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The purpose of this paper is to present in linear spaces some results for new notions called A-left (resp., A-right) ascent and A-left (resp., A-right) descent of linear operators (where A is a given operator) which generalize two important notions in operator theory: ascent and descent. Moreover, if A is a positive operator, we obtain several properties of ascent and descent of an operator in semi-Hilbertian spaces. Some basic properties and many results related to the ascent and descent for a linear operator on a linear space Kaashoek (Math Ann 172:105–115, 1967), Taylor (Math Ann 163:18–49, 1966) are extended to these notions. Some stability results under perturbations by compact operators and operators having some finite rank power are also given for these notions.

We consider relative or subjective optimization problems where the goal function and feasible set are dependent on the current state of the system under consideration. In general, they are formulated as quasi-equilibrium problems, hence finding their solutions may be rather difficult. We describe a rather general class of relative optimization problems in metric spaces, which in addition depend on the starting state. We also utilize quasi-equilibrium type formulations of these problems and show that they admit rather simple descent solution methods. This approach gives suitable trajectories tending to a relatively optimal state. We describe several examples of applications of these problems. Preliminary results of computational experiments confirmed efficiency of the proposed method.

We apply an explicit construction of a simplicial powering in dg-categories, due to Holstein (Properness and simplicial resolutions for the model category dgCat, 2014. arXiv:1403.4381 ) and Arkhipov and Poliakova (Homol Homotopy Appl 22(2):151–162, 2019), as well as our own results on homotopy ends (Arkhipov and Orsted in Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories, 2018. arXiv:1807.03266 ), to obtain an explicit model for the homotopy limit of a cosimplicial system of dg-categories. We apply this to obtain a model for homotopy descent in terms of $$\mathrm {A}_{\infty }$$ -comodules, proving a conjecture by Block et al. (Homol Homotopy Appl 19(2):343–371, 2017) in the process.

We give necessary and sufficient conditions for a cdh sheaf to satisfy Milnor excision, following ideas of Bhatt and Mathew. Along the way, we show that the cdh infinity-topos of a quasi-compact quasi-separated scheme of finite valuative dimension is hypercomplete, extending a theorem of Voevodsky to nonnoetherian schemes. As an application, we show that if E is a motivic spectrum over a field k which is n-torsion for some n invertible in k, then the cohomology theory on k-schemes defined by E satisfies Milnor excision., Comment: 28 pages. v3: final version, to appear in Mathematische Annalen; v2: new title and minor corrections

In this paper we consider generalized Leontief model. We show that under certain condition the generalized Leontief model is solvable by iterative descent method based on infeasible interior point algorithm. We prove the convergence of the method from strictly positive starting point. A numerical example is presented to demonstrate the performance of the algorithm

Discussed here is descent theory in the differential context where everything is equipped with a differential operator. To answer a question personally posed by A. Pianzola, we determine all twisted forms of the differential Lie algebras over $\mathbb{C}(t)$ associated with complex simple Lie algebras. Hopf-Galois Theory, a ring-theoretic counterpart of theory of torsors for group schemes, plays a role when we grasp the above-mentioned twisted forms from torsors., Largely revised; in particular, changed the title and added several articles to the References, see the added Remark 2.2

The Online Order Batching Problem (OOBP) is a variant of the well-known Order Batching Problem (OBP). As in the OBP, the goal of this problem is to collect all the orders that arrive at a warehouse, following an order batching picking policy, while minimizing a particular objective function. Therefore, orders are grouped in batches, of a maximum predefined capacity, before being collected. Each batch is assigned to a single picker, who collects all the orders within the batch in a single route. Unlike the OBP, this variant presents the peculiarity that the orders considered in each instance are not fully available in the warehouse at the beginning of the day, but they can arrive at the system once the picking process has already begun. Then, batches have to be dynamically updated and, as a consequence, routes must too. In this paper, the maximum turnover time (maximum time that an order remains in the warehouse) and the maximum completion time (total collecting time of all orders received in the warehouse) are minimized. To that aim, we propose an algorithm based in the combination of a Greedy Randomized Adaptive Search Procedure and a Variable Neighborhood Descent. The best variant of our method has been tested over a large set of instances and it has been favorably compared with the best previous approach in the state of the art.

