Version Of Construction Research Articles

The p-order maximum principle for an irregular optimal control problem

The general optimal control problem subject to irregular constraints is considered for which the factor of the objective functional in Pontryagin’s function may vanish. It turns out that, in the case of p-regular constraints, this drawback can be overcome and a constructive version of the p-order maximum principle can be formulated.

Complex dimensions of fractals and meromorphic extensions of fractal zeta functions

We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function ζA(s):=∫Aδd(x,A)s−Ndx, where δ>0 is fixed and d(x,A) denotes the Euclidean distance from x to A, has been introduced by the first author in 2009, extending the definition of the zeta function ζL associated with bounded fractal strings L=(ℓj)j≥1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D(ζA) coincides with D:=dim‾BA, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” {Res=D}. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function |At| as t→0+, where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that |At|=tN−D(M+O(tγ)) as t→0+, with γ>0, and to a class of Minkowski nonmeasurable sets, such that |At|=tN−D(G(log⁡t−1)+O(tγ)) as t→0+, where G is a nonconstant periodic function and γ>0. In both cases, we show that ζA can be meromorphically extended (at least) to the open right half-plane {Res>D−γ} and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of ζA evaluated at s=D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line {Res=D}. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of RN, for N≥1 arbitrary. These are compact subsets of RN such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line {Res=D}.

Adaptive mesh point selection for the efficient solution of scalar IVPs

We discuss adaptive mesh point selection for the solution of scalar IVPs. We consider a method that is optimal in the sense of the speed of convergence, and aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.

The Frobenius condition, right properness, and uniform fibrations

We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform) fibrations and that satisfy the (functorial) Frobenius condition. As applications, we obtain a new proof that the Quillen model structure for Kan complexes is right proper, avoiding entirely the use of topological realization and minimal fibrations, and we solve an open problem in the study of Voevodsky's simplicial model of type theory, proving a constructive version of the preservation of Kan fibrations by pushforward along Kan fibrations. Our results also subsume and extend work by Coquand and others on cubical sets.

Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $\mathbb{F} S_n$ of the symmetric group $S_n$ over a field $\mathbb{F}$ of characteristic 0 (or $p > n$). The goal is to obtain a constructive version of the isomorphism $\psi\colon \bigoplus_\lambda M_{d_\lambda} (\mathbb{F}) \longrightarrow \mathbb{F} S_n$ where $\lambda$ is a partition of $n$ and $d_\lambda$ counts the standard tableaux of shape $\lambda$. Young showed how to compute $\psi$; to compute its inverse, we use an efficient algorithm for representation matrices discovered by Clifton. In §2, we discuss constructive methods based on §1 which allow us to analyze the polynomial identities satisfied by a specific (non)associative algebra: fill and reduce algorithm, module generators algorithm, Bondari's algorithm for finite dimensional algebras. In §3, we study the multilinear identities satisfied by the octonion algebra $\mathbb{O}$ over a field of characteristic 0. For $n \le 6$ we compare our computational results with earlier work of Racine, Hentzel \&\ Peresi, Shestakov \&\ Zhukavets. Going one step further, we verify computationally that every identity in degree 7 is a consequence of known identities of lower degree; this result is our main original contribution. This gap (no new identities in degree 7) motivates our concluding conjecture: the known identities for $n \le 6$ generate all of the octonion identities in characteristic 0.

The common invariant subspace problem and Tarski’s theorem

This article presents a computable criterion for the existence of a common invariant subspace of $n\times n$ complex matrices $A_{1}, \dots ,A_{s}$ of a fixed dimension $1\leq d\leq n$. The approach taken in the paper is model-theoretic. Namely, the criterion is based on a constructive proof of the renowned Tarski's theorem on quantifier elimination in the theory $\ACF$ of algebraically closed fields. This means that for an arbitrary formula $\varphi$ of the language of fields, a quantifier-free formula $\varphi'$ such that $\varphi\lra\varphi'$ in $\ACF$ is given explicitly. The construction of $\varphi'$ is elementary and based on the effective Nullstellensatz. The existence of a common invariant subspace of $A_{1},\dots,A_{s}$ of dimension $d$ can be expressed in the first-order language of fields, and hence, the constructive version of Tarski's theorem yields the criterion. In addition, some applications of this criterion in quantum information theory are discussed.

