Congruence Research Articles

Robust stabilization for networked systems with transmission delay via integral Lyapunov functional and congruence transformation method

Uncertainties and information transmission delay exist in many real-world applications. The control design of such systems is challenging even for existing advanced control theory. Besides, if there is actuator saturation in the system and networked control strategy is considered, the problem will become even more complicated. In order to solve the problem, the stochastic Takagi-Sugeno (T-S) fuzzy delay-dependent static output feedback control is proposed in this paper, and strictly dissipative analysis is addressed for the stochastic T-S fuzzy singular information networked control systems. In the propose control scheme, the T-S fuzzy model and stochastic Bernoulli theory are employed in the controller design. The stability conditions of the closed-loop control system are summarized in the linear matrix inequalities (LMIs), then the closed-loop system is regular impulse free, stochastically admissible and strictly dissipative. Both fuzzy-basis-dependent and delay-dependent stability analysis, with the consideration of strictly dissipative performance index, are conducted to develop stability conditions in terms of LMIs based on Lyapunov stability theory. Via the LMIs optimization constraints, the nonconvex problem caused by fuzzy-basis-dependent can be solved. Finally, the simulation examples are provided to verify the effectiveness of the proposed control strategy.

A Principal Bundle Framework for Generalised Notions of Formation and Rigidity

In this work, the notions of formation and rigidity (first developed for formation control in Euclidean spaces) are recast in a general framework of principal fiber bundles and a control law is derived for second order systems for driving the system to a given formation. The Euclidean inter-agent distance measurements and congruence transformations are generalised as output maps and Lie group actions respectively, to capture more generic scenarios.

Some congruences for $ \ell $-regular partitions with certain restrictions

<abstract><p>Let $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $ denote the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct odd parts and the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct even parts, respectively. Our first goal in this note was to prove two congruence relations for $ {\rm{pod}}_\ell(n) $. Furthermore, we found a formula for the action of the Hecke operator on a class of eta-quotients. As two applications of this result, we obtained two infinite families of congruence relations for $ {\rm pod}_5(n) $. We also proved a congruence relation for $ {\rm{ped}}_\ell(n) $. In particular, we established a congruence relation modulo 2 connecting $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $.</p></abstract>

A recursion relation for the number of Goldbach partitions of an even integer

The contour integral representation of the number of Goldbach partitions of an even integer, G(n), is extended to an integral with a support function that equals a linear combination of integers {G(m)}. A support function is found such that there is a nontrivial integral relation relating number of Goldbach partitions of n and m < n. The proof of the existence of a partition of any even integer greater than or equal to four into the sum of two primes follows from a recursion relation, resulting from an integral identity, that yields a non-zero lower bound for G(n). A partition of n, given a partition of n/2 , is derived from an algorithm based on solutions to congruence relations related to the roots of unity. This result provides a method for generating a prime partition of every even integer greater than or equal to 6 in addition to the above demonstration of its existence.

Dual ideal theory on L-algebras

<abstract><p>This paper aims to study bounded algebras in another perspective-dual ideals of bounded $ L $-algebras. As the dual concept of ideals in $ L $-algebras, dual ideals are designed to characterize some significant properties of bounded $ L $-algebras. We begin by providing a definition of dual ideals and discussing the relationships between ideals and dual ideals. Then, we prove that these dual ideals induce congruence relations and quotient $ L $-algebras on bounded $ L $-algebras. Naturally, in order to construct the first isomorphism theorem between bounded $ L $-algebras, the relationship between dual ideals and morphisms between bounded $ L $-algebras is investigated and that the kernels of any morphisms between bounded $ L $-algebras are dual ideals is proven. Fortunately, although the first isomorphism theorem between arbitrary bounded $ L $-algebras fails to be proven when using dual ideals, the theorem was proven when the range of morphism was good. Another main purpose of this study is to use dual ideals to characterize several kinds of bounded $ L $-algebras. Therefore, first, the properties of dual ideals in some special bounded $ L $-algebras are studied; then, some special bounded $ L $-algebras are characterized by dual ideals. For example, a good $ L $-algebra is a $ CL $-algebra if and only if every dual ideal is $ C $ dual ideal is proven.</p></abstract>

Constructing Symmetric Equality Algebras

In this paper, we introduce the notion of strong fuzzy filter on hyper equality algebras and investigate some equivalence definitions of it. Then by using this notion we constructed a symmetric equality algebra and define a special form of classes. By using these, we define the concept of a fuzzy hyper congruence relation on hyper equality algebra and we prove that the quotient is made by it is an equality algebra. Also, by using a fuzzy equivalence relation on hyper equality, we introduce a fuzzy hyper congruence relation and prove that this fuzzy hyper congruence is regular and finally we prove that the quotient structure that is made by it is a symmetric hyper equality algebra.

