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Synchronous reluctance motors are arousing lively interest as a possible alternative to the less efficient induction motors. An open issue is the effective tuning of the inner current loops because of the nonlinearity that cannot be overlooked. The present paper uses a relay feedback approach to perform autotuning without resorting to any parameter knowledge. The tuning is iterated at different working points, to get a uniform current control bandwidth everywhere. Unlike many solutions in the field, the algorithm is truly autonomous, in the sense that it also suggests a correct value for the bandwidth specification. The paper includes both simulation and experimental results, obtained on a laboratory prototype.

An important application of spectral decomposition (SD) is to identify subsurface geological anomalies such as channels and karst caves, which may be buried in full-band seismic data. However, the classical SD methods including the wavelet transform (WT) are often limited by relatively low time–frequency resolution, which is responsible for false high horizon-associated space resolution probably indicating more geological structures, especially when close geological anomalies exist. To address this issue, we impose a constraint of minimizing an lp (0 p l1-norm constraint, the modified ISD (MISD) using an lp-norm constraint can yield a more compact spectrum contributing to detect the distributions of close geological features. We design a 3D synthetic dataset involving frequency-close thin geological anomalies and the other 3D non-stationary dataset involving time-close anomalies to demonstrate the effectiveness of MISD. The application of 4D spectrum on a 3D real dataset with an area of approximately 230 km2 illustrates its potential for detecting deep channels and the karst slope fracture zone.

We study an iterated temporal and contemporaneous aggregation of N independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index α ∈ (0, 2). We show that limits of finite-dimensional distributions of appropriately centered and scaled aggregated partial-sum processes exist when first taking the limit as N → ∞and then the time scale n→ ∞. The limit process is an α-stable process if α ∈ (0, 1) ∪ (1, 2) and a deterministic line with slope 1 if α = 1.

High order stop-loss transforms provide a risk measure that enables some flexibility on the weight given to high or low values of the risk. We interpret stop-loss transforms as iterated distributions and prove a recursive representation for risks expressed as convolutions. We apply this to the case of gamma distributions with integer shape parameter, the Erlang distributions, proving that high order stop-loss transforms are equivalent to the tails of the exponential distribution. The latter result is also extended to general gamma distributions. Furthermore, we prove that this equivalence to exponential tails does not hold in general, by proving that the stop-loss transform for a Weilbull distribution degenerates, unless, of course, in the exponential case.

Previous studies of the Huckel-London-Pople-McWeeny (HLPM) ring-currents and bond-currents associated with the neutral and the dianionic altans, and iterated altans, of corannulene and coronene, are here extended to the hexaanions of (altan)n-corannulene and (altan)n-coronene, with n = 1, 2, and 3. The magnetic properties of some of these species were originally studied by Monaco and Zanasi using the ipso-centric approach, which is ab initio in nature. The "topological" HLPM method, by contrast, constitutes a parameter-free, quasi graph-theoretical model requiring knowledge only of the molecular graph of the conjugated system being considered and the areas assumed for its constituent rings. The HLPM calculations are partially presented by means of a novel illustrative device that we call "difference diagrams". One aim of this work is to compare predictions of this very rudimentary HLPM formalism with those of the much more sophisticated ipso-centric ab initio approach when they are both applied to the calculation of ring-current properties in this family of neutral and anionic (especially, hexaanionic) conjugated systems. For those altans and iterated altans for which ab initio calculations are available for comparison, encouraging qualitative, and even quantitative, agreement is found between the predictions of the ab initio and HLPM approaches. Other structures for which HLPM calculations are reported here but which have not yet been studied by ab initio means are offered with some confidence for possible subsequent evaluation using the ipso-centric (or any other ab initio) formalism.

Harvey Friedman’s gap condition on embeddings of finite labelled trees plays an important role in combinatorics (proof of the graph minor theorem) and mathematical logic (strong independence results). In the present paper we show that the gap condition can be reconstructed from a small number of well-motivated building blocks: It arises via iterated applications of a uniform Kruskal theorem.

The robust optimization based dynamic economic dispatch (DED) model has been extensively studied to address uncertainties of net loads induced by renewable energy resource variabilities and demand fluctuations. However, the robust DED model is computationally expensive, when solved by the column-and-constraint generation (CCG) approach that iterates between a master problem and a max-min subproblem. This letter proposes a feasible region projection-based approach to equivalently reformulate the robust DED as a single-level linear programming (LP) model that can be effectively solved while guaranteeing solution optimality. Numerical studies show the proposed approach is one order of magnitude faster than CCG.

