Metin Demiralp

Feeling Blue or Turquoise? Emotional Differentiation in Major Depressive Disorder

Some individuals have very specific and differentiated emotional experiences, such as anger, shame, excitement, and happiness, whereas others have more general affective experiences of pleasure or discomfort that are not as highly differentiated. Considering that individuals with major depressive disorder (MDD) have cognitive deficits for negative information, we predicted that people with MDD would have less differentiated negative emotional experiences than would healthy people. To test this hypothesis, we assessed participants’ emotional experiences using a 7-day experience-sampling protocol. Depression was assessed using structured clinical interviews and the Beck Depression Inventory-II. As predicted, individuals with MDD had less differentiated emotional experiences than did healthy participants, but only for negative emotions. These differences were above and beyond the effects of emotional intensity and variability.

Chemical kinetic functional sensitivity analysis: Derived sensitivities and general applications

This paper considers possible applications of sensitivity analysis to kinetics problems defined in the space–time domain. At the outset it is assumed that elementary sensitivity functional densities δui/δαj are available, where ui is the ith chemical species concentration and αj is the jth system parameter function. Emphasis is placed on the use of functional calculus to manipulate the elementary sensitivities for addressing a variety of physical questions. In this fashion, a family of derived sensitivities are generated from the elementary set. An extensive list of sensitivity applications is presented along with a discussion as an aid to future work.

High-dimensional model representations generated from low order terms—lp-RS-HDMR

High‐dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. RS‐HDMR is a particular form of HDMR based on random sampling (RS) of the input variables. The component functions in an HDMR expansion are optimal choices tailored to the n‐variate function f(x) being represented over the desired domain of the n‐dimensional vector x. The high‐order terms (usually larger than second order, or equivalently beyond cooperativity between pairs of variables) in the expansion are often negligible. When it is necessary to go beyond the first and the second order RS‐HDMR, this article introduces a modified low‐order term product (lp)‐RS‐HDMR method to approximately represent the high‐order RS‐HDMR component functions as products of low‐order functions. Using this method the high‐order truncated RS‐HDMR expansions may be constructed without directly computing the original high‐order terms. The mathematical foundations of lp‐RS‐HDMR are presented along with an illustration of its utility in an atmospheric chemical kinetics model.

Chemical kinetic functional sensitivity analysis: Elementary sensitivities

Sensitivity analysis is considered for kinetics problems defined in the space–time domain. This extends an earlier temporal Green’s function method to handle calculations of elementary functional sensitivities dui/daj where ui is the ith species concentration and aj is the jth system parameter. The system parameters include rate constants, diffusion coefficients, initial conditions, boundary conditions, or any other well‐defined variables in the kinetic equations. These parameters are generally considered to be functions of position and/or time. Derivation of the governing equations for the sensitivities and the Green’s funciton are presented. The physical interpretation of the Green’s function and sensitivities is given along with a discussion of the relation of this work to earlier research.

A new approach for data partitioning through high dimensional model representation

A multivariate function f(x 1, …, x N ) can be evaluated via interpolation if its values are given at a finite number nodes of a hyperprismatic grid in the space of independent variables x 1, x 2, …, x N . Interpolation is a way to characterize an infinite data structure (function) by a finite number of data approximately. Hence it leaves an infinite arbitrariness unless a mathematical structure with finite number of flexibilities is imposed for the unknown function. Imposed structure has finite dimensionality. When the dimensionality increases unboundedly, the complexities grow rapidly in the standard methods. The main purpose here is to partition the given multivariate data into a set of low-variate data by using high dimensional model representation (HDMR) and then, to interpolate each individual data in the set via Lagrange interpolation formula. As a result, computational complexity of the given problem and needed CPU time to obtain the results through a series of programs in computers decrease.

