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In this paper, we give a description of differential operators on tensor products A ⊗ 핂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 핂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 핂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B, respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , ퟎ ) = ( V ⁢ ( f ) , ퟎ ) ⊆ ( ℂ r , ퟎ ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n. [ABSTRACT FROM AUTHOR]

Given Banach space operators A, B, let δ A , B denote the generalised derivation δ (X) = (L A - R B) (X) = A X - X B and ▵ A , B the length two elementary operator ▵ A , B (X) = (I - L A R B) (X) = X - A X B . This note considers the structure of m-symmetric operators δ ▵ A 1 , B 1 , ▵ A 2 , B 2 m (I) = (L ▵ A 1 , B 1 - R ▵ A 2 , B 2 ) m (I) = 0 . It is seen that there exist scalars λ i ∈ σ a (B 1) , 1 ≤ i ≤ 2 , such that δ λ 1 A 1 , λ 2 A 2 m (I) = 0 . Translated to Hilbert space operators A and B this implies that if δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 , then there exists λ ¯ ∈ σ a (B ∗) such that δ (λ A) ∗ , λ A m (I) = 0 = δ λ ¯ B , λ B ∗ m (I) . We prove that the operator δ ▵ A ∗ , B ∗ , ▵ A , B m is compact if and only if (i) there exists a real number α and finite sequnces (i) { a j } j = 1 n ⊆ σ (A) , { b j } j = 1 n ⊆ σ (B) such that a j b j = 1 - α , 1 ≤ j ≤ n ; (ii) decompositions ⊕ j = 1 n H j and ⊕ j = 1 n H J of H such that ⊕ j = 1 n (A - a j I) | H j and ⊕ j = 1 n (B - b j I) | H j are nilpotent. If δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 implies δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 to imply δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 is that λ A and λ ¯ B satisfy the commutativity property for scalars lambda ¯ ∈ σ a (B ∗) . An analogous result is seen to hold for the operators ▵ δ A ∗ , B ∗ , δ A , B m and ▵ δ A ∗ , B ∗ , δ A , B m (I) . Perturbation by commuting nilpotents is considered. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we discuss Lie nilpotency and Lie solvability of non-abelian tensor product of multiplicative Lie algebras. As our main results, we provide upper bounds on the Lie nilpotency class and Lie solvability length of quotient of tensor product of multiplicative Lie algebras. [ABSTRACT FROM AUTHOR]

In graph theory, graph labeling is an essential area of study because labeled graphs offer useful mathematical models for coding theory, cryptography, astronomy, radar, database administration and communication networks. Consider a bijection for a graph G of order n, f : V (G) → {1, 2, ..., n}. The weight of a vertex z of G, expressed as w(z), is defined as the sum of labels assigned to all vertices adjacent to vertex z in G. If the weights are distinct for every unique pair of vertices y, z in V (G), then the labeling f is referred to as distance antimagic. A distance antimagic graph is any graph G that accepts such a labeling. Distance antimagic labeling on various basic graph products are discussed in this paper. We explore results on (a, d)-distance antimagic labeling for the lexicographic product G * H and distance antimagic labeling for the cartesian product G * H, tensor product G x H and strong product G x H in this work, where the graphs G and H are cycle related graphs, paths or complete graphs. Also, computer-aided algorithms are designed to verify that vertex weights are distinct. [ABSTRACT FROM AUTHOR]

In this paper we study the modular structure of the Foulkes module H (2 m) of the symmetric group S 2 m acting on set partitions of a set of size 2m into m sets each of size 2, defined over a field of positive characteristic p. We aim to understand the decomposition of the tensor powers of H (2 m) . In particular, we study the asymptotics of the non-projective part of tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds. [ABSTRACT FROM AUTHOR]

A graph G with p vertices and q edges has antimagic labeling if there is a bijection from the edge set of the graph to the label set { 1 , 2 , ... , q } such that p vertices must have distinct vertex sums, where the vertex sums are determined by adding up all the edge labels incident to each vertex v in V (G). Hartsfield and Ringel conjectured that every connected graph is antimagic, except P 2 . In this study, we identified a class of connected graphs that lend credence to this conjecture. In this paper, we proved that the tensor product of a Wheel Graph, Helm Graph, and Flower graph with Star is antimagic. [ABSTRACT FROM AUTHOR]