A large series of studies on evolving agents to play mathematical games has demonstrated that many factors can significantly impact which agents arise, when those agents arise during evolution, and how robust they are in their play against other agents. Some or all of these factors have been shown to be relevant in the iterated prisoner's dilemma, the snowdrift game, and a fairly complex game called divide-the-dollar. This study demonstrates the impact or representation and agent resource allocation for a new game called coordination prisoner's dilemma. This paper demonstrates protocols from a recently published book for analysis of agent behavior and extends the work to another game, the first three-move game so treated. A new representation for agents playing mathematical games is introduced, a linear genetic programming register machine. New metrics for agent behavior including total exploitation, strategic variability, and action entropy are introduced. It is found that varying the representation and resource levels within a representation changes the types of game playing agents produced by evolution for coordination prisoner's dilemma.

On-line social networks, such as in Facebook and Twitter, are often studied from the perspective of friendship ties between agents in the network. Adversarial ties, however, also play an important role in the structure and function of social networks, but are often hidden. Underlying generative mechanisms of social networks are predicted by structural balance theory, which postulates that triads of agents, prefer to be transitive, where friends of friends are more likely friends, or anti-transitive, where adversaries of adversaries become friends. The previously proposed Iterated Local Transitivity (ILT) and Iterated Local Anti-Transitivity (ILAT) models incorporated transitivity and anti-transitivity, respectively, as evolutionary mechanisms. These models resulted in graphs with many observable properties of social networks, such as low diameter, high clustering, and densification. We propose a new, generative model, referred to as the Iterated Local Model (ILM) for social networks synthesizing both transitive and anti-transitive triads over time. In ILM, we are given a countably infinite binary sequence as input, and that sequence determines whether we apply a transitive or an anti-transitive step. The resulting model exhibits many properties of complex networks observed in the ILT and ILAT models. In particular, for any input binary sequence, we show that asymptotically the model generates finite graphs that densify, have clustering coefficient bounded away from 0, have diameter at most 3, and exhibit bad spectral expansion. We also give a thorough analysis of the chromatic number, domination number, Hamiltonicity, and isomorphism types of induced subgraphs of ILM graphs.

A new characterization of $\text {CMO}(\mathbb R^n)$ is established replying upon local mean oscillations. Some characterizations of iterated compact commutators on weighted Lebesgue spaces are given, which are new even in the unweighted setting for the first order commutators.

We present an extension of the proximal point method with Bregman distances to solve variational inequality problems (VIP) on Hadamard manifolds with null sectional curvature. Under some natural assumptions, as for example, the existence of solutions of the VIP and the monotonicity of the multivalued vector field, we prove that the sequence of the iterates given by the method converges to a solution of the problem. Furthermore, this convergence is linear or superlinear with respect to the Bregman distance.

A filter proximal bundle algorithm is presented for nonsmooth nonconvex constrained optimization problems. The new algorithm is based on the proximal bundle method and utilizes the improvement function to regularize the constraint. At every iteration by solving a convex piecewise-linear subproblem a trial point is obtained. The process of the filter technique is employed either to accept the trial point as a serious iterate or to reject it as a null iterate. Under some mild and standard assumptions and for every possible choice of a starting point, it is shown that every accumulation point of the sequence of serious iterates is feasible. In addition, there exists at least one accumulation point which is stationary for the improvement function. Finally, some encouraging numerical results show that the proposed algorithm is effective.

In this article, we consider the problem of finding zeros of monotone inclusions of three operators in real Hilbert spaces, where the first operator’s inverse is strongly monotone and the third is linearly composed, and we suggest an extended splitting method. This method allows relative errors and is capable of decoupling the third operator from linear composition operator well. At each iteration, the first operator can be processed with just a single forward step, and the other two need individual computations of the resolvents. If the first operator vanishes and linear composition operator is the identity one, then it coincides with a known method. Under the weakest possible conditions, we prove its weak convergence of the generated primal sequence of the iterates by developing a more self-contained and less convoluted techniques. Our suggested method contains one parameter. When it is taken to be either zero or two, our suggested method has interesting relations to existing methods. Furthermore, we did numerical experiments to confirm its efficiency and robustness, compared with other state-of-the-art methods.

Optimization methods for difference-of-convex programs iteratively solve convex subproblems to define iterates. Although convex, depending on the problem’s structure, these subproblems are very often challenging and require specialized solvers. This work investigates a new methodology that defines iterates as approximate critical points of significantly easier difference-of-convex subproblems approximating the original one. Since there is considerable freedom to choose such more accessible subproblems, several algorithms can be designed from the given approach. In some cases, the resulting algorithm boils down to a straightforward process with iterates given in an analytic form. In other situations, decomposable subproblems can be chosen, opening the way for parallel computing even when the original program is not decomposable. Depending on the problem’s assumptions, a possible variant of the given approach is the Josephy–Newton method applied to the system of (necessary) optimality conditions of the original difference-of-convex program. In such a setting, local convergence with superlinear and even quadratic rates can be achieved under certain conditions.

