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In this paper, we prove some q-numerical radius power inequalities for a product of operators on a complex Hilbert space. We introduce also the notion of the q-center for bounded operators, and we give the relationship between this q-center, the q-numerical radius and the center of the q -numerical range. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder's fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the problem of stabilization of a degenerate parabolic equation with a Dirichlet control, evolving in bounded domain 퓞 ⊂ ℝd, d ≥ 2. We apply the proportional control design technique based on the spectrum of the linear operator which governs the evolution equation. The stabilizing feedback control, we design here, is linear, of finite-dimensional structure, easily manageable from the computational point of view. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some Korovkin type approximation theorems for double sequences of positive linear operators on two-dimensional weighted spaces via statistical type convergence method with respect to power series method. Additionally, we calculate the rate of convergence. As an application, we provide an approximation using the generalization of Gadjiev-Ibragimov operators for Pp-statistical convergence. Our results are meaningful and stronger than those previously given for two-dimensional weighted spaces. [ABSTRACT FROM AUTHOR]

This study takes a detailed look at various inequalities related to the Euclidean operator radius. It examines groups of n-tuple operators, studying how they add up and multiply together. It also uncovers a unique power inequality specific to the Euclidean operator radius. The research broadens its scope to analyze how n-tuple operators, when used as parts of 2 × 2 operator matrices, illustrate inequalities connected to the Euclidean operator radius. By using the Euclidean numerical radius and Euclidean operator norm for n-tuple operators, the study introduces a range of new inequalities. These inequalities not only set limits for the addition, multiplication, and Euclidean numerical radius of n-tuple operators but also help in establishing inequalities for the Euclidean operator radius. This process involves carefully examining the Euclidean numerical radius inequalities of 2 × 2 operator matrices with n-tuple operators. Additionally, a new inequality is derived, focusing specifically on the Euclidean operator norm of 2 × 2 operator matrices. Throughout, the research keeps circling back to the idea of finding and understanding symmetries in linear operators and matrices. The paper highlights the significance of symmetry in mathematics and its impact on various mathematical areas. [ABSTRACT FROM AUTHOR]

In this paper, a modification of general linear positive operators introduced by Ibragimov and Gadjiev in 1970 is constructed. It is shown that this modification preserves exponential mappings and also contains modified Bernstein-, Szász- and Baskakov-type operators as special cases. The convergence properties of corresponding operators on [ 0 , ∞) and in exponentially weighted spaces are investigated. Finally, the quantitative Voronovskaja theorem in terms of modulus of continuity for functions having exponential growth is examined. [ABSTRACT FROM AUTHOR]

In this paper, a new general approach is developed to construct and study Lebesgue‐type decompositions of linear operators or relations T$T$ in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue‐type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of T$T$. The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue‐type decomposition for pairs of quadratic forms. The approach built in this paper uses so‐called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak. [ABSTRACT FROM AUTHOR]

The conventional approach of the local convergence analysis of an iterative method on R m , with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in R m spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects. [ABSTRACT FROM AUTHOR]