Let sl (2) ⋉ h n , n ≥ 1 , be the Galilean Lie algebra over a field of characteristic zero, here h n is the Heisenberg Lie algebra of dimension 2 n + 1 , and sl (2) acts on h n so that, sl (2) -modules, h n ≃ V (2 n - 1) ⊕ V (0) (here V(k) denotes the irreducible sl (2) -module of highest weight k). In this paper, we study the tensor product of two uniserial representations of sl (2) ⋉ h n . We obtain the sl (2) -module structure of the socle of V ⊗ W and we describe the space of intertwining operators Hom sl (2) ⋉ h n (V , W) , where V and W are uniserial representations of sl (2) ⋉ h n . The structure of the radical of V ⊗ W follows from that of the socle of V ∗ ⊗ W ∗ . The result is subtle and shows how difficult is to obtain the whole socle series of arbitrary tensor products of uniserials. In contrast to the serial associative case, our results for sl (2) ⋉ h n reveal that these tensor products are far from being a direct sum of uniserials; in particular, there are cases in which the tensor product of two uniserial (sl (2) ⋉ h n) -modules is indecomposable but not uniserial. Recall that a foundational result of T. Nakayama states that every finitely generated module over a serial associative algebra is a direct sum of uniserial modules. This article extends a previous work in which we obtained the corresponding results for the Lie algebra sl (2) ⋉ a m where a m is the abelian Lie algebra of dimension m + 1 and sl (2) acts so that a m ≃ V (m) as sl (2) -modules. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists an almost resolvable k-cycle decomposition of (Ku × Kg)(λ), (where × represents tensor product of graphs) for all even integers k ≥ 8 with some possible exceptions. [ABSTRACT FROM AUTHOR]

Kobayashi–Pevzner discovered in [Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs, Selecta Math. (N.S.) 22(2) (2016) 847–911, MR 3477337] that the failure of the multiplicity-one property in the fusion rule of Verma modules of 2 occurs exactly when the Rankin–Cohen brackets vanish, and classified all the corresponding parameters. In this paper, we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras . [ABSTRACT FROM AUTHOR]

Numerous works have been dedicated to the topic of m -isometries, including [2–6,14,18–20,27,47,48]. In this article, we introduce the concept of (m , N A) -isometry, where A is a non-zero operator and m is a positive integer, as an extension of the m -isometry class created by J. Alger and M. Stankus in the 1980s. We present some algebraic and spectral characteristics of (m , N A) -isometries. Additionally, we investigate the product of an (m , N A) -isometry by an (n , N B) -isometry, which enhances and broadens the previous work of Gu et al. on m -isometries [40]. Finally, we apply our main findings to elementary operators defined on the Hilbert-Schmidt class, which can be identified with a tensor product. This provides a new, less complicated, and non-combinatorial proof of Theorem 2.10 of [30]. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying two classes of finite dimensional -associative algebras, which are extensions of a family of -dimensional Kac-Paljutkin type semi-simple Hopf algebras . All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations. [ABSTRACT FROM AUTHOR]

A shortest [Formula: see text]-[Formula: see text] path between two vertices u and v of a graph G is a [Formula: see text]-[Formula: see text] geodesic of G. Let I[u, v] denote the set of all internal vertices lying on some [Formula: see text]-[Formula: see text] geodesic of G. For a nonempty subset S of [Formula: see text], let [Formula: see text]. If [Formula: see text], then S is a geodetic set of G. The cardinality of a minimum geodetic set of G is the geodetic number of G and it is denoted by [Formula: see text] In this paper, the exact geodetic numbers of the product graphs [Formula: see text] and [Formula: see text] are obtained, where T is a tree, [Formula: see text] denotes the complement of the complete graph [Formula: see text] and, [Formula: see text] and [Formula: see text] denote the tensor product and lexicographic product $($also called the wreath product$)$ of graphs, respectively.

In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $ A $ on a complex Hilbert space $ \mathcal{X} $, which is called jointly $ A $-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $ m $-tuples of operators that admit adjoint operators with respect to $ A $. Mainly, we prove that if $ \mathbf{B} = (B_1, \cdots, B_m) $ is a jointly $ A $-hyponormal $ m $-tuple of commuting operators, then $ \mathbf{B} $ is jointly $ A $-normaloid. This result allows us to establish, for a particular case when $ A $ is the identity operator, a sharp bound for the distance between two jointly hyponormal $ m $-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $ A $-$ p $-hyponormal operators with $ 0 < p < 1 $. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.

