Nontrivial Construction Research Articles

Critical periods in planar polynomial centers near a maximum number of cusps

We provide the best lower bound for the number of critical periods of planar polynomial centers known up to now. The new lower bound is obtained in the Hamiltonian class and considering a single period annulus. This lower bound doubles the previous one from the literature, and we end up with at least n2−2 (resp. n2−2n−1) critical periods for planar polynomial systems of odd (resp. even) degree n. Key idea is the perturbation of a vector field with many cusp equilibria, whose construction is by itself a nontrivial construction that uses elements of catastrophe theory.

Mixed-norm Herz spaces and their applications in related Hardy spaces

In this paper, the authors introduce a class of mixed-norm Herz spaces, [Formula: see text], which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on [Formula: see text]. Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on [Formula: see text]. As applications, the authors develop various real-variable theory of Hardy spaces associated with [Formula: see text] by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of [Formula: see text] and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem.

Oscillation of the Remainder Term in the Prime Number Theorem of Beurling, “Caused by a Given ζ-Zero”

Abstract Continuing previous studies of the Beurling zeta function, here, we prove two results, generalizing long existing knowledge regarding the classical case of the Riemann zeta function and some of its generalizations. First, we address the question of Littlewood, who asked for explicit oscillation results provided a zeta-zero is known. We prove that given a zero $\rho _0$ of the Beurling zeta function $\zeta _{{\mathcal {P}}}$ for a given number system generated by the primes ${\mathcal {P}}$, the corresponding error term $\Delta (x):=\psi _{{\mathcal {P}}}(x)-x$, where $\psi _{{\mathcal {P}}}(x)$ is the von Mangoldt summatory function shows oscillation in any large enough interval, as large as $\frac {\pi /2-\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. The somewhat mysterious appearance of the constant $\pi /2$ is explained in the study. Finally, we prove as the next main result of the paper the following: given $\varepsilon>0$, there exists a Beurling number system with primes ${\mathcal {P}}$, such that $|\Delta (x)| \le \frac {\pi /2+\varepsilon }{|\rho _0|}x^{\Re \rho _0}$. In this 2nd part, a nontrivial construction of a low norm sine polynomial is coupled by the application of the wonderful recent prime random approximation result of Broucke and Vindas, who sharpened the breakthrough probabilistic construction due to Diamond, Montgomery, and Vorhauer.

LIPSCHITZ EQUIVALENCE OF SELF-SIMILAR SETS AND FINITE-STATE AUTOMATON

The study of Lipschitz equivalence of fractals is a very active topic in recent years. Most of the studies in literature concern totally disconnected fractals. In this paper, using neighbor automata of self-similar sets and a transducer, we construct a bi-Lipschitz map between two fractal squares which are not totally disconnected. This is the first nontrivial construction of this type. We also show that this map is measure-preserving.

Sparse hop spanners for unit disk graphs

A unit disk graph G on a given set P of points in the plane is a geometric graph where an edge exists between two points p,q∈P if and only if |pq|≤1. A spanning subgraph G′ of G is a k-hop spanner if and only if for every edge pq∈G, there is a path between p,q in G′ with at most k edges. We obtain the following results for unit disk graphs in the plane.(i)Every n-vertex unit disk graph has a 5-hop spanner with at most 5.5n edges. We analyze the family of spanners constructed by Biniaz (2020) and improve the upper bound on the number of edges from 9n to 5.5n.(ii)Using a new construction, we show that every n-vertex unit disk graph has a 3-hop spanner with at most 11n edges.(iii)Every n-vertex unit disk graph has a 2-hop spanner with O(nlog⁡n) edges. This is the first nontrivial construction of 2-hop spanners.(iv)For every sufficiently large positive integer n, there exists a set P of n points on a circle, such that every plane hop spanner on P has hop stretch factor at least 4. Previously, no lower bound greater than 2 was known.(v)For every finite point set on a circle, there exists a plane (i.e., crossing-free) 4-hop spanner. As such, this provides a tight bound for points on a circle.(vi)The maximum degree of k-hop spanners cannot be bounded from above by a function of k for any positive integer k.