In this paper, we introduced double cyclic codes over Rr × Rs, where R = Z2 + uZ2 = {0, 1, u, 1 + u} is the ring with four elements and u² = 0. We first determined the generator polynomials of R-double cyclic codes for odd integers r and s, then gave the generators of duals of free double cyclic codes over Rr × Rs. By defining a linear Gray map, we looked at the binary images of R-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from R-double cyclic codes. Moreover, we studied self-dual R-double cyclic codes and presented an example of a self-dual R-double cyclic code. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of three-composite nonconvex nonsmooth optimization problems, where one of nonsmooth functions is further composed with linear operator. Based on the variable smoothing method, as well as first-order methods with suitable majorization techniques, we propose a proximal variable smoothing gradient (ProxVSG) method for solving this kind of problem. The ProxVSG can be implemented efficiently, thanks to the fact that at each iteration, one just separately computes the proximal mapping of each nonsmooth function, rather than that of the sum of these nonsmooth functions. Furthermore, within our broad and flexible analysis framework, we propose a new proximal variable smoothing incremental aggregated gradient (ProxVSIAG) generalizations of the ProxVSG. In ProxVSIAG, an incremental aggregated estimate of the gradient is used, instead of the full gradient. Under suitable assumptions, we prove a complexity of O (ϵ - 3) to achieve an ϵ -approximate solution. Preliminary numerical experiments show the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Mechanized harvesting is the key technology to solving the high cost and low efficiency of manual harvesting, and the key to realizing mechanized harvesting lies in the accurate and fast identification and localization of targets. In this paper, a lightweight YOLOv5s model is improved for efficiently identifying grape fruits and stems. On the one hand, it improves the CSP module in YOLOv5s using the Ghost module, reducing model parameters through ghost feature maps and cost-effective linear operations. On the other hand, it replaces traditional convolutions with deep convolutions to further reduce the model's computational load. The model is trained on datasets under different environments (normal light, low light, strong light, noise) to enhance the model's generalization and robustness. The model is applied to the recognition of grape fruits and stems, and the experimental results show that the overall accuracy, recall rate, mAP, and F1 score of the model are 96.8%, 97.7%, 98.6%, and 97.2% respectively. The average detection time on a GPU is 4.5 ms, with a frame rate of 221 FPS, and the weight size generated during training is 5.8 MB. Compared to the original YOLOv5s, YOLOv5m, YOLOv5l, and YOLOv5x models under the specific orchard environment of a grape greenhouse, the proposed model improves accuracy by 1%, decreases the recall rate by 0.2%, increases the F1 score by 0.4%, and maintains the same mAP. In terms of weight size, it is reduced by 61.1% compared to the original model, and is only 1.8% and 5.5% of the Faster-RCNN and SSD models, respectively. The FPS is increased by 43.5% compared to the original model, and is 11.05 times and 8.84 times that of the Faster-RCNN and SSD models, respectively. On a CPU, the average detection time is 23.9 ms, with a frame rate of 41.9 FPS, representing a 31% improvement over the original model. The test results demonstrate that the lightweight-improved YOLOv5s model proposed in the study, while maintaining accuracy, significantly reduces the model size, enhances recognition speed, and can provide fast and accurate identification and localization for robotic harvesting. [ABSTRACT FROM AUTHOR]

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of 'admissible' unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations. [ABSTRACT FROM AUTHOR]

In this paper, we study a Riemann-Liouville-type fractional Riemann-Stieltjes integral boundary value problem under some conditions regarding the spectral radius of the relevant linear operator. The existence of nontrivial solutions is obtained using topological degree, and our results improve and generalize some results in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions Τ J (α i) . Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f ∈ Τ J (α i) . In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass. [ABSTRACT FROM AUTHOR]

In this paper, the Parseval K - g -frames are constructed from a given K - g -frame by scaling the elements of the K - g -frame with the help of diagonal operators, and these frames are named scalable K - g -frames. Also, we prove some properties of scalable K - g -frames and construct new scalable K - g -frames from a given K - g -frame. The necessary and sufficient conditions for a K - g -frame to be scalable are given. Further, equivalent conditions for the scalability of K - g -frames and the K -frames induced by K - g -frames are obtained. Finally, it is shown that the direct sum of two scalable K - g -frames is again a scalable K - g -frame for some suitable bounded linear operator K. [ABSTRACT FROM AUTHOR]

Our paper delves into exploring skew cyclic codes over a generalized class of rings denoted by T = Tr. We define Tr = Fq[u1, u2, ..., ur]/(ui3 - ui, uiuj -ujui)i,jr=1, q = pm and p is some odd prime. Our study introduces a Gray map for the ring T and explores its properties. Using a decomposition theorem, we analyze the structural features of skew cyclic codes over T . Additionally, we offer a formula to find the count of skew cyclic codes of length n over the ring T under specific conditions. Further, we derive a criterion to get Linear Complementary Dual (LCD) codes over T from skew cyclic codes. Moreover, we present a technique for deriving quantum codes from a particular class of skew cyclic codes over T which contain their dual. [ABSTRACT FROM AUTHOR]