In this paper, according to some suitable features of three-term conjugate gradient methods and excellent theoretical properties of the quasi-Newton methods, a new spectral three-term conjugate gradient is proposed. A modified secant condition is used to compute a suitable spectral parameter. The new search direction ensures the sufficient descent condition without any line search. It is established that the new scheme possesses the global convergence, under the strong Wolfe conditions. Preliminary numerical experiments show the efficiency of the new method, dealing with unconstrained optimization problems.

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the $p$-adic de Rham period map $j^{dr}_n$ on elliptic and hyperelliptic curves over number fields via their universal unipotent connections $\mathcal{U}$. Several algorithms forming part of the computation of finite level versions $j^{dr}_n$ of the unipotent Albanese maps are presented. The computation of the logarithmic extension of $\mathcal{U}$ in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on $\mathcal{U}$ over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated $p$-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well., Comment: 60 pages

For X a smooth projective variety over a field k, we consider the problem of Galois descent for higher Brauer groups. More precisely, we extend a finiteness result of Colliot-Thelene and Skorobogatov (J Reine Angew Math 682:141–165, 2013) to higher Brauer groups.

In this paper, we develop a new hybrid conjugate gradient method that inherits the features of the Liu and Storey (LS), Hestenes and Stiefel (HS), Dai and Yuan (DY) and Conjugate Descent (CD) conjugate gradient methods. The new method generates a descent direction independently of any line search and possesses good convergence properties under the strong Wolfe line search conditions. Numerical results show that the proposed method is robust and efficient.

Spectral conjugate gradient methods are considered as an efficient family of conjugate gradient methods to solve unconstrained optimization problems. In this work, based on the scaled memoryless BFGS update, a spectral Polak–Ribiere–Polyak conjugate gradient method is proposed. It is established that the method ensures the sufficient descent property independent of line search techniques as well as convexity of the objective function. The global convergence of the method is shown with the strong Wolfe line search strategy for general functions. Finally, the practical competencies of the method are investigated by numerical experiments on a set of test functions of CUTEr library. The results demonstrate computational efficiency of the proposed method.

In this paper, the spectral algorithm for nonlinear equations (SANE) is adapted to the problem of finding a zero of a given tangent vector field on a Riemannian manifold. The generalized version of SANE uses, in a systematic way, the tangent vector field as a search direction and a continuous real-valued function that adapts this direction and ensures that it verifies a descent condition for an associated merit function. In order to speed up the convergence of the proposed method, we incorporate a Riemannian adaptive spectral parameter in combination with a non-monotone globalization technique. The global convergence of the proposed procedure is established under some standard assumptions. Numerical results indicate that our algorithm is very effective and efficient solving tangent vector field on different Riemannian manifolds and competes favorably with a Polak-Ribi\'ere-Polyak Method recently published and other methods existing in the literature.

For the large-scale optimization problems, we propose a new conjugate parameter by modifying the denominator of the Polak–Ribiere–Polyak formula, and give its non-negative form. Under the weak Wolfe line search, their corresponding algorithms perform superior to their congener methods, respectively. To guarantee its global convergence, we further introduce a restart condition and a restart direction to improve the proposed method. Under usual assumptions and using the strong Wolfe line search to yielded the step-length, the improved method is sufficient descent and globally convergent. Numerical experiments for the improved method and its comparisons are carried out, and the corresponding numerical results and performance profiles are reported, which showed that the improved method is practicable and efficient for the large-scale optimization problems.

We prove an analogue of Jacquet's conjecture on the local converse theorem for \ell-adic families of co-Whittaker representations of GL_n(F), where F is a finite extension of Q_p and \ell does not equal p. We also prove an analogue of Jacquet's conjecture for a descent theorem, which asks for the smallest collection of gamma factors determining the subring of definition of an \ell-adic family. These two theorems are closely related to the local Langlands correspondence in \ell-adic families., 32 pages

We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney’s embedding theorem allows us to work in a reduced-dimensional embedding space. A numerical continuation method applied to the reduced-dimensional embedded manifold is used to drive the procedure.