Corrigendum and addendum to "Non-autonomous quasilinear elliptic equations and Ważewski's principle"

In this addendum we fill a gap in a proof and we correct some results appearing in \cite{F6}. In the original paper \cite{F6} we classified positive solutions for the following equation \begin{equation*} \Delta_{p}u+K(r) u^{\sigma-1}=0 \end{equation*} where $r=|x|$, $x \in \RR^n$, $n> p> 1$, $\sigma ={n p}/({n-p})$ and $K(r)$ is a function strictly positive and bounded. In fact \cite{F6} had two main purposes. First, to establish asymptotic conditions which are sufficient for the existence of ground states with fast decay and to classify regular and singular solutions: these results are correct but need some non-trivial further explanations. Second to establish some computable conditions on $K$ which are sufficient to obtain multiplicity of ground states with fast decay in a non-perturbation context. Also in this case the original argument contained a flaw: here we correct the assumptions of \cite{F6} by performing a new nontrivial construction. A third purpose of this addendum is to generalize results of \cite{F6} to a slightly more general equation \begin{equation*} \Delta_p u+ r^{\delta}K(r) u^{\sigma(\delta)-1}=0 \end{equation*} where $\delta> -p$, and $\sigma(\delta) ={p(n+\delta)}/({n-p})$.

Generalization of the Cauchy–Poisson Method and the Construction of Timoshenko-Type Equations

We consider a generalization of the Cauchy–Poisson method to the n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. The presented investigation generalizes and supplements the results obtained earlier. In the Euclidean space, we introduce certain restrictions for the derivatives. The principle of hyperbolic degeneracy in terms of parameters is formulated and its realization in the form of necessary and sufficient conditions is presented. In a special case of fourdimensional space (in which the operators are preserved up to the sixth order), we obtain a generalized hyperbolic equation for the bending vibrations of plates with coefficients that depend only on the Poisson ratio. This equation includes, as special cases, the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As the development of Maxwell's and Einstein's investigations of the propagation of perturbations with finite velocity in continuous media, we can mention the nontrivial construction of Timoshenko’s equation for the bending vibrations of a beam.

Complexity of detectability, opacity and A-diagnosability for modular discrete event systems

Modular discrete event systems are modeled as a parallel composition of finite automata. While deciding weak detectability, opacity, and A-diagnosability for monolithic systems is PSPACE-complete, the complexity for modular systems is unknown. We show that for modular systems the problems are EXPSPACE-complete, and hence there is neither a polynomial-time nor a polynomial-space algorithm solving them. While the upper bound is a natural modification of the PSPACE algorithms for monolithic systems, the lower bound requires a novel and nontrivial construction. We further discuss a case where the complexity drops to PSPACE-complete.

Development and Application of the Cauchy–Poisson Method to Layer Elastodynamics and the Timoshenko Equation

We consider a generalization of the Cauchy–Poisson method to an n-dimensional Euclidean space and its application to the construction of hyperbolic approximations. In Euclidean space, constraints on derivatives are introduced. The principle of hyperbolic degeneracy in terms of parameters is formulated and its implementation in the form of necessary and sufficient conditions is given. As the particular case of a four-dimensional space with preserving operators up to the sixth order a generalized hyperbolic equation is obtained for bending vibrations of plates with coefficients dependent only on the Poisson number. As special cases, this equation includes all the well-known Bernoulli–Euler, Kirchhoff, Rayleigh, and Timoshenko equations. As a development of Maxwell’s and Einstein’s research on the propagation of perturbations with finite velocity in a continuous medium, Tymoshenko’s non-trivial construction of the equation for bending vibrations of a beam is noted.

On the Square Subgroup of a MixedSI-Group

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

What prevents gravitational collapse in string theory?

It is conventionally believed that if a ball of matter of mass [Formula: see text] has a radius close to [Formula: see text][Formula: see text]GM then it must collapse to a black hole. But string theory microstates (fuzzballs) have no horizon or singularity, and they do not collapse. We consider two simple examples from classical gravity to illustrate how this violation of our intuition happens. In each case, the ‘matter’ arises from an extra compact dimension, but the topology of this extra dimension is not trivial. The pressure and density of this matter diverge at various points, but this is only an artifact of dimensional reduction; thus, we bypass results like Buchadahl’s theorem. Such microstates give the entropy of black holes, so these topologically nontrivial constructions dominate the state space of quantum gravity.

Second Generation of Aldol Reaction.