For a general complex unital Banach algebraA, the spectrumalways commutes : for all a, b ∈ A, σ(ab) \ {0} = σ(ba) \ {0}. In this paper, we prove that the above commutative property is not true if we replace the usual spectrum by condition spectrum. Further we study the similar question for more general spec- trum called Ransford spectrum. [ABSTRACT FROM AUTHOR]

The research of biomedical data is crucial for disease diagnosis, health management, and medicine development. However, biomedical data are usually characterized by high dimensionality and class imbalance, which increase computational cost and affect the classification performance of minority class, making accurate classification difficult. In this paper, we propose a biomedical data classification method based on feature selection and data resampling. First, use the minimal-redundancy maximal-relevance (mRMR) method to select biomedical data features, reduce the feature dimension, reduce the computational cost, and improve the generalization ability; then, a new SMOTE oversampling method (Spectral-SMOTE) is proposed, which solves the noise sensitivity problem of SMOTE by an improved spectral clustering method; finally, the marine predators algorithm is improved using piecewise linear chaotic maps and random opposition-based learning strategy to improve the algorithm's optimization seeking ability and convergence speed, and the key parameters of the spectral-SMOTE are optimized using the improved marine predators algorithm, which effectively improves the performance of the over-sampling approach. In this paper, five real biomedical datasets are selected to test and evaluate the proposed method using four classifiers, and three evaluation metrics are used to compare with seven data resampling methods. The experimental results show that the method effectively improves the classification performance of biomedical data. Statistical test results also show that the proposed PRMPA-Spectral-SMOTE method outperforms other data resampling methods. [ABSTRACT FROM AUTHOR]

Statistical convergence is more extensive than the classical convergence and has recently drawn the recognition of many researchers. The Korovkin-type approximation theorems are usually based on the convergence analysis of sequences of positive linear operators. Gradually, such approximation theorems are extended over more general sequence spaces with several settings via different kinds of statistical summability techniques. In this paper, we introduce presumably a new statistical Riesz-Euler product summability technique to prove a Korovkin-type approximation theorem. Moreover, we demonstrate another result for the rate of statistical convergence under our proposed summability technique. [ABSTRACT FROM AUTHOR]

This paper deals with the equation - Δ u + μ u = f on high-dimensional spaces R m , where the right-hand side f (x) = F (T x) is composed of a separable function F with an integrable Fourier transform on a space of a dimension n > m and a linear mapping given by a matrix T of full rank and μ ≥ 0 is a constant. For example, the right-hand side can explicitly depend on differences x i - x j of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression ‖ T t y ‖ 2 deviates on the unit sphere ‖ y ‖ = 1 the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges. [ABSTRACT FROM AUTHOR]

An asset pricing model with two types of chartists and fundamentalists and trend followers is considered, it is driven by a two-dimensional piecewise linear discontinuous map with three subsystems. There are great differences in the dynamic behaviour between expected offset (the expectations of trend followers offset the difference between the expectations of Type 1 traders in bull and bear markets) and expected non-offset. It is proven that there is no chaos in the dynamic of system with expected offset. However, chaos may exist in the dynamic of system with expected non-offset. We present a systematic approach to the problem of analysing the bifurcation phenomena associated with the appearance/disappearance of cycles, which may be related to several bifurcations. The multi-stability regions in parameter plane and related basins of multi-attractors in phase space are investigated. This paper aims to uncover the endogenous law of the unpredictability and excessive volatility in financial markets. [ABSTRACT FROM AUTHOR]