A probabilistic evolution approach trilogy, part 1: quantum expectation value evolutions, block triangularity and conicality, truncation approximants and their convergence

This is the first one of three companion papers focusing on the “probabilistic evolution approach (PEA)” which has been developed for the solution of the explicit ODE involving problems under certain consistent impositions. The main purpose here is the determination of the expectation value of a given operator in quantum mechanics by solving only ODEs, not directly using the wave function. To this end we first define a basis operator set over the Kronecker powers of an appropriately defined “system operator vector”. We assume that the target operator’s commutator with the system’s Hamiltonian can be expressed in terms of the above-mentioned basis operators. This assumption leads us to an infinite set of linear homogeneous ODEs over the expectation values of the basis operators. Its coefficient matrix is in block Hessenberg form when the target operator has no singularity, and beyond that, it may become block triangular when certain conditions over the system’s potential function are satisfied. The initial conditions are the basic determining agents giving the probabilistic nature to the solutions of the obtained infinite set of ODEs. They may or may not have fluctuations depending on the nature of the probability density. All these issues are investigated in a phenomenological and constructive theoretical manner in this paper. The remaining two papers are devoted to further details of PEA in quantum mechanics, and, the application of PEA to systems defined by Liouville equation.

A probabilistic foundation for dynamical systems: theoretical background and mathematical formulation

In this paper we describe a probabilistic framework for describing dynamical systems. The approach is inspired by quantum dynamical expectation dynamics. Specifically, an abstract evolution operator corresponding to the Hamiltonian in quantum dynamics is constructed. The evolution of this operator defining PDE’s solution is isomorphic to the functional structure of the wave function as long as its initial form permits. This operator enables us to use one of the most important probabilistic concepts, namely expectations. The expectation dynamics are governed by equations which are constructed via commutator algebra. Based on inspiration from quantum dynamics, we have used both the independent variables and the symmetric forms of their derivatives. For construction of the expectation dynamics, the algebraic independent variable operators which multiply their operands by the corresponding independent variable suffice. In our descriptions, we remain at the conceptual level in a self-consistent manner. The phenomenological implications and the tremendous potential of this approach for scientific discovery and advancement is described in the companion to this paper.

Variational time‐dependent perturbation scheme based on the hydrodynamic analogy to Schrödinger's equation

A variational principle correct through the second order is presented for time‐dependent perturbations about a bound state. The formulation is based on the hydrodynamic analogy to quantum mechanics and is obtained by linearization of a general principle within the spirit of the acoustic approximation in classical fluid mechanics. The method requires the knowledge of the wavefunction for the unperturbed state only, unlike in the standard procedure of expansion in terms of a complete set of unperturbed wavefunctions. In dealing with the changes in the amplitude and phase of the wavefunction which are nonoscillatory, the method is expected to offer advantages over the usual variational perturbation methods which utilize the oscillatory wavefunctions directly. The method is applied to the calculation of the dynamic polarizability of the hydrogen atom and reasonable accuracy is obtained for an overly simple choice of basis functions.

A probabilistic foundation for dynamical systems: phenomenological reasoning and principal characteristics of probabilistic evolution

This paper is the second in a series of two. The first paper has been devoted to the detailed explanation of the mathematical formulation of the underlying theoretical framework. Specifically, the first paper shows that it is possible to construct an infinite linear ODE set, which describes a probabilistic evolution. The evolution is probabilistic because the unknowns are expectations, with appropriate initial conditions. These equations, which we name, Probabilistic Evolution Equations (PEE) are linear at the level of ODEs and initial conditions. In this paper, we first focus on the phenomenological reasoning that lead us to the derivation of PEE. Second, the aspects of the PEE construction is revisited with a focus on the spectral nature of the probabilistic evolution. Finally, we postulate fruitful avenues of research in the fields of dynamical causal modeling in human neuroimaging and effective connectivity analysis. We believe that this final section is a prime example of how the rigorous methods developed in the context of mathematical chemistry can be influential in other fields and disciplines.

A probabilistic evolution approach trilogy, part 3: temporal variation of state variable expectation values from Liouville equation perspective

This is the third and therefore the final part of a trilogy on probabilistic evolution approach. The work presented here focuses on the probabilistic evolution determination for the state variables of a many particle system from classical mechanical point of view. Probabilistic evolution involves the expected value evolutions for all natural number Kronecker powers of the state variables, positions and momenta. We use the phase space distribution of the Liouville equation perspective to construct the expected values of the state variables’ Kronecker powers to define unknown temporal functions. The infinite number homogeneous linear ODEs with an infinite constant coefficient matrix are constructed by following the same steps as in the previous two works on quantum mechanics. The only difference is in the definitions of the expected values here. We also focus on a system of many harmonic oscillators to illustrate the block triangularity.