Mechanical devices for tracing curves offer a tangible approach to geometry, complementing symbolic abstraction with physical construction. Starting from the historical machine for the real exponential, we introduce a planar mechanism that solves f = f' in the complex domain, with a natural extension of the real case. To enhance accessibility and visualization, the behavior of the machine is illustrated using Dynamic Geometry simulations.

A fat Ψ-space is a Tychonoff space obtained by replacing the isolated points of a Ψ-space with more structured topological components, commonly referred to as ”building blocks”. This paper surveys the principal known constructions of such spaces and introduces a new example developed within this framework.

In 2013 Zhi-Wei Sun conjectured that p(n) is never a power of an integer when n > 1.We confirm this claim in many cases.We also observe that integral powers appear to repel the partition numbers.If k > 1 and k (n) is the distance between p(n) and the nearest kth power, then for every d 0 we conjecture that there are at most finitely many n for which k (n) d.More precisely, for every > 0, we conjecture thatIn k-power aspect with d fixed, we also conjecture that if k is sufficiently large, thenIn other words, 1 generally appears to be the closest kth power among the partition numbers.

Exact analytical expressions are derived for the long time components of the dimensionless velocities and shear stresses corresponding to the modified Stokes problems for incompressible upper-convected Maxwell fluids whose viscosity exponentially depends on pressure.The influence of magnetic field and of the gravitational acceleration is taken into account and some known results from the literature are recovered as limiting cases.Obtained solutions can be used as tests for numerical methods that are developed for more complex flow problems and to find the required time to reach the steady state.Graphical representations showed that the fluids with pressure dependent viscosity flow more quickly in comparison with ordinary fluids.

We prove that the solutions of the damped Klein-Gordon equation with a monotone perturbation cannot fulfill the finite extinction time property, even if the perturbation is a non-Lipschitz (or multivalued) function of the unknown u.This contrasts with the case of the nonlinear Schrdinger damped equation (recent results dealing with this same monotone expressions but with a purely imaginary coefficient), and with the case of nonlinear parabolic equations with strong absorption (for which the finite extinction time property is well-known since the middle of the seventies of the last century).

A nonlocal integro-multi-point boundary value problem associated to a Hilfer-Hadamard fractional integro-differential inclusion is studied. The existence of solutions is established in the case where the set-valued maps have non-convex values.

In this note we compare the entropy principle and the objectivity arguments in the methodologies of Dunn and Serrin [6] and in the more recent weakly nonlocal thermodynamic analysis of Korteweg-type fluids in [29]. It is concluded that the different ob jectivity approaches lead to the same constitutive functions, and that the difference in the thermodynamically compatible pressure tensors of perfect Korteweg fluids is due to different symmetry requirements.

Trace conjunction integrals are introduced and studied. They appear in trace conjunction inequalities which unify the Hardy inequality on a halfspace and the classical Gagliardo trace inequality. At the endpoint they satisfy a Bourgain-Brezis-Mironescu formula for smooth maps, which raises some new open problems.

In the framework of general nonlinear theory of six-parameter shells we derive pointwise necessary conditions for energy minimizers.We consider conservative problems and exploit the property that the second variation of the potential energy is non-negative if an equilibrium state represents an energy minimizer.Then, using variational calculus we derive the relevant Legendre-Hadamard condition in the theory of shells.Finally, we apply the necessary Legendre-Hadamard inequality to several isotropic strain energy functions proposed previously in the literature on shells.

We consider a variational inequality in a reflexive Banach space X, governed by a history-dependent operator.The existence of a unique solution to the inequality can be proven by using a fixed point argument.Based on this fixed point property, we provide necessary and sufficient conditions which guarantee the uniform convergence of a sequence of functions to the solution of the variational inequality.We then exploit this result in the study of both a penalty method and the well-posedness analysis of the problem.Moreover, we present an example which arises in Contact Mechanics.It concerns the study of a mathematical model which describes the contact of a viscoelastic membrane with a foundation.