Let Cp be the Schatten p-class of ℓ2 for 1

Iterative algorithms requiring the computationally expensive in general inversion of linear operators are difficult to implement. This is the reason why hybrid Newton-like algorithms without inverses are developed in this paper to solve Banach space-valued nonlinear equations. The inverses of the linear operator are exchanged by a finite sum of fixed linear operators. Two types of convergence analysis are presented for these algorithms: the semilocal and the local. The Fréchet derivative of the operator on the equation is controlled by a majorant function. The semi-local analysis also relies on majorizing sequences. The celebrated contraction mapping principle is utilized to study the convergence of the Krasnoselskij-like algorithm. The numerical experimentation demonstrates that the new algorithms are essentially as effective but less expensive to implement. Although the new approach is demonstrated for Newton-like algorithms, it can be applied to other single-step, multistep, or multipoint algorithms using inverses of linear operators along the same lines. [ABSTRACT FROM AUTHOR]

Minor perturbations to a linear operator may drastically change its spectrum, and hence the difficulty of deciding whether or not a numerically computed quantity is zero causes problems in spectral theory. The purpose of this expository paper is to illustrate such instability phenomena by some examples with Toeplitz-like operators and matrices. [ABSTRACT FROM AUTHOR]

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a λ -TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative λ -TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the λ -TD shuffle product. We then show that the free commutative λ -TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative λ -TD algebra is connected and filtered, and thus is a left counital Hopf algebra. [ABSTRACT FROM AUTHOR]

Let be a 2-torsion free unital generalized matrix algebra with center Z () , and Φ be a linear mapping on satisfying the condition X , Y ∈ , X Y = Y X = 0 ⇒ [ Φ (X) , Y ] = 0. This paper is devoted to the study of the structure of Φ under some mild assumptions on . We provide the necessary and sufficient conditions for Φ to be in the form Φ (X) = λ X + μ (X) (∀ X ∈ ), where λ ∈ Z () and μ : → Z () is a linear mapping. Then we apply our results to characterize linear mappings on that are commutant preservers or double commutant preservers. [ABSTRACT FROM AUTHOR]

In this paper, we study connections between composition operators on Sobolev spaces and mappings defined by p-moduli inequalities (p-capacity inequalities). We prove that weighted moduli inequalities lead to composition operators on corresponding Sobolev spaces and conversely, that composition operators on Sobolev spaces imply weighted moduli inequalities. [ABSTRACT FROM AUTHOR]

In this paper we focus on different--global, semilocal, and local--versions of Hoffmantype inequalities expressed in a variational form. In a first stage our analysis is developed for generic multifunctions between metric spaces, and we finally deal with the feasible set mapping associated with linear semi-infinite inequality systems (finitely many variables and possibly infinitely many constraints) parameterized by their right-hand sides. The Hoffman modulus is shown to coincide with the Lipschitz upper semicontinuity modulus and the supremum of calmness moduli when confined to multifunctions with a convex graph and closed images in a reflexive Banach space, which is the case for our feasible set mapping. Moreover, for this particular multifunction a formula--involving only the system's left-hand side--of the global Hoffman constant is derived, providing a generalization to our semi-infinite context of finite counterparts developed in the literature. In the particular case of locally polyhedral systems, the paper also provides a point-based formula for the (semilocal) Hoffman modulus in terms of the calmness moduli at certain feasible points (extreme points when the nominal feasible set contains no lines), yielding a practically tractable expression for finite systems. [ABSTRACT FROM AUTHOR]

In this paper, we characterize bounded linear operators A,B on a complex Hilbert space such that inf λ∈C ||A+B-λ I|| = inf λ∈C ||A-λ I||+ inf λ∈C ||B-λ I||, where I is the identity operator. We also establish some inequalities satisfied by the distance from scalar operators for products of two complex Hilbert space operators. [ABSTRACT FROM AUTHOR]

The paper is devoted to study the uniqueness problem of linear delay-differential operator of a meromorphic function sharing two sets or small function together with values with its c-shift and q-shift operator. Results of this paper drastically improve two recent results of Meng-Liu [J. Appl. Math. Inform. 37(1-2)(2019), 133-148] and Qi-Li-Yang [Comput. Methods Funct. Theory, 18(2018), 567-582]. In addition to this, one of our results improves and extends that of Qi-Yang [Comput. Methods Funct. Theory, 20(2020), 159-178]. [ABSTRACT FROM AUTHOR]