Effect handlers encourage programmers to abstract over repeated patterns of complex control flow. As of today, this abstraction comes at a significant price in performance. In this paper, we aim to achieve abstraction without regret for effect handlers. We present a language for effect handlers in _capability-passing style_ (λ Cap ) and an implementation of this language as a translation to simply-typed lambda calculus in _iterated continuation-passing style_. A suite of benchmarks indicates that the novel combination of capability-passing style and iterated CPS enables significant speedups over existing languages with effect handlers or control operators. Our implementation technique is general and allows us to generate code in any language that supports first-class functions. We then identify a subset of programs for which we can further improve the performance and guarantee full elimination of the effect handler abstraction. To formally capture this subset, we refine λ Cap to λ λ Cap with a more restrictive type system. We present a type-directed translation for λ λ Cap that inserts staging annotations and prove that no abstractions or applications related to effect handlers occur in the translated program. Using this second translation we observe additional speedups in some of the benchmarks.

Key predistribution is an adequate technique to solve the (symmetric) key management problem in distributed systems. Owing to the large network size N and limited memory (z) in devices, a given key is shared by a number of nodes (r). This leads to a generic degradation of a network's resilience. Several works dedicate to reduce this deterioration by applications of lightweight mechanisms. Few such prominent approaches are the q-composite method by Chan et al., hashed random preloaded subset by Ramkumar et al. and one-way function iterates by Cichon et al. or its special case of hash chains (HCs) by Bechkit et al. The authors introduce the generic concepts of bidirectional HCs (2HCs) and q-composite 2HCs. A thorough qualitative investigations of vital system parameters indicate trade-offs between significant improvement in a network's resilience and nominal increment in space and computational overheads in constituent devices. In the process, they correct an error in resilience computations in the works of Bechkit et al. Instantiations of their indigenous concepts to three prominent schemes that represent individual types of key predistribution schemes (one random and two deterministic) concretely quantify their analyses.

We consider the transformations of independent random variables over a linearly ordered finite set (a chain) by the join and meet operations. We investigate the possibility of approximating an arbitrary probability distribution on a chain by means of a (possibly iterated) application of the join and meet operations to independent random variables with distributions from a given set. We establish some conditions under which the approximation is impossible and the conditions when it becomes possible.

Physical knots observed in various contexts – from DNA biology to vortex dynamics and condensed matter physics – are found to undergo topological simplification through iterated recombination of knot strands following a common, qualitative pattern that bears remarkable similarities across fields. Here, by interpreting evolutionary processes as geodesic flows in a suitably defined knot polynomial space, we show that a new measure of topological complexity allows accurate quantification of the probability of decay pathways by selecting the optimal unlinking pathways. We also show that these optimal pathways are captured by a logarithmic best-fit curve related to the distribution of minimum energy states of tight knots. This preliminary approach shows great potential for establishing new relations between topological simplification pathways and energy cascade processes in nature. What makes a given knot's unlinking pathway more likely to appear in certain topological decay processes—from DNA to vortex dynamics? Here, the authors introduce a measure of topological complexity as distance in a suitably defined knot polynomial space, allowing them to quantify the probability of decay pathways as geodesic flows in this space.

Let ( $${\cal X}$$ , d,μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderon-Zygmund operator and $$\vec b: = ({b_1}, \ldots ,{b_m})$$ be a finite family of $$\widetilde{{\rm{RBMO}}}(\mu)$$ functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator $${{\rm{T}}_{\Pi \vec b}}$$ generated by T and $$\vec b$$ are obtained.

The Fisher information matrix related to an order statistic and its concomitant used to order a bivariate random sample are obtained in the case of the shape-parameter vector of an iterated Farlie–...

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.

We study the existence of limiting laws of rare events corresponding to the entrance of the orbits on certain target sets in the phase space. The limiting laws are obtained when the target sets shrink to a Cantor set of zero Lebesgue measure. We consider both the presence and absence of clustering, which is detected by the Extremal Index, which turns out to be very useful to identify the compatibility between the dynamics and the fractal structure of the limiting Cantor set. The computation of the Extremal Index is connected to the box dimension of the intersection between the Cantor set and its iterates.

We study the dynamics of a geometric piecewise map defined on the set of three pairwise nonparallel, nonconcurrent lines in R2. The geometric map of study may be analogized to the billiard map with...

We consider the phase retrieval problem, where the goal is to reconstruct an $n$ -dimensional complex vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, if ${m}\geq Mn^{3/2}\log ^{7/2}n$ for some $M>0$ , then the classical algorithm of alternating minimization with random initialization succeeds with high probability as $n,m\rightarrow \infty $ . This is a step toward proving the conjecture in, which conjectures that the algorithm succeeds when $m=O(n)$ . The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.

