The article investigates a boundary value problem for a hyperbolic equation with the Riemann–Liouville fractional derivative, which is periodic in one variable. Such equations are widely used in modeling complex physical processes with memory effects, including viscoelasticity, anomalous diffusion, and thermoviscoelasticity phenomena, where classical integer-order models fail to adequately describe the hereditary properties of materials and transport processes. To solve this problem, an iterative algorithm is proposed based on domain decomposition and the reduction of the original problem to a system of integro-differential equations. A theorem on the existence and uniqueness of the solution is proved, and an estimate of the convergence rate of the method is obtained using matrix analysis and a strengthened Gronwall–Bellman inequality. It is established that the choice of the decomposition step plays a key role in ensuring the stability of the algorithm. The conducted analysis extends the class of problems for which efficient computational algorithms can be constructed and may serve as a foundation for studying more complex nonlinear cases and problems in irregular domains.
Many studies find reflective thinking predicts less belief in God or less religiosity — so-called analytic atheism. However, the most widely used tests of reflection confound reflection with ancillary abilities such as numeracy, some studies do not detect analytic atheism in every country, experimentally encouraging reflection makes some non-believers more open to believing in God, and one of the most common sources of online research participants seems to produce lower data quality. So analytic atheism may be less than universal or partially explained by confounding factors. To test this, we developed better measures, controlled for more confounds, and employed more recruitment methods. All four studies detected signs of analytic atheism above and beyond confounds (N > 70,000 people from five of six continental regions). We also discovered analytic apostasy: the better a person performed on reflection tests, the greater their odds of losing their religion since childhood — even when controlling for confounds. Analytic apostasy even seemed to explain analytic atheism: apostates were more reflective than others and analytic atheism was undetected after excluding apostates. Religious conversion was rare and unrelated to reflection, suggesting reflection’s relationships to conversion and deconversion are asymmetric. Detected relationships were usually small, indicating reflective thinking is a reliable albeit marginal predictor of apostasy.
Rosemary Zielinski, Patrick McGlynn, Cédric Simenel
Abstract Though theoretical treatments of quantum tunnelling within single-particle quantum mechanics are well-established, at present, there is no quantum field-theoretic description (QFT) of tunnelling. Due to the single-particle nature of quantum mechanics, many-particle effects arising from quantum field theory are not accounted for. Such many-particle effects, including pair-production, have proved to be essential in resolving the Klein-paradox. This work seeks to determine how quantum corrections affect the tunnelling probability through an external field. We investigate a massive neutral scalar field, which interacts with an external field in accordance with relativistic quantum mechanics. To consider QFT corrections, we include another massive quantised neutral scalar field coupling to the original via a cubic interaction. This study formulates an all-order recursive expression for the loop-corrected scalar propagator, which contains only the class of vertex-corrected Feynman diagrams. This equation applies for general external potentials. Though there is no closed-form analytic solution, we also demonstrate how to approximate the QFT corrections if a perturbative coupling to the quantised field is assumed.
Astrophysics, Nuclear and particle physics. Atomic energy. Radioactivity
The study of boundary value problems for elliptic equations is of both theoretical and applied interest. A thorough study of model physical and spectral problems requires an explicit and effective representation of the problem solution. Integral representations of solutions of problems of differential equations are one of the main tools of mathematical physics. Currently, the integral representation of the Green function of classical problems for the Laplace equation for an arbitrary domain is obtained only in a two-dimensional domain by the Riemann conformal mapping method. Starting from the three-dimensional case, these classical problems are solved only for spherical sectors and for the regions lying between the faces of the hyperplane. The problem of constructing integral representations of general boundary value problems and studying their spectral problems remains relevant. In this work, using the boundary condition of the Newtonian (volume) potential and the spectral property of the potential of a simple layer, the Green function of the Dirichlet problem for the Laplace equation was constructed.