International audience; Direct search is a methodology for derivative-free optimization whose iterations are characterized by evaluating the objective function using a set of polling directions. In deterministic direct search applied to smooth objectives, these directions must somehow conform to the geometry of the feasible region, and typically consist of positive generators of approximate tangent cones (which then renders the corresponding methods globally convergent in the linearly constrained case). One knows however from the unconstrained case that randomly generating the polling directions leads to better complexity bounds as well as to gains in numerical efficiency, and it becomes then natural to consider random generation also in the presence of constraints. In this paper, we study a class of direct-search methods based on sufficient decrease for solving smooth linearly constrained problems where the polling directions are randomly generated (in approximate tangent cones). The random polling directions must satisfy probabilistic feasible descent, a concept which reduces to probabilistic descent in the absence of constraints. Such a property is instrumental in establishing almost-sure global convergence and worst-case complexity bounds with overwhelming probability. Numerical results show that the randomization of the polling directions can be beneficial over standard approaches with deterministic guarantees, as it is suggested by the respective worst-case complexity bounds.

This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although prior work has established convergence results for algorithms that integrate both descent and negative curvature steps, there has not yet been extensive numerical evidence showing that such methods offer consistent performance improvements. In this paper, we present new frameworks for combining descent and negative curvature directions: alternating two-step approaches and dynamic step approaches. The aspect that distinguishes our approaches from ones previously proposed is that they make algorithmic decisions based on (estimated) upper-bounding models of the objective function. A consequence of this aspect is that our frameworks can, in theory, employ fixed stepsizes, which makes the methods readily translatable from deterministic to stochastic settings. For deterministic problems, we show that instances of our dynamic framework yield gains in performance compared to related methods that only follow descent steps. We also show that gains can be made in a stochastic setting in cases when a standard stochastic-gradient-type method might make slow progress.

Primal–dual hybrid gradient (PDHG) and alternating direction method of multipliers (ADMM) are popular first-order optimization methods. They are easy to implement and have diverse applications. As first-order methods, however, they are sensitive to problem conditions and can struggle to reach the desired accuracy. To improve their performance, researchers have proposed techniques such as diagonal preconditioning and inexact subproblems. This paper realizes additional speedup about one order of magnitude. Specifically, we choose general (non-diagonal) preconditioners that are much more effective at reducing the total numbers of PDHG/ADMM iterations than diagonal ones. Although the subproblems may lose their closed-form solutions, we show that it suffices to solve each subproblem approximately with a few proximal-gradient iterations or a few epochs of proximal block-coordinate descent, which are simple and have closed-form steps. Global convergence of this approach is proved when the inner iterations are fixed. Our method opens the choices of preconditioners and maintains both low per-iteration cost and global convergence. Consequently, on several typical applications of primal–dual first-order methods, we obtain 4–95 $$\times $$ speedup over the existing state-of-the-art.

In this paper, we propose a hybrid conjugate gradient (CG) method based on the approach of convex combination of Fletcher–Reeves (FR) and Polak–Ribiere–Polyak (PRP) parameters, and Quasi-Newton’s update. This is made possible by using self-scaling memory-less Broyden’s update together with a hybrid direction consisting of two CG parameters. However, an important property of the new algorithm is that, it generates a descent search direction via non-monotone type line search. The global convergence of the algorithm is established under appropriate conditions. Finally, numerical experiments on some benchmark test problems, demonstrate the effectiveness of the proposed algorithm over some existing alternatives.

In this paper, a new subspace minimization conjugate gradient method based on modified secant equation is proposed and analyzed. For a classical subspace minimization conjugate gradient method, the search direction is derived by minimizing an approximate quadratic model of objective function in a two-dimensional subspace. Generally, the approximate Hessian matrix in the above quadratic model is required to satisfy the standard secant equation, while we consider an approximate Hessian matrix which satisfies the modified secant equation. We give some rules such that if these rules are satisfied, we choose the standard secant equation, otherwise we choose the modified one. We can prove that the proposed directions satisfy the sufficient descent property under some extra conditions. We also present a modified nonmonotone Wolfe line search and establish the global convergence of the proposed method for general nonlinear functions under mild assumptions. Numerical comparisons are given with famous CG_DESCENT (5.3) (Hager and Zhang in SIAM J Optim 16(1):170–192, 2005) and SMCG_BB (Liu and Liu in J Optim Theory Appl 180(3):879–906, 2019), and show that the proposed algorithm is very promising.