The affine subdivision of a simplex $\Delta$ is a certain collection of $(n+1)!$ smaller $n$-simplices whose union is $\Delta$. Barycentric subdivision is a well know example of affine subdivision(see ). Richard Schwartz(2003) proved that the infinite process of iterated barycentric subdivision on a tetrahedron produces a dense set of shapes of smaller tetrahedra. We prove that the infinite iteration of several kinds of affine subdivision on a tetrahedron produce dense sets of shapes of smaller tetrahedra, respectively.

Improving and extending some ideas of Gottlob Frege from 1874 (on a generalization of the notion of the composition iterates of a function), we consider the composition iterates ϕn of a relation ϕ on X, defined by ϕ 0 = Δ x , ϕ n = ϕ ∘ ϕ n - 1 if n ∈ 𝕅 , and ϕ ∞ = ∪ n = 0 ∞ ϕ n . {\varphi ^0} = {\Delta _x},\,\,{\varphi ^n} = \varphi \circ {\varphi ^{n - 1}}{\rm{ if n}} \in \mathbb{N,}\,\,{\rm{and }}\,\,{\varphi ^\infty } = \bigcup\limits_{n = 0}^\infty {{\varphi ^n}} . In particular, by using the relational inclusion ϕn ◦ϕm ⊆ ϕn+m with n, m ∈ 𝕅 ¯ 0 \mathbb{\bar {N}_0}} , we show that the function α, defined by α ( n ) = ϕ n for n ∈ 𝕅 ¯ 0 , \alpha \left( n \right) = {\varphi ^{\rm{n}}}\,\,\,{\rm{for n}} \in {{\rm\mathbb{\bar N}}_{\rm{0}}}, satisfies the Cauchy problem α ( n ) ∘ α ( m ) ⊆ α ( n + m ) , α ( 0 ) = Δ x . \alpha \left( n \right) \circ \alpha \left( {\rm{m}} \right) \subseteq \alpha \left( {{\rm{n}} + {\rm{m}}} \right),\,\,\,\alpha \left( 0 \right) = {\Delta _{\rm{x}}}. Moreover, the function f, defined by f ( n , A ) = α ( n ) [ A ] for n ∈ 𝕅 ¯ 0 and A ⊆ X , {\rm{f}}\left( {{\rm{n}},{\rm{A}}} \right) = \alpha \left( {\rm{n}} \right)\left[ {\rm{A}} \right]\,\,\,{\rm{for}}\,{\rm{n}} \in {{\rm\mathbb{\bar {N}}}_{\rm{0}}}\,\,{\rm{and}}\,{\rm{A}} \subseteq {\rm{X,}} satisfies the translation problem f ( n , f ( m , A) ) ⊆ f ( n + m , A ) , f ( 0 , A ) = A . {\rm{f}}\left( {{\rm{n}},f(m,{\rm{A)}}} \right) \subseteq {\rm{f}}\left( {{\rm{n}} + {\rm{m,A}}} \right),\,\,\,{\rm{f}}\left( {0,{\rm{A}}} \right) = {\rm{A}}{\rm{.}} Furthermore, the function F, defined by F ( A , B ) = { n ∈ 𝕅 ¯ 0 : A ⊆ f ( n , B ) } for A , B ⊆ X , {\rm{F}}\left( {{\rm{A}},{\rm{B}}} \right) = \left\{ {{\rm{n}} \in {{{\rm\mathbb{\bar {N}}}}_{\rm{0}}}:\,\,{\rm{A}} \subseteq {\rm{f}}\left( {{\rm{n}},{\rm{B}}} \right)} \right\}\,\,{\rm{for}}\,\,{\rm{A,B}} \subseteq {\rm{X,}} satisfies the Sincov problem F ( A , B ) + F ( B , C ) ⊆ F ( A , C ) , 0 ∈ F ( A , A ) . {\rm{F}}\left( {{\rm{A}},{\rm{B}}} \right) + {\rm{F}}\left( {{\rm{B}},{\rm{C}}} \right) \subseteq {\rm{F}}\left( {{\rm{A,C}}} \right),\,\,\,\,0 \in {\rm{F}}\left( {{\rm{A}},{\rm{A}}} \right). Motivated by the above observations, we investigate a function F on the product set X2 to the power groupoid 𝒫(U) of an additively written groupoid U which is supertriangular in the sense that F ( x , y ) + F ( y , z ) ⊆ F ( x , z ) {\rm{F}}\left( {{\rm{x}},{\rm{y}}} \right) + {\rm{F}}\left( {{\rm{y}},{\rm{z}}} \right) \subseteq {\rm{F}}\left( {{\rm{x}},{\rm{z}}} \right) for all x, y, z ∈ X. For this, we introduce the convenient notations R ( x , y ) = F ( y , x ) and S ( x , y ) = F ( x , y ) + R ( x , y ) , {\rm{R}}\left( {{\rm{x}},{\rm{y}}} \right) = {\rm{F}}\left( {{\rm{y}},{\rm{x}}} \right)\,\,\,{\rm{and}}\,\,{\rm{S}}\left( {{\rm{x}},{\rm{y}}} \right) = {\rm{F}}\left( {{\rm{x}},{\rm{y}}} \right) + {\rm{R}}\left( {{\rm{x}},{\rm{y}}} \right), and Φ ( x ) = F ( x , x ) and Ψ ( x ) ∪ y ∈ X S ( x , y ) . \Phi \left( {\rm{x}} \right) = {\rm{F}}\left( {{\rm{x}},{\rm{x}}} \right)\,\,{\rm{and}}\,\,\Psi \left( {\rm{x}} \right)\bigcup\limits_{{\rm{y}} \in {\rm{X}}} {{\rm{S}}\left( {{\rm{x}},{\rm{y}}} \right).} Moreover, we gradually assume that U and F have some useful additional properties. For instance, U has a zero, U is a group, U is commutative, U is cancellative, or U has a suitable distance function; while F is nonpartial, F is symmetric, skew symmetric, or single-valued.

