Static Conditions

This research theme investigates equilibrium problems where the classical assumption of equilibrium condition (f(x,x) = 0) is missing, extending the framework to generalized equilibrium problems with bifunctions defined on distinct sets and involving compatibility conditions. It addresses existence of solutions under relaxed continuity, convexity, and pseudomonotonicity assumptions, expanding applicability to variational inequalities and quasi-equilibrium problems.

This theme focuses on the analysis of shadow systems obtained as limits of reaction-diffusion systems when some components diffuse infinitely fast. It investigates the spatial structure, monotonicity, stability, and asymptotic behavior of stationary and bounded solutions, elucidating conditions under which solutions become spatially homogeneous or monotone, and characterizing instability of non-monotone time-periodic solutions.

This area investigates quantitative measures of energy stored and dissipated in non-equilibrium steady states (NESS), the minimization principles potentially governing these states, and extensions of thermodynamic potentials like embedded energy analogous to Helmholtz free energy. It includes analyzing different energy input methods, formulation of steady-state energy balances, and implications for thermodynamic variables and stability in NESS.

This theme explores the algebraic and dynamical characterizations of non-equilibrium stationary states in quantum statistical mechanics, encompassing notions such as weighted detailed balance, local Kubo-Martin-Schwinger (KMS) conditions, and dynamical detailed balance within weak coupling limit frameworks. It addresses how these concepts generalize equilibrium conditions to capture statistical properties of stationary non-equilibrium systems.

This research area examines thermodynamic stability criteria using geometric and matrix analytical methods grounded in contact geometry and multivariable calculus. It rigorously analyzes classical stability principles like Le Chatelier-Brown, the role of second derivatives of thermodynamic potentials, and determines precise mathematical conditions for stability and instability boundaries in equilibrium thermodynamics.

This theme addresses the modeling of real-world imperfect or non-ideal boundary conditions of structures such as beams and shells, and their effects on natural frequencies and mode shapes. It includes developing analytical and numerical finite difference or finite element approaches to approximate these imperfect constraints, improving the prediction accuracy of dynamic responses over classical idealized boundary condition assumptions.

This theme clarifies the concept of initial conditions in fractional calculus, distinguishing initial condition problems from related extrapolation or prediction problems. It addresses inconsistencies in the use of one-sided Laplace transforms for initial conditions in fractional derivatives of Riemann-Liouville and Caputo types, presents rigorous formulations using bilateral Laplace transforms and fractional jump formulas, and develops solutions for fractional autoregressive-moving average systems consistent with state-space representations.

This research explores methods to enforce thermodynamic consistency in continuum mechanics constitutive models via compatibility with entropy balance and Clausius-Duhem inequality. It emphasizes explicit representation of entropy production as a constitutive function, develops representation formulas to solve for rates, and applies these tools to model nonlinear heat conduction, viscoelasticity, and wave propagation with thermodynamic rigor.

This theme investigates alternative methods for conducting comparative statics without total differentiation of first-order conditions, focusing on economic optimization problems. It shows that for certain objectives, derivative effects on functions of endogenous variables can be obtained directly via implicit differentiation of constraint functions and utility gradients, offering simplifications in economic analysis.