,p)=Φ(p)+V(t,q), where Φ is a G-function in the sense of Trudinger, V is C^2-smooth, periodic in the time variable, has a single well of infinite depth at a point ξ and a unique strict global maximum 0 at the origin. Under a strong force type condition aroud the singular point ξ, we prove...
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The Palais–Smale condition for the Hamiltonian action on a mixed regularity space of loops in cotangent bundles and applications
PublicationWe show that the Hamiltonian action satisfies the Palais-Smale condition over a “mixed regular- ity” space of loops in cotangent bundles, namely the space of loops with regularity H^s, s ∈ (1/2, 1), in the baseand H^{1−s} in the fiber direction. As an application, we give a simplified proof of a theorem of Hofer-Viterbo on the existence of closed characteristic leaves for certain contact type hypersufaces in cotangent bundles.
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If Gravity is Geometry, is Dark Energy just Arithmetic?
PublicationArithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, R^4 and (−L/2,L/2)^4, are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms...
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Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics
PublicationNewtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β1⊕β2=tanh(tanh−1(β1)+tanh−1(β2)), although multiplication β1⊙β2=tanh(tanh−1(β1)⋅tanh−1(β2)),...
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Information Processing and Fechner’s Problem as a Choice of Arithmetic
PublicationFechner’s law and its modern generalizations can be regarded as manifestations of alternative forms of arithmetic, coexisting at stimulus and sensation levels. The world of sensations may be thus described by a generalization of the standard mathematical calculus.
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Fractional Order Dynamic Positioning Controller
PublicationImproving the performance of Dynamic Positioning System in such applications as station keeping, position mooring and slow speed references tracking requires improving the position and heading control precision. These goals can be achieved through the improvement of the ship control system. Fractional-order calculus is a very useful tool which extends classical, integer-order calculus and is used in contemporary modeling and control...
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Determination of the shape of the CFGFT cylindrical column based on laboratory tests
PublicationAnalyses were carried out on glass-fibre-reinforced polymer tube columns with reference to laboratory tests. The angles of the glass fibre beams were 20◦, 55◦ and 85◦. The study employed non-classical operational calculus. Various modulated harmonic signal shapes were considered for columns and tubes at buckling. The buckling loads were assessed and compared for different models.
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Crystallization of space: Space-time fractals from fractal arithmetic
PublicationFractals such as the Cantor set can be equipped with intrinsic arithmetic operations (addition, subtraction, multiplication, division) that map the fractal into itself. The arithmetics allows one to define calculus and algebra intrinsic to the fractal in question, and one can formulate classical and quantum physics within the fractal set. In particular, fractals in space-time can be generated by means of homogeneous spaces associated...
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Variational Method of Finding Streamlines in Ring Cascades for Creeping Flows
PublicationThis paper presents a new, analytical method of finding streamlinesfor creeping flows inside a ring cascade which is composed of an infinite number of infinitely thin blades. An analytical solution has been obtained through minimisation of a dissipation functional by means of variational calculus method. The necessary condition for optimum of a functional gives the Stokes equation if some additional assumptions are introduced....
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Arithmetic Loophole in Bell's Theorem: Overlooked Threat to Entangled-State Quantum Cryptography
PublicationBell’s theorem is supposed to exclude all local hidden-variable models of quantum correlations. However,an explicit counterexample shows that a new class of local realistic models, based on generalized arith-metic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical oftwo-electron singlet states. Observable probabilities are consistent with the usual arithmetic employedby macroscopic observers...
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Chemistry and Nanochemistry 2024
e-Learning CoursesWe start from General Chemistry, including: calculus in chemistry, bonds, balancing the chemical reactions, redox reactions, electrochemistry, pH, acids, bases, salts, oxides, organic molecules, and many informations about synthesis, analysis and applications in nanochemistry.
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Chemistry and Nanochemistry 2023
e-Learning CoursesWe start from General Chemistry, including: calculus in chemistry, bonds, balancing the chemical reactions, redox reactions, electrochemistry, pH, acids, bases, salts, oxides, organic molecules, and many informations about synthesis, analysis and applications in nanochemistry.
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Numerical Investigation of Nuclear Reactor Kinetic and Heat Transfer Fractional Model with Temperature Feedback
PublicationAbstract—In the paper, the numerical results concerning the kinetics and proposed heat exchange models in nuclear reactor based on fractional calculus are presented for typical inputs. Two fractional models are proposed and compared with the model based on ordinary derivative. The first fractional model is based on one of the generalized Cattaneo equations. The second one is based on replacing the ordinary to fractional order of...
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Analysis of Transformation Methods of Hydroacoustic and Optoelectronic Data Based on the Tombolo Measurement Campaign in Sopot
PublicationMeasurements in the coastal zone are carried out using various methods, including Global Navigation Satellite Systems (GNSS), hydroacoustic and optoelectronic methods. Therefore, it is necessary to develop coordinate transformation models that will enable the conversion of data from the land and marine parts to one coordinate system. The article presents selected issues related to the integration of geodetic and hydrographic data....
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Simulation of Signal Propagation Along Fractional-Order Transmission Lines
PublicationIn this paper, the simulation method of signal propagation along fractional-order (FO) transmission lines is presented. Initially, fractional calculus and the model of FO transmission line are introduced. Then, the algorithm allowing for simulation of the nonmonochromatic wave propagation along FO transmission lines is presented. It employs computations in the frequency domain, i.e., an analytical excitation is transformed to the...
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Applications of Tensor Analysis in Continuum Mechanics
PublicationA tensor field is a tensor-valued function of position in space. The use of tensor fields allows us to present physical laws in a clear, compact form. A byproduct is a set of simple and clear rules for the representation of vector differential operators such as gradient, divergence, and Laplacian in curvilinear coordinate systems. The tensorial nature of a quantity permits us to formulate transformation rules for its components...
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Chronic cholecystitis - Male, 64 - Tissue image [6260730021881081]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Chronic cholecystitis - Female, 55 - Tissue image [6210730043073411]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Chronic cholecystitis - Male, 52 - Tissue image [6210730043071791]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.
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Chronic cholecystitis - Male, 56 - Tissue image [6210730043077541]
Open Research DataThis is the histopathological image of GALLBLADDER tissue sample obtained in Medical University Gdańsk and deposited in ZMDL-GUMED. The sample image was taken using: Pannoramic 250 3DHistech slide scanner (20x magnification) and saved to DICOM format.