For a positive integer k , a proper k -coloring of a graph G is a mapping f : V (G) → { 1 , 2 , ... , k } such that f (u) ≠ f (v) for each edge u v of G. The smallest integer k for which there is a proper k -coloring of G is called the chromatic number of G , denoted by χ (G). A locally identifying coloring (for short, lid-coloring) of a graph G is a proper k -coloring of G such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer k such that G has a lid-coloring with k colors is called locally identifying chromatic number (for short, lid-chromatic number) of G , denoted by χ l i d (G). This paper studies the lid-coloring of the Cartesian product and tensor product of two graphs. We prove that if G and H are two connected graphs having at least two vertices, then (a) χ l i d (G □ H) ≤ χ (G) χ (H) − 1 and (b) χ l i d (G × H) ≤ χ (G) χ (H). Here G □ H and G × H denote the Cartesian and tensor products of G and H , respectively. We determine the lid-chromatic numbers of C m □ P n , C m □ C n , P m × P n , C m × P n and C m × C n , where C m and P n denote a cycle and a path on m and n vertices, respectively. [ABSTRACT FROM AUTHOR]

We consider the tensor products of square matrices of different sizes and introduce the stretching maps, which can be viewed as a generalized matricization. Stretching maps conserve algebraic properties of the tensor product, but are not necessarily injective. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, this leads to the averaging of the tensor product and possibly could be used to compress the data.

In this paper, the connection between the ϵ -pseudospectrum of the tensor product operator A1 ⊗ A2 and the ϵ -pseudospectrums of operators A1 and A2 has been investigated and some results are given about this connection under certain conditions. [ABSTRACT FROM AUTHOR]

We investigate two different ways of recovering a Grothendieck category as a filtered bicolimit of small categories and the compatibility of both with the tensor product of Grothendieck categories. Firstly, we show that any locally presentable linear category (and in particular any Grothendieck category) can be recovered as the filtered bicolimit of its subcategories of α-presentable objects, with α varying in the family of small regular cardinals. We then prove that the tensor product of locally presentable linear categories (and in particular the tensor product of Grothendieck categories) can be recovered as a fltered bicolimit of the Kelly tensor product of α-cocomplete linear categories of the corresponding subcategories of α-presentable objects. Secondly, we show that one can recover any Grothendieck category as a filtered bicolimit of its linear site presentations. We then prove that the tensor product of Grothendieck categories, in contrast with the first case, cannot be recovered in general as a filtered bicolimit of the tensor product of the corresponding linear sites. Finally, we show how the first presentation can be helpful when computing tensor products of cocontinuous linear functors between Grothendieck categories. [ABSTRACT FROM AUTHOR]

This paper is committed in characterizing the preserving maps for the class of separable matrices and a subclass of infinitely divisible matrices which are also separable. For this, an association between the classes of separable matrices and infinitely divisible matrices is established. Also, several properties and attributes of the above mentioned classes are stated and proved. [ABSTRACT FROM AUTHOR]

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules. [ABSTRACT FROM AUTHOR]

In this paper, we present a new analytical scheme to generate atomic solutions of partial differential equations. The theory of tensor product in Banach spaces coupled with some features of atomics operators are utilized to attain our results. Some demonstrative examples are given for completeness. [ABSTRACT FROM AUTHOR]

In this paper we introduce some results on hyponormal operators acting on a complex Hilbert space H. We give sufficient conditions for which the sum and product of two hyponormal operators is a hyponormal operator. Also, we study the sum direct and tensor product of this class. [ABSTRACT FROM AUTHOR]

This paper characterize important concepts such as the Riesz bases and pair of dual frames for tensor product of Hilbert spaces and introduce some special properties of pair of dual frames in these spaces. We show that Riesz bases and pair of dual frames in H ⊗ K are preserved under invertible and unitary operators on H . Finally, We present a representation for Hilbert–Schmidt operators by a pair of dual frames. [ABSTRACT FROM AUTHOR]

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant k G C . [ABSTRACT FROM AUTHOR]

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra A and a compact space X, let A be a Banach A-valued function algebra on X and let A be the subalgebra of A consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra A and the associated algebras A and A. That A inherits a certain regularity condition P to A and A is the easy part of the problem. We investigate the converse and show that, under certain conditions, A receives P form A and A. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we discuss the A-vertex magicness of some products of graphs, where A is a non-trivial additive Abelian group with identity 0. We characterize A-vertex magicness of the Cartesian product of Pn and Pm, where A has at least three elements, and we also discuss the A -vertex magicness of co-normal and tensor product of two graphs. Finally, we give a new method to construct an infinite number of A-vertex magic graphs from existing ones using the join operation. As a consequence we obtain an earlier result in [Balamoorthy et al. in AKCE Int. J. Graphs Comb. (2022) 19 (3), 268–275]