Franz and Georg: Cantor's Mathematics of the Infinite in the Work of Kafka

The infinite plays a significant role in many of Franz Kafka's stories. In this note we show that a version of the middle-thirds Cantor set construction appears in Kafka's Great Wall of China. His description of the Wall's construction mirrors an iterated system of functions whose limit set is the Cantor set; we present the relevant mathematical details and a close reading of the text of the story to illustrate this metaphor.

A Practical View on Renaming

We revisit variable renaming from a practitioner's point of view, presenting concepts we found useful in dealing with operational semantics of pure Prolog. A concept of relaxed core representation is introduced, upon which a concept of prenaming is built. Prenaming formalizes the intuitive practice of renaming terms by just considering the necessary bindings, where now some passive "bindings" x/x may be necessary as well. As an application, a constructive version of variant lemma for implemented Horn clause logic has been obtained. There, prenamings made it possible to incrementally handle new (local) variables.

The Realization Problem for some wild monoids and the Atiyah Problem

The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if it cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid we consider is the monoid M \mathcal M obtained by successive refinements of the identity x 0 + y 0 = x 0 + z 0 x_0+y_0=x_0+z_0 . This monoid is known to be realizable by the algebra A = K [ F ] A= K[\mathcal F] of the monogenic free inverse monoid F \mathcal F , for any choice of field K K , but A A is not an exchange ring. We show that, for any uncountable field K K , M \mathcal M is not realizable by a regular K K -algebra, but that a suitable universal localization Σ − 1 A \Sigma ^{-1}A of A A provides an exchange, non-regular, K K -algebra realizing M \mathcal M . For any countable field F F , we show that a skew version of the above construction gives a regular F F -algebra realizing M \mathcal M . Finally, we develop some connections with the Atiyah Problem for the lamplighter group. We prove that the algebra A A can be naturally seen as a ∗ * -subalgebra of the group algebra k G kG over the lamplighter group G = Z 2 ≀ Z G= \mathbb {Z}_2\wr \mathbb {Z} , for any subfield k k of C \mathbb {C} closed under conjugation, and we determine the structure of the ∗ * -regular closure of A A in U ( G ) \mathcal U(G) . Using this, we show that the subgroup of R \mathbb R generated by the von Neumann dimensions of matrices over k G kG contains Q \mathbb {Q} .

On explicit descent of marked curves and maps

We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to D\`ebes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.

Analysis of the Combustion Air Preheater from the Aluminum Melting Furnaces

Abstract This paper presents solutions and the equipment for preheating combustion air from scrap aluminum melting furnaces through flue gas heat recovery. For sizing convection pre-heaters, there has been developed a mathematical model which has been transcribed into a computer program in C + +. A constructive version of the pre-heater was drawn up and a recovery heat exchanger was manufactured and mounted on an aluminum melting furnace. Both the functional parameters values and the reasons causing the pre-heater worning out, as well as the steps taken for sizing and the achievement of a new air pre-heater able to bear the operating conditions of the aluminum melting furnace are shown.

Exact triangle for fibered Dehn twists

We use quilted Floer theory to generalize Seidel’s long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply the sequence to construct versions of the Floer and Khovanov–Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.

Analysis of the heat transfer in double and triple concentric tube heat exchangers

The tubular heat exchangers (shell and tube heat exchangers and concentric tube heat exchangers) represent an important category of equipment in the petroleum refineries and are used for heating, pre-heating, cooling, condensation and evaporation purposes. The paper presents results of analysis of the heat transfer to cool a petroleum product in two types of concentric tube heat exchangers: double and triple concentric tube heat exchangers. The cooling agent is water. The triple concentric tube heat exchanger is a modified constructive version of double concentric tube heat exchanger by adding an intermediate tube. This intermediate tube improves the heat transfer by increasing the heat area per unit length. The analysis of the heat transfer is made using experimental data obtained during the tests in a double and triple concentric tube heat exchanger. The flow rates of fluids, inlet and outlet temperatures of water and petroleum product are used in determining the performance of both heat exchangers. Principally, for both apparatus are calculated the overall heat transfer coefficients and the heat exchange surfaces. The presented results shows that triple concentric tube heat exchangers provide better heat transfer efficiencies compared to the double concentric tube heat exchangers.