Beyond the Name: Analysis of Cohesion Factor in Name Congruence and Socioemotional Wealth Relation of Family Business

In the forthcoming study, our overarching objective is to embark on a comprehensive exploration of the intricate linkages between the family name and the socioemotional wealth intrinsic to family businesses. This investigation places a specific emphasis on unraveling the mediating role played by family cohesion in shaping these complex dynamics. Our theoretical endeavor seeks to make a distinctive contribution to the scholarly landscape by providing nuanced insights into the multifaceted relationships within family businesses. By anchoring our study in the foundation of established literature, our aim extends beyond expanding empirical understanding to substantively contribute to the theoretical discourse that shapes the broader landscape of family business dynamics. As we delve into this exploration, we anticipate our findings will not only enrich our understanding of the interplay between family names, family cohesion, and socioemotional wealth but also offer valuable insights for practitioners, policymakers, and scholars invested in the success and resilience of family-owned enterprises.

Congruence Relations and Direct Decomposition of Uninorms on Bounded Lattices

In this paper, the concept of [Formula: see text]-compatible congruence and its related properties are studied. We also focus on homomorphism theorems for uninorms. Decomposition of uninorms is investigated via factor congruences, at first. Subsequently, we investigate decomposition of conjunctive and disjunctive uninorms, respectively. After all, we determine necessary and sufficient conditions for decomposition of some uninorms that are neither conjunctive nor disjunctive. In addition to all these, we shed light on the application of our results with the examples.

Rooted Divergence-Preserving Branching Bisimilarity is a Congruence for Guarded CCS

Branching bisimilarity is a well-known equivalence relation for labelled transition systems. Based on this equivalence relation, with an additional simple rootedness condition, a congruence relation for calculus of communication system (CCS) processes can be obtained. However, neither branching bisimilarity nor the corresponding congruence relation preserves divergence, and it is still a question whether, based on a divergence-preserving variant of branching bisimilarity, a divergence-preserving congruence relation for CCS processes can be obtained by introducing the same simple rootedness condition. In this article, we present a partial solution by showing that rooted divergence-preserving branching bisimilarity is preserved under the usual CCS operators, including prefixing, summation, parallel composition, relabelling, restriction, and (weakly) guarded recursion.

LMIs-based exponential stabilization for interval delay systems via congruence transformation: Application in chaotic Lorenz system

This paper investigates the stochastic Takagi-Sugeno (T-S) fuzzy uncertainty-dependent state feedback sampled-data control, asymptotical stability and exponential stability analysis for a class of stochastic T-S fuzzy system with measurable uncertainties, mixed interval time varying delays and Gaussian white noise. Firstly, the T-S fuzzy model and stochastic Bernoulli theory are employed to approximate the system plant. Secondly, the stochastic T-S fuzzy uncertainty-dependent state feedback sampled-data controller is designed via uncertainty-dependent parameters and the hold times series of sampled-data. Thirdly, the Bernoulli probability distribution is employed to describe the random occurrence characteristic of multipath packet dropouts. Three classes of delay-dependent stability conditions in terms of linear matrix inequalities (LMIs) are derived for the closed-loop system. The multiple integral fuzzy-basis-dependent Lyapunov functional and augmented state variable are employed to derive the delay-dependent stability conditions, then the closed-loop system is asymptotically stable and exponentially stable. The congruence transformation method and prescribed H-infinity performance functional are employed to derive the less conservative delay-dependent stability conditions, and the prescribed H-infinity performance is guaranteed. The relaxation matrix and congruence transformation method are employed to design the linear transformation equalities and congruence transformation matrices, then the controller gain matrix is determined and computation complexity of solving LMIs is reduced. Finally, the proposed method is applied in the chaotic Lorenz system to show the effectiveness and advantage of proposed method.

Abelian Groups Derived From Hypergroups

We introduce a new strongly regular relation α on a given group G and show that α is a congruence relation on G, with respect to module the commutator subgroup of G. Then we show that the composition of this relation with the fundamental relation β* is equal to the fundamental and γ are is equal to the relation α. and we conclude that if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of α on ρ, results in a strongly regular relation such that its quotients is an abelian group.

Berger-Coburn-Lebow representation for pure isometric representations of product system over [formula omitted

We obtain Berger-Coburn-Lebow (BCL)-representation for pure isometric covariant representation of product system over N02. Then the corresponding complete set of (joint) unitary invariants is studied, and the BCL-representations are compared with other canonical multi-analytic descriptions of the pure isometric covariant representation. We characterize the invariant subspaces for the pure isometric covariant representation. Also, we study the connection between the joint defect operators and Fringe operators, and the Fredholm index is introduced in this case. Finally, we introduce the notion of congruence relation to classify the isometric covariant representations of the product system over N02.

On fuzzy congruence relation in residuated lattices

In this paper, we characterize some properties of fuzzy congruence relations and obtain a fuzzy congruence relation generated by a fuzzy relation in residuated lattices. For this purpose, two various types of fuzzy relations (regular and irregular) are introduced. In order to obtain a fuzzy congruence relation generated by an irregular fuzzy relation it must convert to a regular fuzzy relation.