In this paper, we establish a new concentration inequality and almost complete convergence of the value of the process of autoregressive Hilbertian of order one (ARH (1)), which directly stems from works of Serge Guillas, Denis Bosq, that is defined by Xt = ρ(Xt-1) + ζt; t ε ... where the random variables are all Hilbertian, ρ is a linear operator on a space of separable Hilbert and ζt which constitute a widely orthant dependent error (WOD, in short) after recalling some results on the finite-dimensional model of this type, we introduce the mathematical and statistical tools which will be used afterwards. Then we build an estimator of the operator and we establish its asymptotic properties. [ABSTRACT FROM AUTHOR]

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, "how about the situation in the differential game?", "will the same phenomenon appear in SLQ?". This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity. [ABSTRACT FROM AUTHOR]

In this paper, we present some further refinements of the Tan-Xie inequality for sector matrices and its applications due to Nasiri and Furuichi [J. Math. Inequal., 15 (2021), 1425-1434]. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T ¯ . The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T ¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T ¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider general Markov chains with discrete time in an arbitrary measurable (phase) space. Markov chains are given by a classical transition function that generates a pair of conjugate linear Markov operators in a Banach space of measurable bounded functions and in a Banach space of bounded finitely additive measures. We study sequences of Cesaro means of powers of Markov operators on the set of finitely additive probability measures. It is proved that the set of all limit measures (points) of such sequences in the weak topology generated by the preconjugate space is non-empty, weakly compact, and all of them are invariant for this operator. We also show that the well-known Doeblin condition (D) for the ergodicity of a Markov chain is equivalent to condition (∗) : all invariant finitely additive measures of the Markov chain are countably additive, i.e., there are no invariant purely finitely additive measures. We give all the proofs for the most general case. [ABSTRACT FROM AUTHOR]

The application of a complex fuzzy logic system based on a linear conjunctive operator represents a significant advancement in the field of data analysis and modeling, particularly for studying physical scenarios with multiple options. This approach is highly effective in situations where the data involved is complex, imprecise and uncertain. The linear conjunctive operator is a key component of the fuzzy logic system used in this method. This operator allows for the combination of multiple input variables in a systematic way, generating a rule base that captures the behavior of the system being studied. The effectiveness of this method is particularly notable in the study of phenomena in the actual world that exhibit periodic behavior. The foremost aim of this paper is to contribute to the field of fuzzy algebra by introducing and exploring new concepts and their properties in the context of conjunctive complex fuzzy environment. In this paper, the conjunctive complex fuzzy order of an element belonging to a conjunctive complex fuzzy subgroup of a finite group is introduced. Several algebraic properties of this concept are established and a formula is developed to calculate the conjunctive complex fuzzy order of any of its powers in this study. Moreover, an important condition is investigated that determines the relationship between the membership values of any two elements and the membership value of the identity element in the conjunctive complex fuzzy subgroup of a group. In addition, the concepts of the conjunctive complex fuzzy order and index of a conjunctive complex fuzzy subgroup of a group are also presented in this article and their various fundamental algebraic attributes are explored structural. Finally, the conjunctive complex fuzzification of Lagrange's theorem for conjunctive complex fuzzy subgroups of a group is demonstrated. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of elliptic problems in a periodically perforated domain with L 1 data and nonlinear Robin conditions on the boundary of the holes. Using the framework of renormalized solutions, which is well adapted to this situation, we show a convergence result for the truncated energy in the quasilinear case. When the operator is linear, we also prove a corrector result. Since we cannot expect to have solutions belonging to H 1 , the main difficulty is to express the corrector result through the truncations of the solutions, together with the fact that the definition of a renormalized solution contains test functions which are nonlinear functions of the solution itself. In this paper, we consider a class of elliptic problems in a periodically perforated domain with data and nonlinear Robin conditions on the boundary of the holes. Using the framework of renormalized solutions, which is well adapted to this situation, we show a convergence result for the truncated energy in the quasilinear case. When the operator is linear, we also prove a corrector result. Since we cannot expect to have solutions belonging to , the main difficulty is to express the corrector result through the truncations of the solutions, together with the fact that the definition of a renormalized solution contains test functions which are nonlinear functions of the solution itself. [ABSTRACT FROM AUTHOR]