At first, we construct a connection between the Atangana–Baleanu and the Riemann–Liouville fractional integrals of a function with respect to a monotone function with nonsingular kernel. By examining this relationship and the iterated form of Prabhakar fractional model, we are able to find some new Hermite–Hadamard inequalities and related results on integral inequalities for the two models of fractional calculus which are defined using monotone functions with nonsingular kernels.

This paper is devoted to generalizations of Fubini theorem, Transformation theorem, and generalized Minkowski inequality for the so called pseudo-integral. The approach is based on the relation of double pseudo-integrals and iterated pseudo-integrals. Since the pseudo-integral covers Lebesgue integral, Sugeno's fuzzy integral, and Zhao's (N)fuzzy integral (with a respect to special nonadditive measure: ∨-measure), the obtained results generalize the corresponding previously known results.

This paper is part of the investigation of some operations on words and languages with motivations coming from DNA biochemistry, namely three variants of hairpin completion and three variants of hairpin reduction. Since not all the hairpin completions or reductions of semilinear languages remain semilinear, we study sufficient conditions for semilinear languages to preserve their semilinearity property after applying the non-iterated hairpin completion or hairpin reduction. A similar approach is then applied to the iterated variants of these operations. Along these lines, we define the hairpin reduction root of a language and show that the hairpin reduction root of a semilinear language is not necessarily semilinear except the universal language. A few open problems are finally discussed.

This paper is devoted to the study of the operators L having the property that there exists $$c\in [0,1)$$ with $$|Lf|_{Lip}\le c|f|_{Lip}$$ . A class of convolution type operators acting on spaces of continuous periodic functions for which there exists the above mentioned constant c is described. We investigate the asymptotic behaviour of the iterates of such operators and give rates of convergence of the iterates toward the limit. Some examples involving the convolution operators of de la Vallee Poussin, Jackson and Fejer–Korovkin are presented. The well-known connection between de la Vallee Poussin operators and Durrmeyer operators with Chebyshev weights is used in order to obtain new estimates of the rates of convergence of the iterates of these operators.