The construction of the ghost number zero and one vertex operators for the 11D pure spinor superparticle will be revisited. In this sense, an alternative way of defining the ghost number one vertex operator will be given after introducing a ghost number -2 operator made out of physical operators defined on the 11D non-minimal pure spinor superspace. This procedure will make explicit and transparent the relation between the ghost number three and one vertex operators. In addition, using a non-Lorentz covariant b-ghost, ghost number zero and two vertex operators satisfying standard descent equations will be presented in full form., 16 pages. Added reference. Small modification to the published version

In this paper, we first propose a new three-term conjugate gradient (CG) method, which is based on the least-squares technique, to determine the CG parameter, named LSTT. And then, we present two improved variants of the LSTT CG method, aiming to obtain the global convergence property for general nonlinear functions. The least-squares technique used here well combines the advantages of two existing efficient CG methods. The search directions produced by the proposed three methods are sufficient descent directions independent of any line search procedure. Moreover, with the Wolfe–Powell line search, LSTT is proved to be globally convergent for uniformly convex functions, and the two improved variants are globally convergent for general nonlinear functions. Preliminary numerical results are reported to illustrate that our methods are efficient and have advantages over two famous three-term CG methods.

In this paper, we propose a self adaptive spectral conjugate gradient-based projection method for systems of nonlinear monotone equations. Based on its modest memory requirement and its efficiency, the method is suitable for solving large-scale equations. We show that the method satisfies the descent condition$F_{k}^{T}d_{k}\leq -c\|F_{k}\|^{2}, c>0$FkTdk≤−c∥Fk∥2,c>0, and also prove its global convergence. The method is compared to other existing methods on a set of benchmark test problems and results show that the method is very efficient and therefore promising.

We show that the complete Segal model structure extends to a model structure on bimplicial presheaves on a small site $\mathscr{C}$, for which the weak equivalences are local (or stalkwise) weak equivalences. This model structure can be realized as a left Bousfield localization of the Jardine model structure on the simplicial presheaves on a site $\mathscr{C}/ \Delta^{op}$. Furthermore, it is shown that this model structure is Quillen equivalent to the model structure of the author's previous preprint entitled 'the Local Joyal Model Structure'. This Quillen equivalence extends an equivalence between the complete Segal space and Joyal model structures, due to Joyal and Tierney., Comment: This is the pre copy edited, post peer review version of an article to appear in 'Applied Categorical Structures'

A coring (A,C) consists of an algebra A and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring., Comment: 46 pages

We establish several inequalities for manifolds with positive scalar curvature and, more generally, for the scalar curvature bounded from below. In so far as geometry is concerned these inequalities appear as generalisations of the classical bounds on the distances between conjugates points in surfaces with positive sectional curvatures. The techniques of our proofs is based on the Schoen–Yau descent method via minimal hypersurfaces, while the overall logic of our arguments is inspired by and closely related to the torus splitting argument in Novikov’s proof of the topological invariance of the rational Pontryagin classes.

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.

We prove that algebraic K-theory satisfies `pro-descent' for abstract blow-up squares of noetherian schemes. As an application we derive Weibel's conjecture on the vanishing of negative K-groups., 48 pages, final version

Given an integer $n\geq2$ , in this article we provide a complete description of all additive surjective maps on the algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space, preserving in both directions the set of Drazin invertible operators of index $n$ .

We present an approximate bundle method for solving nonsmooth equilibrium problems. An inexact cutting-plane linearization of the objective function is established at each iteration, which is actually an approximation produced by an oracle that gives inaccurate values for the functions and subgradients. The errors in function and subgradient evaluations are bounded and they need not vanish in the limit. A descent criterion adapting the setting of inexact oracles is put forward to measure the current descent behavior. The sequence generated by the algorithm converges to the approximately critical points of the equilibrium problem under proper assumptions. As a special illustration, the proposed algorithm is utilized to solve generalized variational inequality problems. The numerical experiments show that the algorithm is effective in solving nonsmooth equilibrium problems.

We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifications of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings., 25 pages

In this paper, we present a new conjugate gradient method, in which the search direction is computed by minimizing a selected approximate model in a two-dimensional subspace. That is, if the objective function is not close to a quadratic, the search direction is generated by a conic model. Otherwise, a quadratic model is considered. The direction of the proposed method is proved to possess the sufficient descent property. With the modified nonmonotone line search, we establish a global convergence of the proposed method under appropriate assumptions. R-linear convergence of the proposed method is also analyzed. Numerical results using two different test function collections show that the proposed algorithm is efficient.