This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at 0 , β and ∞ where β represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves. This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at , and where represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves. [ABSTRACT FROM AUTHOR]

This note draws conclusions that arise by combining two recent papers, by Anuj Dawar, Erich Grädel and Wied Pakusa, published at ICALP 2019, and by Moritz Lichter, published at LICS 2021. In both papers, the main technical results rely on the combinatorial and algebraic analysis of the invertible-map equivalences |${\equiv ^{\text {IM}}_{k, Q}}$| on certain variants of Cai–Fürer–Immerman structures (CFI-structures for short). These |${\equiv ^{\text {IM}}_{k, Q}}$| -equivalences, for a natural number |$k$| and a set of primes |$Q$|⁠ , refine the well-known Weisfeiler–Leman equivalences used in algorithms for graph isomorphism. The intuition is that two graphs |$G{\equiv ^{\text {IM}}_{k, Q}}H$| cannot be distinguished by iterative refinements of equivalences on |$k$| -tuples defined via linear operators on vector spaces over fields of characteristic |$p \in Q$|⁠. In the first paper it has been shown, using considerable algebraic machinery, that for a prime |$q \notin Q$|⁠ , the |${\equiv ^{\text {IM}}_{k, Q}}$| equivalences are not strong enough to distinguish between non-isomorphic CFI-structures over the field |$\mathbb {F}_q$|⁠. In the second paper, a similar but not identical construction for CFI-structures over the rings |$\mathbb {Z}_{2^i}$| has, again by rather involved combinatorial and algebraic arguments, been shown to be indistinguishable with respect to |${\equiv ^{\text {IM}}_{k, \{2\}}}$|⁠. Together with an earlier work on rank logic, this second result suffices to separate rank logic from polynomial time. We show here that the two approaches can be unified to prove that CFI-structures over the rings |$\mathbb {Z}_{2^i}$| are in fact indistinguishable with respect to |${\equiv ^{\text {IM}}_{k, {\mathbb {P}}}}$|⁠ , for the set |${\mathbb {P}}$| of all primes. In particular, this implies the following two results. First, there is no fixed |$k$| such that the invertible-map equivalence |${\equiv ^{\text {IM}}_{k, {\mathbb {P}}}}$| coincides with isomorphism on all finite graphs. Second, no extension of fixed-point logic by linear-algebraic operators over fields can capture polynomial time. [ABSTRACT FROM AUTHOR]

In our previous paper, it is shown that topology of G-fuzzy normed linear space is generated by two types of open balls: one is elliptic and the other is circular. In the theoretical aspect of functional analysis, will this type of exception happen or not? To address this problem in this paper, firstly, G-fuzzy bounded linear operators as well as G-fuzzy bounded linear functionals are defined which are the key elements of functional analysis. Then, operator G-fuzzy norms are introduced for both the cases using the idea of quasi-G-norm family. The definition of operator G-fuzzy norm is quite different from the existing operator fuzzy norm. Completeness of operator G-fuzzy norm is investigated. Lastly, Hahn-Banach theorem in G-fuzzy setting is studied using all the above concepts. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present Korovkin theorems for positive linear operators of two variables from Hω(K) into CB(K) via the power series method. In addition, we give an example that our new approximation result works but its classical case does not work. Furthermore, we obtain the rate of convergence of these operators. [ABSTRACT FROM AUTHOR]

This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We first motivate the use of representing nonlinear systems as linear Koopman operator systems by illustrating the improved model-based control performance with an actuated Van der Pol system. Information-theoretic methods are then applied to the Koopman operator formulation of dynamical systems where we derive a controller for active learning of robot dynamics. The active learning controller is shown to increase the rate of information about the Koopman operator. In addition, our active learning controller can readily incorporate policies built on the Koopman dynamics, enabling the benefits of fast active learning and improved control. Results using a quadcopter illustrate single-execution active learning and stabilization capabilities during free fall. The results for active learning are extended for automating Koopman observables and we implement our method on real robotic systems. [ABSTRACT FROM AUTHOR]