The purpose of this work is to advance the current state of mathematical knowledge regarding fixed point theorems of functions. Such ideas have historically enjoyed many applications, for example, to the qualitative and quantitative understanding of differential, difference and integral equations. Herein, we extend an established result due to Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42] that involves two metrics to ensure wider classes of functions admit a unique fixed point. In contrast to the literature, a key strategy herein involves placing assumptions on the iterations of the function under consideration, rather than on the function itself. In taking this approach we form new advances in fixed point theory under two metrics and establish interesting connections between previously distinct theorems, including those of Rus [Studia Univ. Babes-Bolyai Math., 22, 1977, 40–42], Caccioppoli [Rend. Acad. Naz. Linzei. 11, 1930, 31–49] and Bryant [Am. Math. Month. 75, 1968, 399–400]. Our results make progress towards a fuller theory of fixed points of functions under two metrics. Our work lays the foundations for others to potentially explore applications of our new results to form existence and uniqueness of solutions to boundary value problems, integral equations and initial value problems. References Almuthaybiri, S. S. and C. C. Tisdell. ``Global existence theory for fractional differential equations: New advances via continuation methods for contractive maps''. Analysis, 39(4):117–128, 2019. doi:10.1515/anly-2019-0027 Almuthaybiri, S. S. and C. C. Tisdell. ``Sharper existence and uniqueness results for solutions to third-order boundary value problems, mathematical modelling and analysis''. Math. Model. Anal. 25(3):409–420, 2020. doi:10.3846/mma.2020.11043 Banach, S. ``Sur les operations dans les ensembles abstraits et leur application aux equations integrales''. Fund. Math., 3:133–181 1922. doi:10.4064/fm-3-1-133-181 Brouwer, L. E. J. ``Ueber Abbildungen von Mannigfaltigkeiten''. Math. Ann. 71:598, 1912. doi:10.1007/BF01456812 Bryant, V. W. ``A remark on a fixed point theorem for iterated mappings'' Am. Math. Month. 75: 399–400, 1968. doi:10.2307/2313440 Caccioppoli, R. ``Un teorema generale sullesistenza de elemente uniti in una transformazione funzionale''. Rend. Acad. Naz. Linzei. 11:31–49, 1930. Goebel, K., and W. A. Kirk. Topics in metric fixed point theory. Cambridge University Press, 1990, doi:10.1017/CBO9780511526152 Leray, J., and J. Schauder. ``Topologie et equations fonctionnelles''. Ann. Sci. Ecole Norm. Sup. 51:45–78, 1934. doi:10.24033/asens.836 O'Regan, D. and R. Precup. Theorems of Leray–Schauder type and applications, Series in Mathematical Analysis and Applications, Vol. 3. CRC Press, London, 2002. doi:10.1201/9781420022209 Rus, I. A. ``On a fixed point theorem of Maia''. Studia Univ. Babes-Bolyai Math. 22:40–42, 1977. Schaefer, H. H. ``Ueber die Methode der a priori-Schranken''. Math. Ann. 129:415–416, 1955. doi:10.1007/bf01362380 Tisdell, C. C. ``When do fractional differential equations have solutions that are bounded by the Mittag-Leffler function?'' Fract. Calc. Appl. Anal. 18(3):642–650, 2015. doi:10.1515/fca-2015-0039 Tisdell, C. C. ``A note on improved contraction methods for discrete boundary value problems.'' J. Diff. Eq. Appl. 18(10):1773–1777, 2012. doi:10.1080/10236198.2012.681781 Tisdell, C. C. ``On the application of sequential and fixed-point methods to fractional differential equations of arbitrary order.'' J. Int. Eq. Appl. 24(2):283–319, 2012. doi:10.1216/JIE-2012-24-2-283 Ehrnstrom, M., Tisdell, C. C. and E. Wahlen. ``Asymptotic integration of second-order nonlinear difference equations.'' Glasg. Math. J. 53(2):223–243, 2011. doi:10.1017/S0017089510000650 Erbe, L., A. Peterson and C. C. Tisdell. ``Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales.'' Nonlin. Anal. 69(7):2303–2317, 2008. doi:10.1016/j.na.2007.08.010 Tisdell, C. C. and A. Zaidi. ``Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling.'' Nonlin. Anal. 68(11):3504–3524, 2008. doi:10.1016/j.na.2007.03.043 Tisdell, C. C. ``Improved pedagogy for linear differential equations by reconsidering how we measure the size of solutions.'' Int.. J. Math. Ed. Sci. Tech. 48(7):1087–1095, 2017. doi:10.1080/0020739X.2017.1298856 Tisdell, C. C. ``On Picard's iteration method to solve differential equations and a pedagogical space for otherness.'' Int. J. Math. Ed. Sci. Tech. 50(5):788–799, 2019. doi:10.1080/0020739X.2018.1507051 Zeidler, E. Nonlinear functional analysis and its applications. Springer-Verlag, New York, 1986. doi:10.1007/978-1-4612-4838-5

ARAL, Ali/0000-0002-2024-8607 WOS:000544789400001 The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present some qualitative and quantitative results concerning the approximation by such Boolean sums. The general results are illustrated by examples. Scientific and Technological Research Council of Turkey, "2221-Fellowships for Visiting Scientists and Scientists on Sabbatical Leave."; TUBITAK (The Scientific and Technological Research Council of Turkey) 1002-ProjectTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [119F191] The work was done jointly, while the third author visited Selcuk University during February 2019, supported by The Scientific and Technological Research Council of Turkey, "2221-Fellowships for Visiting Scientists and Scientists on Sabbatical Leave." The first author has been partially supported within TUBITAK (The Scientific and Technological Research Council of Turkey) 1002-Project 119F191.

We consider a large class of so-called dynamical Belyi maps and study the Galois groups of iterates of such maps. From the combinatorial invariants of the maps, we construct a useful presentation of the geometric Galois groups as subgroups of automorphism groups of regular trees, in terms of iterated wreath products. Using results on the reduction of dynamical Belyi maps modulo certain primes, we obtain results on the corresponding arithmetic Galois groups of iterates. These lead to results on the behavior of the arithmetic Galois groups under specialization, with applications to dynamical sequences.