Let $X$ be a Banach space, $\mathcal{B}(X)$ the algebra of bounded linear operators on $X$ and $(J, \|{\cdot}\|_J)$ an admissible Banach ideal of $\mathcal{B}(X)$. For $T\in\mathcal{B}(X)$, let $L_{J, T}$ and $R_{J, T}\in\mathcal{B}(J)$ denote the left and right multiplication defined by $L_{J, T}(A)=TA$ and $R_{J, T}(A)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in\mathcal{B}(X)$, and their elementary operators $L_{J, T}$ and $R_{J, T}$. In particular, we give necessary and sufficient conditions for $L_{J, T}$ and $R_{J, T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J, T}$ is recurrent if and only if $Tøplus T$ is recurrent on $Xøplus X$. Moreover, we introduce the notion of a mixing recurrent operator and we show that $L_{J, T}$ is mixing recurrent if and only if $T$ is mixing recurrent.

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for regular completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation (σ , V) modulo N (V ~) is hyponormal modulo N (V ~) . [ABSTRACT FROM AUTHOR]

Tensor ring (TR) decomposition has been widely applied as an effective approach in a variety of applications to discover the hidden low‐rank patterns in multidimensional and higher‐order data. A well‐known method for TR decomposition is the alternating least squares (ALS). However, solving the ALS subproblems often suffers from high cost issue, especially for large‐scale tensors. In this paper, we provide two strategies to tackle this issue and design three ALS‐based algorithms. Specifically, the first strategy is used to simplify the calculation of the coefficient matrices of the normal equations for the ALS subproblems, which takes full advantage of the structure of the coefficient matrices of the subproblems and hence makes the corresponding algorithm perform much better than the regular ALS method in terms of computing time. The second strategy is to stabilize the ALS subproblems by QR factorizations on TR‐cores, and hence the corresponding algorithms are more numerically stable compared with our first algorithm. Extensive numerical experiments on synthetic and real data are given to illustrate and confirm the above results. In addition, we also present the complexity analyses of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this note, considering the main properties of closed ideals of C* ­-algebras, we will determine the structure of closed ideals of the C* ­-algrbra C(X, A), the space of all continuous functions from compact Hausdorff space X to C* ­-algebra A. Indeed, we will show that for every closed ideal I of C(X, A), there is some closed subset F of topological space X × prim(A), such that I = {f ∈ C(X, A) : ∀ (x, P) ∈ F, f(x) ∈ P}. [ABSTRACT FROM AUTHOR]

A m-quasi class A k , denoted as M ∈ Q (A k , m) , is defined as an operator M acting on a complex Hilbert space H , provided that the following condition holds true: M ∗ m | M k + 1 | 2 k + 1 M m ≥ M ∗ m | M | 2 M m , where both k and m are positive integers. In this research, we unveil fundamental structural characteristics of these operators. Leveraging these findings, we can establish that P is self-adjoint if P represents the Riesz idempotent associated with the isolated point λ in the spectrum of M ∈ Q (A k , m) . Additionally, we provide a necessary and sufficient condition for M ⊗ N to belong to Q A k when both M and N are non-zero. [ABSTRACT FROM AUTHOR]

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over N 0 k. As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system. [ABSTRACT FROM AUTHOR]

We show that if Auslander's depth formula holds for non-zero Tor-independent modules over Cohen-Macaulay local rings of dimension 1, then it holds for such modules over any Cohen-Macaulay local ring. More generally, we show that the depth formula for non-zero Tor-independent modules that have finite Cohen-Macaulay dimension over depth 1 local rings implies the depth formula for such modules over any positive depth local ring. [ABSTRACT FROM AUTHOR]

We study projective and injective tensor products of Banach L 0 -modules over a σ -finite measure space. En route, we extend to Banach L 0 -modules several technical tools of independent interest, such as quotient operators, summable families, and Schauder bases. [ABSTRACT FROM AUTHOR]

The geometric Satake correspondence can be regarded as a geometric construction of rational representations of a complex connected reductive group G. In their study of this correspondence, Mirkovi'c and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig. [ABSTRACT FROM AUTHOR]