Due to the fact that the Navier-Stokes equations are involved in the formulation of a large number of interesting problems that are important from an applied point of view, these equations have been the object of attention of mechanics, mathematicians and other scientists for several decades in a row. But despite this, many problems for the Navier-Stokes equation remain unexplored to this day. In this work, we are exploring the solvability of a boundary value problem for a two-dimensional Navier-Stokes system in a non-cylindrical degenerating domain, namely, in a cone with its vertex at the origin. Previously, we studied cases of the linearized Navier-Stokes system or non-degenerating cylindrical domains, so this work is a logical continuation of our previous research in this direction. To the above-mentioned degenerate domain we associate a family of non-degenerate truncated cones, which, in turn, are formed by a oneto-one transformation into cylindrical domains, where for the problem under consideration we established uniform a priori estimates with respect to changes in the index of the domains. Further, using a priori estimates and the Faedo-Galerkin method, we established the existence, uniqueness of solution in Sobolev classes, and its regularity as the smoothness of the given functions increases.
Abstract Collagen fibers, one of the key load-bearing components of the extracellular matrix, contribute significantly to tissue integrity through their mechanical properties of strain-dependent stiffening. This study investigated the effects of bacterial collagenase on the mechanical behavior of individual bovine lung collagen fibers in the presence or absence of mechanical forces, with a focus on potential implications for emphysema, a condition associated with collagen degradation and alveolar wall rupture. Tensile tests were conducted on individual collagen fibers isolated from bovine lung tissue. The rate of degradation was characterized by the change in fiber Young’s modulus during 60 min of digestion under various mechanical conditions mimicking the mechanical stresses on the fibers during breathing. Compared to digestion without mechanical forces, a significantly larger drop of fiber modulus was observed in the presence of static or intermittent mechanical forces. Fiber yield stress was also reduced after digestion indicating compromised fiber failure. By incorporating fibril waviness obtained by scanning electron microscopic images, an analytic model allowed estimation of fibril modulus. A computational model that incorporated waviness and the results of tensile tests was also developed to simulate and interpret the data. The simulation results provided insights into the mechanical consequences of bacterial collagenase and mechanical forces on collagen fibers, revealing both fibril softening and rupture during digestion. These findings shed light on the microscale changes in collagen fiber structure and mechanics under enzymatic digestion and breathing-like mechanical stresses with implications for diseases that are impacted by collagen degradation such as emphysema.
The purpose of this paper is to study resolutions of locally analytic representations of a $p$-adic reductive group $G$. Given a locally analytic representation $V$ of $G$, we modify the Schneider-Stuhler complex (originally defined for smooth representations) so as to give an `analytic' variant ${\mathcal S}^A_\bullet(V)$. The representations in this complex are built out of spaces of analytic vectors $A_σ(V)$ for compact open subgroups $U_σ$, indexed by facets $σ$ of the Bruhat-Tits building of $G$. These analytic representations (of compact open subgroups of $G$) are then resolved using the Chevalley-Eilenberg complex from the theory of Lie algebras. This gives rise to a resolution ${\mathcal S}^{\rm CE}_{q,\bullet}(V) \rightarrow {\mathcal S}^A_q(V)$ for each representation ${\mathcal S}^A_q(V)$ in the analytic Schneider-Stuhler complex. In a last step we show that the family of representations ${\mathcal S}^{\rm CE}_{q,j}(V)$ can be given the structure of a Wall complex. The associated total complex ${\mathcal S}^{\rm CE}_\bullet(V)$ has then the same homology as that of ${\mathcal S}^A_\bullet(V)$. If the latter is a resolution of $V$, then one can use ${\mathcal S}^{\rm CE}_\bullet(V)$ to find a complex which computes the extension group $\underline{Ext}^n_G(V,W)$, provided $V$ and $W$ satisfy certain conditions which are satisfied when both are admissible locally analytic representations.
This paper investigates the notion of Krylov complexity, a measure of operator growth, within the framework of 1-matrix quantum mechanics (1-MQM). Krylov complexity quantifies how an operator evolves over time by expanding it in a series of nested commutators with the Hamiltonian. We analyze the Lanczos coefficients derived from the correlation function, revealing their linear growth even in this integrable system. This growth suggests a link to chaotic behavior, typically unexpected in integrable systems. Our findings in both ground and thermal states of 1-MQM provide new insights into the nature of complexity in quantum mechanical models and lay the groundwork for further studies in more complex holographic theories.