The multi-projection methods for solving the Fredholm-Hammerstein integral equation is proposed in this paper. We obtain the similar super-convergence results as in Mandal and Nelakanti (J Comput Appl Math 319:423–439, 2017) with a smooth kernel using piecewise polynomials of degree $$\le r-1,$$ i.e., for both the multi-Galerkin and multi-collocation methods have order of convergence $$\mathcal O (h^{3r})$$ in uniform norm, where h is the norm of the partition. We have also considered iterated version of these methods and prove that both iterated multi-Galerkin and iterated multi-collocation methods have order of convergence $$\mathcal O(h^{4r})$$ in uniform norm. Numerical examples are given to illustrate the theoretical results.

Researchers are increasingly interested in using sensor technology to collect accurate activity information and make individualized inference about treatments, exposures, and policies. How to optimally combine population data with data from an individual remains an open question. Multisource exchangeability models (MEMs) are a Bayesian approach for increasing precision by combining potentially heterogeneous supplemental data sources into analysis of a primary source. MEMs are a potentially powerful tool for individualized inference but can integrate only a few sources; their model space grows exponentially, making them intractable for high-dimensional applications. We propose iterated MEMs (iMEMs), which identify a subset of the most exchangeable sources prior to fitting a MEM model. iMEM complexity scales linearly with the number of sources, and iMEMs greatly increase precision while maintaining desirable asymptotic and small sample properties. We apply iMEMs to individual-level behavior and emotion data from a smartphone app and show that they achieve individualized inference with up to 99% efficiency gain relative to standard analyses that do not borrow information.

The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.

We consider in this paper the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the real line with a sufficiently smooth kernels, using Hermite polynomials as basis functions. We obtain optimal convergence results in iterated Galerkin method in weighted L 2 − norm. We also discuss multi-Galerkin methods and we obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods. Numerical results are presented to confirm the theoretical results.

The Newton-Kantorovich/Radau pseudospectral method is implemented in a homotopy scheme to solve the rigid spacecraft time-optimal three-axis reorientation problem. First, the reorientation differential equations associated with the current homotopic variable are linearized about the nominal (previous homotopic variable) solution. Then, the Radau pseudospectral method is used to solve the resulting linearized differential equations. By successively updating the nominal solution with the Newton-Kantorovich approach, the linearized solving procedure is iterated to achieve high-order precision. Because no numerical integration is needed and only linear algebraic equations are solved, the proposed method is computationally efficient. Finally, performance of the proposed method is illustrated through numerical examples.

Barndorff-Nielsen and Shephard (2001) proposed a class of stochastic volatility models in which the volatility follows the Ornstein–Uhlenbeck process driven by a positive Levy process without the Gaussian component. The parameter estimation of these models is challenging because the likelihood function is not available in a closed-form expression. A large number of estimation techniques have been proposed, mainly based on Bayesian inference. The main aim of the paper is to present an application of iterated filtering for parameter estimation of such models. Iterated filtering is a method for maximum likelihood inference based on a series of filtering operations, which provide a sequence of parameter estimates that converges to the maximum likelihood estimate. An application to S&P500 index data shows the model perform well and diagnostic plots for iterated filtering ensure convergence iterated filtering to maximum likelihood estimates. Empirical application is accompanied by a simulation study that confirms the validity of the approach in the case of Barndorff-Nielsen and Shephard’s stochastic volatility models.

Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving $$m+1$$ rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality ( $$R\infty AR$$ ), might not be satisfied by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy $$R\infty AR$$ . One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen’s technique of forcing.

Background: This study presents a novel method of constructing a spatiotemporal statistical shape model (st-SSM) for adult brain. St-SSM is an extension of statistical shape model (SSM) in the temporal domain which will represent the statistical variability of shape as well as the temporal change of statistical variance with respect to time. Aims: Expectation-Maximization (EM) based weighted principal component analysis (WPCA) using a temporal weight function is applied where the eigenvalues of each data are estimated by Estep using temporal eigenvectors, and M-step updates Eigenvectors in order to maximize the variance. Both E and M-step are iterated until updating vectors reaches the convergence point. A weight parameter for each subject is allocated in accordance with the subject’s age to calculate the weighted variance. A Gaussian function is utilized to define the weight function. The center of the function is a time point while the variance is a predefined parameter. Methods: The proposed method constructs adult brain st-SSM by changing the time point between minimum to maximum age range with a small interval. Here, the eigenvectors changes with aging. The feature vector of representing adult brain shape is extracted through a level set algorithm. To validate the method, this study employed 103 adult subjects (age: 22 to 93 y.o. with Mean ± SD = 59.32±16.89) from OASIS database. st-SSM was constructed for time point 40 to 90 with a step of 2. Results: We calculated the temporal deformation change between two-time points and evaluated the corresponding difference to investigate the influence of analysis parameter. An application of the proposed model is also introduced which involves Alzheimer’s disease (AD) identification utilizing support vector machine. Conclusion: In this study, st-SSM based adult brain shape feature extraction and classification techniques are introduced to classify between normal and AD subject as an application.