Time invariant linear operators are the building blocks of signal processing. Weighted circular convolution and signal processing framework in a generalized Fourier domain are introduced by Jorge Martinez. In this paper, we prove that under this new signal processing framework, weighted circular convolution also has a generalized time invariant property. We also give an application of this property to algorithm of continuous wavelet transform (CWT). Specifically, we have previously studied the algorithm of CWT based on generalized Fourier transform with parameter 1. In this paper, we prove that the parameter can take any complex number. Numerical experiments are presented to further demonstrate our analyses. [ABSTRACT FROM AUTHOR]

In this paper we consider controlled infinite dimensional systems subjected to stochastic structured nonlinear unbounded multiperturbations. Our objective is to study the maximization of the stability radius by state feedback. We obtain conditions for the existence of suboptimal controllers. The supremal achievable stability radius is characterized via the resolution of a Riccati equation and some linear operator inequalities. Examples are given to illustrate the theory. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the q-numerical radius $ \omega _{q}(.) $ ω q (.) of bounded linear operators on Hilbert spaces. More precisely, first, we show that $ \omega _{q}(.) $ ω q (.) defines a norm which is equivalent to the operator norm. Next, the following compatible generalization of Kittaneh's inequality $$\begin{align*} \frac{1}{4}\left(\frac{q}{2-q^{2}}\right)^{2}\parallel T^{*}T +TT^{*}\parallel & \leq \omega_{q}^{2}(T) \leq \frac{(q+2\sqrt{1-q^2})^2}{2}\\ & \quad\times \Vert T^{*}T+TT^{*}\Vert. \end{align*}$$ 1 4 (q 2 − q 2 ) 2 ∥ T ∗ T + T T ∗ ∥ ≤ ω q 2 (T) ≤ (q + 2 1 − q 2 ) 2 2 × ‖ T ∗ T + T T ∗ ‖. is obtained. Finally, some generalizations of q-numerical radius inequalities for composition of operators are established. [ABSTRACT FROM AUTHOR]

The construction of the fourth-order 2-point Fractional Block Backward Differentiation Formula (2FBBDF(4)) to solve the fractional order differential equations (FDEs) is presented in this paper. The method is developed using the fractional linear multistep method (FLMM) linked with the linear difference operator. This paper aims to approximate the fractional order problems via 2FBBDF(4), which is normally used to solve ordinary differential equations. The criteria for the stability of the method are analyzed in order to solve FDE problems. Consequently, the method is determined to be A-stable for different values of α within the interval (0, 1). Then, Newton's iteration is implemented in this method to solve the problems. Multiple numerical examples of linear, nonlinear, and system FDEs are provided for the scenario where the order α lies within the range of 0 and 1. Ultimately, the numerical results confirm that the proposed method performs at a similar level to the existing methods. [ABSTRACT FROM AUTHOR]

When studying the iterates of certain positive linear operators, systems of linear equations are naturally involved. The last step in investigating the limit of such iterates is represented by a special kind of system of equations. Problems of this type involving several classical operators are studied in the literature. In this paper, we investigate the iterates of new positive linear operators and the corresponding systems of equations. To solve the system we use an iterative algorithm. The approximate solution is used in order to approximate the limit of the iterates of operators. [ABSTRACT FROM AUTHOR]

In this paper, two Halpern-type inertial iteration methods with self-adaptive step size are proposed for estimating the solution of split common null point problems ( S p CNPP ) in such a way that the Halpern iteration and inertial extrapolation are computed simultaneously in the beginning of each iteration. We prove the strong convergence of sequences driven by the suggested methods without estimating the norm of bounded linear operator when certain appropriate assumptions are made. We demonstrate the efficiency of our iterative methods and compare them with some related and well-known results using relevant numerical examples. [ABSTRACT FROM AUTHOR]