Non-homogeneous Riemannian gradient equations for sum of squares of Bures–Wasserstein metric

We investigate Riemannian gradient equations on the open convex cone Pm of all m×m positive definite Hermitian matrices, for the function WA(X)=∑j=1nwjdW2(X,Aj)where dW denotes the Bures–Wasserstein metric. On the special case where the gradient of the weighted sum of squares of the Bures–Wasserstein metrics vanishes, its unique solution is known as the Wasserstein mean of A1,…,An. We discuss the existence and uniqueness of solutions for non-homogeneous Riemannian gradient equations at which vector fields are constant and congruence transformations. Furthermore, we establish some bounds for these solutions.

Extended BCK-Ideal Based on Single-Valued Neutrosophic Hyper BCK-Ideals

This paper introduces the concept of single-valued neutrosophic hyper \(BCK\)-subalgebras as a generalization and alternative of hyper \(BCK\)-algebras and on any given nonempty set constructs at least one single-valued neutrosophic hyper \(BCK\)-subalgebra and one a single-valued neutrosophic hyper \(BCK\)-ideal. In this study level subsets play the main role in the connection between singlevalued neutrosophic hyper \(BCK\)-subalgebras and hyper \(BCK\)-subalgebras and the connection between single-valued neutrosophic hyper \(BCK\)-ideals and hyper \(BCK\)-ideals. The congruence and (strongly) regular equivalence relations are the important tools for connecting hyperstructures and structures, so the major contribution of this study is to apply and introduce a (strongly) regular relation on hyper \(BCK\)-algebras and to investigate their categorical properties (quasi commutative diagram) via single-valued neutrosophic hyper \(BCK\)-ideals. Indeed, by using the single-valued neutrosophic hyper \(BCK\)-ideals, we define a congruence relation on (weak commutative) hyper \(BCK\)-algebras that under some conditions is strongly regular and the quotient of any (single-valued neutrosophic)hyper \(BCK\)-(sub)algebra via this relation is a (single-valued neutrosophic)(hyper \(BCK\)-subalgebra) \(BCK\)-(sub)algebra.

Motion Groupoids and Mapping Class Groupoids

Here {underline{M}} denotes a pair (M, A) of a manifold and a subset (e.g. A=partial M or A=varnothing ). We construct for each {underline{M}} its motion groupoidtextrm{Mot}_{{underline{M}}}, whose object set is the power set {{mathcal {P}}}M of M, and whose morphisms are certain equivalence classes of continuous flows of the ‘ambient space’ M, that fix A, acting on {{mathcal {P}}}M. These groupoids generalise the classical definition of a motion group associated to a manifold M and a submanifold N, which can be recovered by considering the automorphisms in textrm{Mot}_{{underline{M}}} of Nin {{mathcal {P}}}M. We also construct the mapping class groupoidtextrm{MCG}_{{underline{M}}} associated to a pair {underline{M}} with the same object class, whose morphisms are now equivalence classes of homeomorphisms of M, that fix A. We recover the classical definition of the mapping class group of a pair by taking automorphisms at the appropriate object. For each pair {underline{M}} we explicitly construct a functor {textsf{F}}:textrm{Mot}_{{underline{M}}} rightarrow textrm{MCG}_{{underline{M}}}, which is the identity on objects, and prove that this is full and faithful, and hence an isomorphism, if pi _0 and pi _1 of the appropriate space of self-homeomorphisms of M are trivial. In particular, we have an isomorphism in the physically important case {underline{M}}=([0,1]^n, partial [0,1]^n), for any nin {mathbb {N}}. We show that the congruence relation used in the construction textrm{Mot}_{{underline{M}}} can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows—worldlines (e.g. monotonic ‘tangles’). We examine several explicit examples of textrm{Mot}_{{underline{M}}} and textrm{MCG}_{{underline{M}}} demonstrating the utility of the constructions.

Scalable monoids and quantity calculus

We define scalable monoids and prove their fundamental properties. Congruence relations on scalable monoids, direct and tensor products, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are defined and investigated. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. Observations relating to the theory of measurement of physical quantities accompany the results about scalable monoids. We conclude that the algebraic theory of scalable monoids and quantity spaces provides a rigorous foundation for quantity calculus.

Theta-type congruences for colored partitions

We investigate congruence relations of the form pr(ℓmn+t)≡0(modℓ) for all n, where pr(n) is the number of r-colored partitions of n and m,ℓ are distinct primes.

Fuzzy congruence relation on pseudo-UP algebra

Fuzzy congruence relation on pseudo-UP algebra

SELF-DIVISIBLE ULTRAFILTERS AND CONGRUENCES IN

AbstractWe introduceself-divisibleultrafilters, which we prove to be precisely those$w$such that the weak congruence relation$\equiv _w$introduced by Šobot is an equivalence relation on$\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that$w$is self-divisible if and only if$\equiv _w$coincides with the strong congruence relation$\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient$(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$is a profinite group. We also construct an ultrafilter$w$such that$\equiv _w$fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion$\hat {{\mathbb Z}}$of the integers.