Since the discovery of the Mukaiyama aldol reaction more than 40 years ago, several landmark publications have inspired researchers in the field. The Mukaiyama AR is one of the most significant named reactions in organic synthesis. In the past few decades, development of the modern AR has been at the forefront in addressing the challenges of regio-, chemo-, diastereo-, and enantioselectivity in organic synthesis. All of these selectivity challenges maybe present in a single pair of reactants, thus controlling the outcome of such a process has great practical value. More than 10 years ago, our group became involved in this iconic carbon-carbon bond-forming process and attempted to very closely investigate all possible features of the AR to solve several issues still encountered by chemists, most notably the selectivity challenges mentioned above. In this context, our group initiated the second generation of the AR based on a Lewis or Brønsted acid-catalyzed process in conjunction with the use of a "super silyl" (tris(trimethylsilyl)silyl) directing group, which has demonstrated unrivalled properties in controlling the outcome of the AR. Using the extraordinary power of the super silyl group, we were able to develop new methods and concepts that broadly impacted the ability to control the selectivity attributes and thus allowed for a highly stereoselective construction of polyketide, halogenated polyketide, polypropionate, and polyol scaffolds through inter- and/or intramolecular aldolization protocols. Our diastereoselective ARs of super silyl enol ethers and aldehydes have shown great efficiency and modularity in producing exclusively and preferentially syn- or anti-adducts, creating up to four new adjacent stereocenters in a one-pot sequential manner and under mild reaction conditions. The super silyl-directed AR does not only provide a solution to stereochemistry control challenges, but also offers an efficient, modular and high yielding technique toward nontrivial construction of complex architectures with unprecedented ease. We believe that the new Lewis- or Brønsted-acid-catalyzed super-silyl-directed AR processes chronicled in our laboratories have come to maturity and now offer a "road map" for strategic stereoselective synthesis of polyketide-like units. Herein we report our recent achievements in the diastereoselective C-C bond formation, through the super-silyl-directed AR, toward the synthesis of complex and sophisticated hydroxy aldehydes. We would like to note that due to the extremely broad range of work reported in this field, only stereoselective AR involving aldehyde-derived super SEEs will be discussed in this Account.

A combinatorial characterization of smooth LTCs and applications

The study of locally testable codes (LTCs) has benefited from a number of nontrivial constructions discovered in recent years. Yet, we still lack a good understanding of what makes a linear error correcting code locally testable and as a result we do not know what is the rate-limit of LTCs and whether asymptotically good linear LTCs with constant query complexity exist. In this paper, we provide a combinatorial characterization of smooth locally testable codes, which are locally testable codes whose associated tester queries every bit of the tested word with equal probability. Our main contribution is a combinatorial property defined on the Tanner graph associated with the code tester (“well-structured tester”). We show that a family of codes is smoothly locally testable if and only if it has a well-structured tester. As a case study we show that the standard tester for the Hadamard code is “well-structured,” giving an alternative proof of the local testability of the Hadamard code, originally proved by (Blum, Luby, Rubinfeld, J. Comput. Syst. Sci. 47 (1993) 549–595) (STOC 1990). Additional connections to the works of (Ben-Sasson, Harsha, Raskhodnikova, SIAM J. Comput 35 (2005) 1–21) (SICOMP 2005) and of (Lachish, Newman and Shapira, Comput Complex 17 (2008) 70–93) are also discussed. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016

Simple and efficient local codes for distributed stable network construction

In this work, we study protocols so that populations of distributed processes can construct networks. In order to highlight the basic principles of distributed network construction, we keep the model minimal in all respects. In particular, we assume finite-state processes that all begin from the same initial state and all execute the same protocol. Moreover, we assume pairwise interactions between the processes that are scheduled by a fair adversary. In order to allow processes to construct networks, we let them activate and deactivate their pairwise connections. When two processes interact, the protocol takes as input the states of the processes and the state of their connection and updates all of them. Initially all connections are inactive and the goal is for the processes, after interacting and activating/deactivating connections for a while, to end up with a desired stable network. We give protocols (optimal in some cases) and lower bounds for several basic network construction problems such as spanning line, spanning ring, spanning star, and regular network. The expected time to convergence of our protocols is analyzed under a uniform random scheduler. Finally, we prove several universality results by presenting generic protocols that are capable of simulating a Turing Machine (TM) and exploiting it in order to construct a large class of networks. We additionally show how to partition the population into ksupernodes, each being a line of $$\log k$$logk nodes, for the largest such k. This amount of local memory is sufficient for the supernodes to obtain unique names and exploit their names and their memory to realize nontrivial constructions.

ChemInform Abstract: Controlling the Assembly of Cyclotriveratrylene‐Derived Coordination Cages

AbstractReview: 85 refs + subrefs.

Controlling the assembly of cyclotriveratrylene-derived coordination cages.