Mathematical equivalence between statistical mechanics and machine learning theory has been known since the 20th century, and research based on this equivalence has provided novel methodologies in both theoretical physics and statistical learning theory. It is well known that algebraic approaches in statistical mechanics such as operator algebra enable us to analyze phase transition phenomena mathematically. In this paper, we review and prospect algebraic research in machine learning theory for theoretical physicists who are interested in artificial intelligence. If a learning machine has a hierarchical structure or latent variables, then the random Hamiltonian cannot be expressed by any quadratic perturbation because it has singularities. To study an equilibrium state defined by such a singular random Hamiltonian, algebraic approaches are necessary to derive the asymptotic form of the free energy and the generalization error. We also introduce the most recent advance: the theoretical foundation for the alignment of artificial intelligence is now being constructed based on algebraic learning theory. This paper is devoted to the memory of Professor Huzihiro Araki who is a pioneering founder of algebraic research in both statistical mechanics and quantum field theory.
A variational scheme of evolution (minimizing movements) is applied to a sequence of discrete functionals converging, as the mesh size tends to zero, to the prototypical second-order functional with free-discontinuities. At fixed mesh size, a discrete evolution can be defined, depending on a (small) time parameter. We study the limit problem when both the mesh size and the time step tend to zero. The method provides a function which matches the expected evolution of the free-discontinuity limit functional. From a mechanical point of view, the model can be interpreted as the evolution from a non-equilibrium state, of a rod with possible crease discontinuities and fracture.
In the paper, the boundary value problem for the loaded heat equation is solved, and the loaded term is represented as the Riemann-Liouville derivative with respect to the time variable. The domain of the unknown function is the cone. The order of the derivative in the loaded term is less than 1, and the load moves along the lateral surface of the cone, that is in the domain of the desired function. The boundary value problem is studied in the case of the isotropy property in an angular coordinate (case of axial symmetry). The problem is reduced to the Volterra integral equation, which is solved by the method of the Laplace integral transformation. It is also shown by direct verification that the resulting function satisfies the boundary value problem.
The paper proposes a method for solving the problem of optimal performance for linear systems of ordinary differential equations in the presence of phase and integral restrictions, when the initial and final states of the system are elements of given convex closed sets, taking into account the control value restriction. The presented work refers to the mathematical theory of optimal processes from L.S. Pontryagin and his students and the theory of controllability of dynamic systems from R.E. Kalman. We study the problem of optimal speed for linear systems with boundary conditions from given sets close to the presence of phase and integral constraints, as well as constraints on the control value. A theory of the boundary value problem has been created and a method for solving it based on the study of solvability and the construction of a general solution to the Fredholm integral equation of the first kind has been developed. The main results are the distribution of all controls’ sets, each subject of which transfers the trajectory of the system from any initial state to any final state; reducing the initial boundary point to a special initial optimal control problem; constructing a system of algorithms for the gamma-algorithm study on the derivation of problems and rational execution with restrictions on the solution of the optimal speed’ problem with restrictions.
The elastic response of mechanical, chemical, and biological systems is often modeled using a discrete arrangement of Hookean springs, either representing finite material elements or even the molecular bonds of a system. However, to date, there is no direct derivation of the relation between a general discrete spring network\blu{, with arbitrary geometry,} and it's corresponding elastic continuum. Furthermore, understanding the network's mechanical response requires simulations that may be expensive computationally. Here we report a method to derive the exact elastic continuum model of any discrete network of springs, requiring network geometry and topology only. We identify and calculate the so-called "non-affine" displacements. Explicit comparison of our calculations to simulations of different crystalline and disordered configurations, shows we successfully capture the mechanics even of auxetic materials. Our method is valid for residually stressed systems with non-trivial geometries, and is an essential step in generalizing active stresses on such discrete systems. It is easily generalizable to other discrete models, and opens the possibility of a rational design of elastic systems.
In the present paper the initial value problem for the second order ordinary differential equation with damping term and involution is investigated. We obtain equivalent initial value problem for the fourth order ordinary differential equations to the initial value problem for second order linear differential equations with damping term and involution. Theorem on stability estimates for the solution of the initial value problem for the second order ordinary linear differential equation with damping term and involution is proved. Theorem on existence and uniqueness of bounded solution of initial value problem for second order ordinary nonlineardifferential equation with damping term and involution is established.