The problem of convex optimization is studied. Usually in convex optimization the minimization is over a d-dimensional domain. Very often the convergence rate of an optimization algorithm depends on the dimension d. The algorithms studied in this paper utilize dictionaries instead of a canonical basis used in the coordinate descent algorithms. We show how this approach allows us to reduce dimensionality of the problem. Also, we investigate which properties of a dictionary are beneficial for the convergence rate of typical greedy-type algorithms.

This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping x???f(x)$x\rightarrow \partial _{\varepsilon } f(x)$ named ?-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein- ?-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ?-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.

Using the theory of Euler systems and Nekovař’s adaptation of Kolyvagin’s descent techniques, we bound the size of Selmer groups associated to modular forms of even weight larger than 2 twisted by ring class characters of imaginary quadratic fields.

Suppose $R\rightarrow S$ is a faithfully flat ring map. The theory of twisted forms lets one compute, given an $R$-module $M$, how many isomorphism classes of $R$-modules $M^{\prime}$ satisfy $S\otimes_R M\cong S\otimes_R M^{\prime}$. This is really a uniqueness problem. But this theory does not help one to solve the corresponding existence problem: given an $S$-module $N$, does there exists {\em some} $R$-module $M$ such that $S\otimes_R M\cong N$? In this paper we work out (as a special case of a general theorem about existence of coalgebra structures over a comonad) a criterion for the existence of such an $R$-module $M$, under some reasonable hypotheses on the map $R\rightarrow S$., Comment: This version corrects an error in the applications section in the previously posted arxiv version, and also proves the main result in greater generality, as suggested by an anonymous referee. It also contains some example computations which did not appear in the published version

In this paper, we focus on the problem of blind source separation (BSS). To solve the problem efficiently, a new algorithm is proposed. First, a parallel variable dual-matrix model (PVDMM) that considers all the numerical relations between a mixing matrix and a separating matrix is proposed. Different constrained terms are used to construct the cost function for every sub-algorithm. These constrained terms reflect a numerical relation. Therefore, a number of undesired solutions are excluded, the search region is reduced, and the convergence efficiency of the algorithm is ultimately improved. Parallel sub-algorithms are proven to converge to a separable matrix only if the cost function approaches zero. Second, a descent algorithm (DA) as a PVDMM optimizing approach is proposed in this paper as well. Apparently, the efficiency of the algorithm depends completely on the DA. Unlike the traditional descent method, the DA defines step length by solving inequality instead of merely utilizing the Wolfe- or Armijo-type search rule. Stimulation results indicate that the DA can improve computational efficiency. Under mild conditions, the DA has been proven to have strong convergence properties. Numerical results also show that the method is very efficient and robust. Finally, we applied the combinative algorithm to the BSS problem. Computer simulations illustrate its good performance.

Based on an eigenvalue study, the sufficient descent condition of an extended class of the Hager-Zhang nonlinear conjugate gradient methods is established. As an interesting result, it is shown that the search directions of the CG_Descent algorithm satisfy the sufficient descent condition \(d_k^Tg_k

International audience; The paper considers robust parametric optimization problems using multi-point formulations and addresses the issue of the approximation of the gradient of the functional by reduced order models. The question of interest is the impact of such approximations on the search subspace in the multi-point optimization problem. The mathematical concept used to evaluate these approximations is the principal angles between subspaces and practical ways to evaluate these are provided. An additional indicator is provided when a descent minimization algorithm is used. The approach appears also to be an interesting tool for uncertainty quantification of the design in the presence of models of increasing complexity. The application of these concepts is illustrated in the design of the shape of an aircraft robust over a range of transverse winds.

In this paper we use Janelidze’s approach to the classical theory of topological coverings via categorical Galois theory to study coverings in categories of relational algebras. Moreover, we present characterizations of effective descent morphisms in the categories of M-ordered sets and of multi-ordered sets.

Thomas S. Kuhn is famous both for his work on the Copernican Revolution and his ‘paradigm’ view of scientific revolutions. But Kuhn later abandoned the notion of paradigm (and related notions) in favour of a more ‘evolutionary’ view of the history of science. Kuhn’s position therefore moved closer to ‘continuity’ models of scientific progress, for instance ‘chain-of-reasoning’ models, originally championed by D. Shapere. The purpose of this paper is to contribute to the debate around Kuhn’s new ‘developmental’ view and to evaluate these competing models with reference to some major innovations in the history of cosmology, from Copernicanism to modern cosmology. This evaluation is made possible through some unexpected overlap between Kuhn’s earlier discontinuity model and various versions of the later continuity models. It is the thesis of this paper that the ‘chain-of-reasoning’ model accounts better for the cosmological evidence than both Kuhn’s early paradigm model and his later developmental view of the history of science.