The purpose of this paper is to explore the question of embedding a zero-divisor graph and its iterated line graphs on the plane such that the resulting graphs are generalized outerplanar graphs. We give a full characterization of all zero-divisor graphs with respect to their generalized outerplanar index.

We give a criterion to determine when two external rays land at the same point for polynomials with locally connected Julia sets. As an application, we provide an elementary proof of the monotonicity of the core entropy along arbitrary veins of the Mandelbrot set., Comment: 22 pages, 2 figures, writing polished, results unchanged. To appear in Transactions of the American Mathematical Society

We define a cautious version of extensive-form rationalizability for generalized extensive-form games with unawareness that we call prudent rationalizability. It is an extensive-form analog of iterated admissibility. In each round of the procedure, for each tree and each information set of a player a surviving strategy of hers is required to be rational vis-a-vis a belief system with a full-support belief on the opponents' previously surviving strategies that reach that information set. We demonstrate the applicability of prudent rationalizability. In games of disclosure of verifiable information, we show that prudent rationalizability yields unraveling under full awareness but unraveling might fail under unawareness. We compare prudent rationalizability to extensive-form rationalizability. We show that prudent rationalizability may not refine extensive-form rationalizability strategies but conjecture that the paths induced by prudent rationalizable strategy profiles (weakly) refine the set of paths induced by extensive-form rationalizable strategies.

This paper investigates an evolutionary iterated prisoner’s dilemma (IPD) model of multiple agents, in which agents interact in terms of the pair-wise IPD game while adapting their attitudes towards income stream risk. Specifically, agents will become more risk averse (or more risk seeking) if their game payoffs exceed (or fall below) their expectations. In particular, agents use their peers’ average payoffs as expectations (social comparison) when their payoffs are lower than their peers’ averages, but use their own historical payoffs as expectations (historical comparison) when their payoffs are higher than their peers’ averages. Such selective attention to social comparison or historical comparison manifests a desire for continuous improvement of agents. Simulations are conducted to investigate the evolution of cooperation under the selective attention mechanism. Results indicate that agents can sustain a highly cooperative equilibrium when they consider selective attention in adjusting their risk attitudes. This holds true for both the well-mixed and the network-based games, even in the presence of uncertain game payoffs. The reason is that, selective attention can significantly induce agents to adhere to conditional cooperation as well as to identify uncertainty in payoffs, which enhances the risk-averse behavior of agents in the IPD game. As a result, high levels of cooperation can be attained.

Uncertainty quantification in the context of seismic imaging is important for interpretinginverted subsurface models and updating reservoir models. The limited illumination, noisydata and poor initial model in the seismic full waveform inversion (FWI) lead to inversionuncertainties. This is particularly true for anisotropic elastic FWI, which suffers from extra parameter trade-off problems. In this work, we address the uncertainty quantificationof anisotropic elastic FWI problem in the framework of Bayesian inference. Specially, weestimate the uncertainties of the subsurface elastic parameters in the Bayesian anisotropicelastic FWI by combining the iterated extended Kalman filter with an explicit representation of the sensitivity matrix with Green’s functions. The sensitivity matrix is based onthe integral equation approach, which is also within the context of nonlinear inverse scattering theory. We give the results of numerical tests with examples for anisotropic elasticmedia. They show that the proposed Bayesian inversion method can provide reasonablereconstructed results for the elastic coefficients of the stiffness tensor and the framework issuitable for accessing the uncertainties.

The level-k literature classifies subjects into different Lk types by their observed levels of reasoning in games. However, it remains unclear whether the observed level-k behavior is determined by belief or reasoning ability. This study proposes a strategy to identify the ability-bounded Lk subjects, who could not reason more than k steps of iterated best responses and thus have reached the upper bounds of their reasoning capacity. The identification utilizes a combination of simultaneous and sequential ring games. In the sequential games, it requires an extra step to best respond as Lk, and thus the ability-bounded ones would fail the task. Results show that more than half of the L2 and L3 subjects are ability-bounded. Additionally, subjects’ CRT scores, a measure of their cognitive ability, support the separation of the ability-bounded types. The findings suggest that both belief and reasoning ability could determine the observed levels, and thus one must be cautious when trying to infer belief or ability from the existing level-k data.

In this paper, we study the Gelfand–Cetlin systems and polytopes of the co-adjoint SO(n)-orbits. We describe the face structure of Gelfand–Cetlin polytopes and iterated bundle structure of Gelfand–Cetlin fibers in terms of combinatorics on the ladder diagrams. Using this description, we classify all Lagrangian fibers.