Variable (mathematics) | EPFL Graph Search

In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set. Algebraic computations with variables as if they were explicit numbers solve a range of problems in a single computation. For example, the quadratic formula solves any quadratic equation by substituting the numeric values of the coefficients of that equation for the variables that represent them in the quadratic formula. In mathematical logic, a variable is either a symbol representing an unspecified term of the theory (a meta-variable), or a basic object of the theory that is manipulated without referring to its possible intuitive interpretation. History In ancient works such as Euclid's Elements, single letters refer to geometric points and shapes. In the 7th century, Brahmagupta used different colours to represent the unknowns in al

Contribution à l'optimisation des entraînements électriques

Cédric Paroz

The choice of the drive composition is an obligatory stage in designing a product including electric drives. The large variety of the elements composing the electric drive (electronics, driving and kinematic chain) leads to a quasi-unlimited number of possible alternatives. Models and programs make it possible for the engineer to calculate independently the elements of a given composition, but the choice of the composition often remains subjective. We make the assumption that it is possible to automate the choice of an optimal composition. To check this assumption, the criteria, defining this optimum, are expressed and then are expressed in the form of an equation as a function of variable parameters. Finally it is checked that a set of parameters corresponding to the required optimum can be found in an automatic way. To successfully go through of these steps required the understanding of very diverse subjects, in particular: the use of an electric model, the symbolic system evaluation of the smallest root of a polynomial, the important influence of the trajectory and determination of dynainic rigidity. One of the key element resides in the formulation of the characteristics of al1 the elements in terms of admissible torque.

EPFL2003

A Dichotomy for Real Weighted Holant Problems

Sangxia Huang

Holant is a framework of counting characterized by local constraints. It is closely related to other well-studied frameworks such as the counting constraint satisfaction problem (#CSP) and graph homomorphism. An effective dichotomy for such frameworks can immediately settle the complexity of all combinatorial problems expressible in that framework. Both #CSP and graph homomorphism can be viewed as subfamilies of Holant with the additional assumption that the equality constraints are always available. Other subfamilies of Holant such as Holant() and Holant (c) problems, in which we assume some specific sets of constraints to be freely available, were also studied. The Holant framework becomes more expressive and contains more interesting tractable cases with less or no freely available constraint functions, while, on the other hand, it also becomes more challenging to obtain a complete characterization of its time complexity. Recently, a complexity dichotomy for a variety of subfamilies of Holant such as #CSP, graph homomorphism, Holant(), and Holant (c) for Boolean domain was proved. In this paper, we prove a dichotomy for the general Holant framework where all the constraints are real symmetric functions on Boolean inputs. This setting already captures most of the interesting combinatorial counting problems defined by local constraints, such as (perfect) matching. This is the first time a dichotomy is obtained for general Holant problems without any auxiliary functions. One benefit of working with the Holant framework is some powerful new reduction technique such as the holographic reduction. Along the proof of our dichotomy, we introduce a new reduction technique, namely realizing a constraint function by approximating it. This new technique is employed in our proof in a situation where it seems that all previous reduction techniques fail; thus, this new idea of reduction might also be of independent interest. Besides proving a dichotomy and developing a new technique, we also obtained some interesting by-products. We prove a dichotomy for #CSP, restricting to instances where each variable appears a multiple of d times for any d. We also prove that counting the number of Eulerian orientations on 2k-regular graphs is #P-hard for any .

Springer Basel Ag2016

Mapping of Structure-Function Age-Related Connectivity Changes on Cognition Using Multimodal MRI

Maria Giulia Preti, Dimitri Nestor Alice Van De Ville

The relationship between age-related changes in brain structural connectivity (SC) and functional connectivity (FC) with cognition is not well understood. Furthermore, it is not clear whether cognition is represented via a similar spatial pattern of FC and SC or instead is mapped by distinct sets of distributed connectivity patterns. To this end, we used a longitudinal, within-subject, multimodal approach aiming to combine brain data from diffusion-weighted MRI (DW-MRI), and functional MRI (fMRI) with behavioral evaluation, to better understand how changes in FC and SC correlate with changes in cognition in a sample of older adults. FC and SC measures were derived from the multimodal scans acquired at two time points. Change in FC and SC was correlated with 13 behavioral measures of cognitive function using Partial Least Squares Correlation (PLSC). Two of the measures indicate an age-related change in cognition and the rest indicate baseline cognitive performance. FC and SC-cognition correlations were expressed across several cognitive measures, and numerous structural and functional cortical connections, mainly cingulo-opercular, dorsolateral prefrontal, somatosensory and motor, and temporo-parieto-occipital, contributed both positively and negatively to the brain-behavior relationship. Whole-brain FC and SC captured distinct and independent connections related to the cognitive measures. Overall, we examined age-related function-structure associations of the brain in a comprehensive and integrated manner, using a multimodal approach. We pointed out the behavioral relevance of age-related changes in FC and SC. Taken together, our results highlight that the heterogeneity in distributed FC and SC connectivity patterns provide unique information about the variable nature of healthy cognitive aging.

FRONTIERS MEDIA SA2022

A Dichotomy for Real Weighted Holant Problems

Sangxia Huang

Springer Basel Ag2016