International audience; The paper is concerned with the analysis of a new variational model torestore point-like and curve-like singularities in biological images. To this aim we investigate the variational properties of a suitable energy which governs these pathologies. Finally in order to realize numerical experiments we minimize, in the discretesetting, a regularized version of this functional by fast descent gradient scheme.

Yet another proof of the result asserting that a morphism of commutative rings is an effective descent morphism for modules if and only if it is pure is given. Moreover, it is shown that this result cannot be derived from Moerdijk's descent criterion.

We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of integral points on integral models of twists of modular curves over function fields.

The notions of upper and lower exhausters represent generalizations of the notions of exhaustive families of upper convex and lower concave approximations (u.c.a., l.c.a.). The notions of u.c.a.'s and l.c.a.'s were introduced by Pshenichnyi (Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and its Applications, 1980), while the notions of exhaustive families of u.c.a.'s and l.c.a.'s were described by Demyanov and Rubinov in Nonsmooth Problems of Optimization Theory and Control, Leningrad University Press, Leningrad, 1982. These notions allow one to solve the problem of optimization of an arbitrary function by means of Convex Analysis thus essentially extending the area of application of Convex Analysis. In terms of exhausters it is possible to describe extremality conditions, and it turns out that conditions for a minimum are expressed via an upper exhauster while conditions for a maximum are formulated in terms of a lower exhauster (Abbasov and Demyanov (2010), Demyanov and Roshchina (Appl Comput Math 4(2): 114---124, 2005), Demyanov and Roshchina (2007), Demyanov and Roshchina (Optimization 55(5---6): 525---540, 2006)). This is why an upper exhauster is called a proper exhauster for minimization problems while a lower exhauster is called a proper one for maximization problems. The results obtained provide a simple geometric interpretation and allow one to construct steepest descent and ascent directions. Until recently, the problem of expressing extremality conditions in terms of adjoint exhausters remained open. Demyanov and Roshchina (Appl Comput Math 4(2): 114---124, 2005), Demyanov and Roshchina (Optimization 55(5---6): 525---540, 2006) was the first to derive such conditions. However, using the conditions obtained (unlike the conditions expressed in terms of proper exhausters) it was not possible to find directions of descent and ascent. In Abbasov (2011) new extremality conditions in terms of adjoint exhausters were discovered. In the present paper, a different proof of these conditions is given and it is shown how to find steepest descent and ascent conditions in terms of adjoint exhausters. The results obtained open the way to constructing numerical methods based on the usage of adjoint exhausters thus avoiding the necessity of converting the adjoint exhauster into a proper one.

Generalized Nash equilibrium problems are important examples of quasi-equilibrium problems. The aim of this paper is to study a general class of algorithms for solving such problems. The method is a hybrid extragradient method whose second step consists in finding a descent direction for the distance function to the solution set. This is done thanks to a linesearch. Two descent directions are studied and for each one several steplengths are proposed to obtain the next iterate. A general convergence theorem applicable to each algorithm of the class is presented. It is obtained under weak assumptions: the pseudomonotonicity of the equilibrium function and the continuity of the multivalued mapping defining the constraint set of the quasi-equilibrium problem. Finally some preliminary numerical results are displayed to show the behavior of each algorithm of the class on generalized Nash equilibrium problems.

In this note, we prove that for the standard representation Vof the Weyl group W of a semi-simple algebraic group of type An, Bn, Cn, Dn, F4 and G2 over \(\mathbb{C}\), the projective variety \(\mathbb{P}(V^m)/W\) is projectively normal with respect to the descent of \(\mathcal{O}(1)^{\otimes |W|}\), where Vm denote the direct sum of m copies of V.

In this paper, we calculate the $ \phi (\hat{\phi})-$Selmer groups $ S^{(\phi)} (E / \Q) $ and $ S^{(\hat{\varphi})} (E^{\prime} / \Q) $ of elliptic curves $ y^{2} = x (x + \epsilon p D) (x + \epsilon q D) $ via descent theory (see [S, Chapter X]), in particular, we obtain that the Selmer groups of several families of such elliptic curves can be arbitrary large., Comment: 22 pages