A review of the emerging field of cyclotriveratrylene-derived coordination cages is presented. Ligand-functionalised cyclotriveratrylene (CTV) derivatives self-assemble with a range of metal cations to afford coordination cages, polymers and topologically non-trivial constructs, such as [2]catenanes and a self-entangled cube. Increased control over their self-assembly allows for the controlled and predictable formation of well-defined coordination cages for application in host-guest and recognition chemistry, with surfactant binding and single-crystal-to-single-crystal (SCTSC) uptake of small-molecule guests being observed.

Idealized models for FEA derived from generative modeling processes based on extrusion primitives

Shape idealization transformations are very common operations when adapting a CAD component to FEA requirements. Here, an idealization approach is proposed that is based on generative shape processes used to decompose an initial B-Rep solid, i.e., extrusion processes with material addition are used to segment a solid. The corresponding extrusion primitives form the basis of candidate sub-domains for idealization and their connections conveyed through the generative processes they belong to, bringing robustness to set up the appropriate connections between idealized sub-domains. This is made possible because the connections between extrusion primitives have an explicit geometric representation and can be used to bound the connections between idealized sub-domains. Taking advantage of an existing construction tree as available in a CAD software does not help much because it may be complicated to use it for idealization processes because this tree structure is not unique. Using generative processes attached to an object that is no longer reduced to a single construction tree but to a graph containing all non-trivial construction trees, is more useful for the engineer to evaluate variants of idealization. From this automated decomposition, each primitive is subjected to a morphological analysis to define whether it can idealized or not. Subsequently, geometric interfaces between primitives form also a graph that can be used to process the connections between the idealized sub-domains generated from the primitives. These interfaces are taken into account to determine more precisely the idealizable sub-domains and their contours when primitives are incrementally merged to come back to produce the global morphological analysis of the initial object. A user-defined threshold is used to tune the morphological analysis with respect to further user parameters. Finally, the idealizable sub-domains and their connections are processed to locate the mid-surfaces and connect them using generic criteria that the user can tune locally using complementary criteria.

Diagnosis and opacity problems for infinite state systems modeled by recursive tile systems

The analysis of discrete event systems under partial observation is an important topic, with major applications such as the detection of information flow and the diagnosis of faulty behaviors. These questions have, mostly, not been addressed for classical models of recursive systems, such as pushdown systems and recursive state machines. In this paper, we consider recursive tile systems, which are recursive infinite systems generated by a finite collection of finite tiles, a simplified variant of deterministic graph grammars (slightly more general than pushdown systems). Since these systems are infinite-state in general powerset constructions for monitoring do not always apply. We exhibit computable conditions on recursive tile systems and present non-trivial constructions that yield effective computation of the monitors. We apply these results to the classic problems of state-based opacity and diagnosability (off-line verification of opacity and diagnosability, and also run-time monitoring of these properties). For a decidable subclass of recursive tile systems, we also establish the decidability of the problems of state-based opacity and diagnosability.

Smoothing Windows for the Synthesis of Gaussian Stationary Random Fields Using Circulant Matrix Embedding

When generating Gaussian stationary random fields, a standard method based on circulant matrix embedding usually fails because some of the associated eigenvalues are negative. The eigenvalues can be shown to be nonnegative in the limit of increasing sample size. Computationally feasible large sample sizes, however, rarely lead to nonnegative eigenvalues. Another solution is to extend suitably the covariance function of interest so that the eigenvalues of the embedded circulant matrix become nonnegative in theory. Though such extensions have been found for a number of examples of stationary fields, the method depends on nontrivial constructions in specific cases.In this work, the embedded circulant matrix is smoothed at the boundary by using a cutoff window or overlapping windows over a transition region. The windows are not specific to particular examples of stationary fields. The resulting method modifies the standard circulant embedding, and is easy to use. It is shown that this straightforward approach works for many examples of interest, with the overlapping windows performing consistently better. The method even outperforms in the cases where extending the covariance leads to nonnegative eigenvalues in theory, in the sense that the transition region is considerably smaller. The Matlab code implementing the method is included in the online supplementary materials and also publicly available at www.hermir.org.

Large sets of t-designs over finite fields

A t-(n,k,λ;q)-design is a set of k-dimensional subspaces, called blocks, of an n-dimensional vector space V over the finite field with q elements such that each t-dimensional subspace is contained in exactly λ blocks. A partition of the complete set of k-dimensional subspaces of V into disjoint t-(n,k,λ;q) designs is called a large set of t-designs over finite fields. In this paper we give the first nontrivial construction of such a large set with t⩾2.