A Potential Reduction Algorithm for Two-Person Zero-Sum Mean Payoff Stochastic Games

We suggest a new algorithm for two-person zero-sum undiscounted stochastic games focusing on stationary strategies. Given a positive real $$\varepsilon $$ , let us call a stochastic game $$\varepsilon $$ -ergodic, if its values from any two initial positions differ by at most $$\varepsilon $$ . The proposed new algorithm outputs for every $$\varepsilon >0$$ in finite time either a pair of stationary strategies for the two players guaranteeing that the values from any initial positions are within an $$\varepsilon $$ -range, or identifies two initial positions u and v and corresponding stationary strategies for the players proving that the game values starting from u and v are at least $$\varepsilon /24$$ apart. In particular, the above result shows that if a stochastic game is $$\varepsilon $$ -ergodic, then there are stationary strategies for the players proving $$24\varepsilon $$ -ergodicity. This result strengthens and provides a constructive version of an existential result by Vrieze (Stochastic games with finite state and action spaces. PhD thesis, Centrum voor Wiskunde en Informatica, Amsterdam, 1980) claiming that if a stochastic game is 0-ergodic, then there are $$\varepsilon $$ -optimal stationary strategies for every $$\varepsilon > 0$$ . The suggested algorithm is based on a potential transformation technique that changes the range of local values at all positions without changing the normal form of the game.

Fibrational Modal Type Theory

This paper describes a fibrational categorical semantics for the modal necessity-only fragment of constructive modal type theory, both with and without dependent types. Constructive type theory does not usually discuss logical modalities, and modalities tend to be mostly studied within classical logic, not within type theory. But modalities should be very useful in type theory, as they are very useful in modelling theoretical computing systems. Providing constructive versions of modal logics and their associated Curry-Howard modal type theories is also a very productive program, e.g. helpful when dealing with computational effects, staged computation, and functional reactive types, for example. There seems to be renewed interest in the notion of constructive modal type theory (and in notions of linear type theory), in part because of the interest in homotopy type theory. The modal type theory presented here uses dependent types, in the style of Ritter's categorical models of the Calculus of Constructions. To build up to these, we first discuss the kinds of constructive modal type theory in the literature. Then we provide a non-dependent modal type theory, introduced in previous work, that we generalize to dependent types in the following section. Dependent type theories are usually but not always given categorical semantics in terms of fibrations. We provide semantics in terms of fibrations for both the non-dependent and the dependent type systems discussed and prove them sound and complete, thereby providing evidence that the type theory is meaningful. These fibrational models should be also applicable to the homotopy type theory setting.

A verified abstract machine for functional coroutines

Functional coroutines are a restricted form of control mechanism, where each coroutine is represented with both a continuation and an environment. This restriction was originally obtained by considering a constructive version of Parigot's classical natural deduction which is sound and complete for the Constant Domain logic. In this article, we present a refinement of de Groote's abstract machine for functional coroutines and we prove its correctness. Therefore, this abstract machine also provides a direct computational interpretation of the Constant Domain logic.