In this paper, we obtain necessary and sufficient conditions for decomposing tensor product of complete graphs into some connected unicyclic bipartite graphs with eight edges. [ABSTRACT FROM AUTHOR]

In this article, we define the tensor product V ⊗ W of a representation V of a quiver Q with a representation W of an another quiver Q ′ , and show that the representation V ⊗ W is semistable if V and W are semistable. We give a relation between the universal representations on the fine moduli spaces N 1 , N 2 and N 3 of representations of Q , Q ′ and Q ⊗ Q ′ respectively over arbitrary algebraically closed fields. We further describe a relation between the natural line bundles on these moduli spaces when the base is the field of complex numbers. We then prove that the internal product Q ̃ ⊗ Q ′ ̃ of covering quivers is a sub-quiver of the covering quiver Q ⊗ Q ′ ˜. We deduce the relation between stability of the representations V ⊗ W ˜ and V ̃ ⊗ W ̃ , where V ̃ denotes the lift of the representation V of Q to the covering quiver Q ̃. We also lift the relation between the natural line bundles on the product of moduli spaces N 1 ̃ × N 2 ̃ . [ABSTRACT FROM AUTHOR]

The state of moderate and severe food insecurity in Ethiopia has not been significantly reduced for a long time due to cultural, natural, and manmade shocks, which cover most part of the country with considerable magnitude and have adverse effects on the health and economy. This temporal evolution of the wider geographic distribution of the food insecurity levels has not been investigated for the feeding culture and shocks effect in Ethiopia, though previous studies have indicated significant geographic distribution and related factors. In addition, the longtime zone-specific comprehensive drivers were not assessed. Therefore, this study focused on investigating the feeding culture-adjusted food insecurity levels (FCSL) and their evolutional sustainability across zones by identifying the factors causing each level of household's food insecurity tailored to a specific zone.In this study, Ethiopian socio-economic longitudinal data from years 2012, 2014, and 2016 with a sample size of 3835 households were analyzed. An ordinal spatiotemporal model with different interactions under empirical Bayes estimation was adopted, and the type III interaction with Markov random field was selected to reveal the evolution of FCSL sustainability across zones and find-out the causing factors of households to each level of food insecurity tailored to zone-level by analyzing the effects of diverse factors and shocks. The result reveals that a greater portion (52.07 %) of households’ population is moderately and chronically food insecure. Basically, households living in neighboring zones have significantly similar food insecurity levels, and slight improvements were observed over time. This transition over time within neighboring zones, revealed that chronic food insecurity in neighboring zones has transited to moderate food insecurity, particularly in most of the northern and southwestern areas of Ethiopia. However, the interaction term showed that the change in food insecurity levels for households living in neighboring zones is similar at one time point, but the evolution is not sustainable. Therefore, working on major zone-specific factors, such as enhancing zone-level urbanization through improving market and road linkage to rural areas, education, employment, non-agricultural businesses, and promoting integrated farming by conserving soil with consideration of additional confounding factors, will bring change and can sustainably eradicate moderate and chronic food insecurity in zone households through controlling dependency ratios, shocks, and small land size farming. This study brings resilient, manageable, and zone-specific mitigation for households living in food-insecure and vulnerable zones.

The bicategory of Landau-Ginzburg models denoted by ℒ𝒢K possesses adjoints and this helps in explaining a certain duality that exists in the setting of Landau-Ginzburg models in terms of some specified relations. The construction of ℒ𝒢K is reminiscent of, but more complex than, the construction of the bicategory of associative algebras and bimodules. In this paper, we review this complex but very inspiring construction in order to expose it more to pure mathematicians. In particular, we spend some time explaining the intricate construction of unit morphisms in this bicategory from a new vantage point. Besides, we briefly discuss how this bicategory could be constructed in more than one way using the variants of the Yoshino tensor product. Furthermore, without resorting to Atiyah classes, we prove that the left and right unitors in this bicategory have direct right inverses but do not have direct left inverses.

AbstractFor a bounded linear operator, by local spectral theory methods, we study the property [Formula: see text] which means that the difference of the approximate point spectrum with the upper semi-Fredholm spectrum coincides with the set of all finite-range left poles. We will investigate this property under closed proper subspaces of [Formula: see text] also under the tensor product. In addition, the relationships of this property with other spectral properties are studied. Among others, we will obtain several characterizations for the operators that verify the property (bz) and show that the set of these operators is closed.