N.T. Orumbayeva, T.D. Tokmagambetova, Zh.N. Nurgalieva
In this paper, by means of a change of variables, a nonlinear semi-periodic boundary value problem for the Goursat equation is reduced to a linear gravity problem for hyperbolic equations. Reintroducing a new function, the obtained problem is reduced to a family of boundary value problems for ordinary differential equations and functional relations. When solving a family of boundary value problems for ordinary differential equations, the parameterization method is used. The application of this approach made it possible to establish the coefficients of the unique solvability of the semi-periodic problem for the Goursat equation and to propose constructive algorithms for finding an approximate solution.
This article deals with the factorization and solution of nonlocal boundary value problems in a Banach space of the abstract form B1u = Au − SΦ(u) − GΨ(A0u) = f, u ∈ D(B1),where A, A0 are linear abstract operators, S, G are vectors of functions, Φ, Ψ are vectors of linear bounded functionals, and u, f are functions. It is shown that the operator B1 under certain conditions can be factorized into a product of two simpler lower order operators as B1 = BB0. Then the solvability and the unique solution of the equation B1u = f easily follow from the solvability conditions and the unique solutions of the equations Bv = f and B0u = v. The universal technique proposed here is essentially different from other factorization methods in the respect that it involves decomposition of both the equation and boundary conditions and delivers the solution in closed form. The method is implemented to solve ordinary and partial Fredholm integro-differential equations.
In this paper γµ -open sets and γµ -closed sets in a GTS (X, µ) have been studied, where γµ is an operation from µ to P(X). In general, collection of γµ -open sets is smaller than the collection of µ-open sets. The condition under which both are same are also established here. Some properties of such sets have been discussed. Some closure like operators are also defined and their properties are discussed. The relation between similar types of closure operators on the GTS (X, µ) has been established. The condition under which the newly defined closure like operator is a Kuratowski closure operator is given. We have also defined a generalized type of closed sets termed as γµ -generalized closed set with the help of this newly defined closure operator and discussed some basic properties of such sets. As an application, we have introduced some weak separation axioms and discussed some of their properties. Finally, we have shown some preservation theorems of such generalized concepts.
Manu Mannattil, J. M. Schwarz, Christian D. Santangelo
A bar-joint mechanism is a deformable assembly of freely rotating joints connected by stiff bars. Here we develop a formalism to study the equilibration of common bar-joint mechanisms with a thermal bath. When the constraints in a mechanism cease to be linearly independent, singularities can appear in its shape space, which is the part of its configuration space after discarding rigid motions. We show that the free-energy landscape of a mechanism at low temperatures is dominated by the neighborhoods of points that correspond to these singularities. We consider two example mechanisms with shape-space singularities and find that they are more likely to be found in configurations near the singularities than others. These findings are expected to help improve the design of nanomechanisms for various applications.
In this paper we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms which show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator which is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclisity of composition operators.
The displacement field in a circular disc made of a transversely isotropic material is explored in a parametric manner. The disc is assumed to be loaded by a parabolic distribution of compressive radial stresses along two finite arcs of its periphery in the absence of any tangential (frictional) stresses. Advantage is here taken of a recently introduced closed-form analytic solution for the displacement field developed in an orthotropic disc under diametral compression which was achieved adopting the complex potentials technique for rectilinear anisotropic materials as it was formulated in the pioneering work of S.G. Lekhnitskii. The analytic nature of this solution permits thorough, indepth exploration of the influence of some crucial parameters on the qualitative and quantitative characteristics of the deformation of transversely isotropic circular discs compressed between the jaws of the devise suggested by the International Society for Rock Mechanics for the standardized implementation of the Brazilian-disc test. The parameters considered include the anisotropy ratio (i.e., the ratio of the two elastic moduli characterizing the disc material), the angle between the loading axis and the planes of transverse isotropy and the length of the loaded arcs. Strongly non-linear relationships between these parameters and the components of the displacement field are revealed.
Mechanical engineering and machinery, Structural engineering (General)