Costly Circuits, Submodular Schedules and Approximate Carathéodory Theorems

Hybrid switching -- in which a high bandwidth circuit switch (optical or wireless) is used in conjunction with a low bandwidth packet switch -- is a promising alternative to interconnect servers in today's large scale data centers. Circuit switches offer a very high link rate, but incur a non-trivial reconfiguration delay which makes their scheduling challenging. In this paper, we demonstrate a lightweight, simple and nearly-optimal scheduling algorithm that trades-off reconfiguration costs with the benefits of reconfiguration that match the traffic demands. Seen alternatively, the algorithm provides a fast and approximate solution towards a constructive version of Caratheodory's Theorem for the Birkhoff polytope. The algorithm also has strong connections to submodular optimization, achieves a performance at least half that of the optimal schedule and strictly outperforms state of the art in a variety of traffic demand settings. These ideas naturally generalize: we see that indirect routing leads to exponential connectivity; this is another phenomenon of the power of multi-hop routing, distinct from the well-known load balancing effects.

Suppression of Chemotactic Explosion by Mixing

Chemotaxis plays a crucial role in a variety of processes in biology and ecology. In many instances, processes involving chemical attraction take place in fluids. One of the most studied PDE models of chemotaxis is given by Keller-Segel equation, which describes a population density of bacteria or mold which attract chemically to substance they secrete. Solution of Keller-Segel equation can exhibit dramatic collapsing behavior, where the density concentrates positive mass in a measure zero region. A natural question is whether presence of fluid flow can affect singularity formation by mixing the density thus making concentration harder to achieve. In this paper, we consider parabolic-elliptic Keller-Segel equation in two and three dimensions with additional advection term modeling ambient fluid flow. We prove that for any initial data, there exist incompressible fluid flows such that the solution to the equation stays globally regular. On the other hand, it is well known that when the fluid flow is absent, there exist initial data leading to finite time blow up. Thus presence of fluid can prevent the singularity formation. We discuss two classes of flows that have the explosion arresting property. Both classes are known as very efficient mixers. The first class are relaxation enhancing (RE) flow. These flows are stationary. The second class of flows are Yao-Zlatos optimal mixing flows, which are time dependent. The proof is based on the nonlinear version of the relaxation enhancement construction of the paper by Peter Constantin, Alexander Kiselev, Lenya Ryzhik, Andrej Zlatos, and on some variations of global regularity estimate for Keller-Segel model.

A constructive version of the Sylvester–Gallai theorem

The Sylvester–Gallai Theorem, stated as a problem by James Joseph Sylvester in 1893, asserts that for any finite, noncollinear set of points on a plane, there exists a line passing through exactly two points of the set. First, it is shown that for the real plane \({{\mathbb{R}^{2}}}\) the theorem is constructively invalid. Then, a well-known classical proof is examined from a constructive standpoint, locating the nonconstructivities. Finally, a constructive version of the theorem is established for the plane \({{\mathbb{R}^{2}}}\); this reveals the hidden constructive content of the classical theorem. The constructive methods used are those proposed by Errett Bishop.

УСТРОЙСТВА ДЛЯ КОНСТРУКТИВНОЙ АДАПТАЦИИ ДИЗЕЛЕЙ АВТОТРАКТОРНОЙ ТЕХНИКИ К РАБОТЕ НА БИОМИНЕРАЛЬНОМ ТОПЛИВЕ

The research is devoted to possibility of mineral diesel fuel with mixed biodiesel fuel partial replacement, which contains the vegetable oil as a biological component (such as: rapeseed, camelina, radish, mustard, colza, soybean, etc.). Constructive versions of bi-fuel supply system and mixers for diesel engines adaptation for their operation by biodiesel fuel are offered. The advantages of the developed bi-fuel systems and mixers are: universality in relation to different types of diesel engines; polyfuelness; availability of components; competitiveness; possibility of production in the plant conditions, no requirement of big capital investments, the small term of payback. The application of the offered system by the diesel-powered machinery provides the necessary percentage ratio of mixed fuel components depending on the load-speed mode without its essential constructive changes. The using of the mixed biodiesel fuel containing for example 50% of mineral fuel and 50% of camelina oil allows to save 50% of petroleum diesel fuel, to reduce smoke of the exaust gases by 17-20%, to reduce the content of carbon oxide by 35-40% in comparison with the work of the diesel engine on mineral diesel fuel, by insignificant decreasing of the effective power (no more than 6%) and by some increasing of specific fuel consumption of mixed fuel (to 14%).