Bulletin of the Malaysian Mathematical Sciences Society最新文献

The Garden of Eden theorem is a fundamental result in the theory of cellular automata, which establishes a necessary and sufficient condition for the surjectivity of a cellular automaton with a finite alphabet over an amenable group. Specifically, the theorem states that such an automaton is surjective if and only if it is pre-injective, where pre-injectivity requires that any two almost equal configurations with the same image under the automaton must be equal. This paper focuses on establishing the Garden of Eden theorem over a (varphi )-cellular automaton by demonstrating both Moore theorem and Myhill theorem over (varphi )-cellular automata are true. These results have significant implications for the theoretical framework of the Garden of Eden theorem and its applicability across diverse groups or altered versions of the same group. Overall, this paper provides a more comprehensive study of (varphi )-cellular automata and extends the Garden of Eden theorem to a broader class of automata.

Let (T=biggl (begin{matrix} R&{}0 C&{}S end{matrix}biggr )) be a formal triangular matrix ring with C a semidualizing (S, R)-bimodule. It is proven that (1) A left S-module M in Bass class is C-torsionless (resp. C-reflexive) if and only if (biggl (begin{array}{c} textrm{Hom}_{S}(C,M) M end{array}biggr )) is a torsionless (resp. reflexive) left T-module; (2) A left S-module M in Bass class is C-Gorenstein projective if and only if (biggl (begin{array}{c} textrm{Hom}_{S}(C,M) Mend{array}biggr )) is a Gorenstein projective left T-module; (3) If C is a faithfully semidualizing (S, R)-bimodule, then a left S-module M is C-n-tilting if and only if (biggl (begin{array}{c}textrm{Hom}_{S}(C,M) Soplus Mend{array}biggr )) is an n-tilting left T-module.

Let ({p_n}_{nge 1}) and ({ d_n}_{nge 1}) be two sequences of integers such that (|p_n|>|d_n|>0) and ({d_n}_{nge 1}) is bounded. It is proven by Deng and Li that the Moran-type Bernoulli convolution

is a spectral measure if and only if the numbers of factor 2 in the sequence (big {frac{p_1p_2dots p_n}{2d_n}big }_{nge 1}) are different from each other. Unfortunately, there is a gap in the proof of the sufficiency. Here we give a new proof to close the gap.

Let (Asubseteq B) be an extension of integral domains, (B[![X]!]) be the power series ring over B, and (R=A + XB[![X]!]) be a subring of (B[![X]!].) In this paper, we give a complete description of v-invertible v-ideals (with nonzero trace in A) of R. We show that if B is a completely integrally closed domain and I is a fractional divisorial v-invertible ideal of R with nonzero trace over A, then (I = u(J_1 + XJ_2[![X]!])) for some (uin qf(R),) (J_2) an integral divisorial v-invertible ideal of B and (J_1subseteq J_2) a nonzero ideal of A.

A family (Delta ) of subsets of ({1,2,ldots ,n}) is a simplicial complex if all subsets of F are in (Delta ) for any (Fin Delta ,) and the element of (Delta ) is called the face of (Delta .) Let (V(Delta )=bigcup _{Fin Delta } F.) A simplicial complex (Delta ) is a near-cone with respect to an apex vertex (vin V(Delta )) if for every face (Fin Delta ,) the set ((Fbackslash {w})cup {v}) is also a face of (Delta ) for every (win F.) Denote by (f_{i}(Delta )=|{Ain Delta :|A|=i+1}|) and (h_{i}(Delta )=|{Ain Delta :|A|=i+1,nnot in A}|) for every i, and let (text {link}_{Delta }(v)={E:Ecup {v}in Delta , vnot in E}) for every (vin V(Delta ).) Assume that p is a prime and (kgeqslant 2) is an integer. In this paper, some extremal problems on k-wise L-intersecting families for simplicial complexes are considered. (i) Let (L={l_1,l_2,ldots ,l_s}) be a subset of s nonnegative integers. If (mathscr {F}={F_1, F_2,ldots , F_m}) is a family of faces of the simplicial complex (Delta ) such that (|F_{i_1}cap F_{i_2}cap cdots cap F_{i_k}|in L) for any collection of k distinct sets from (mathscr {F},) then (mleqslant (k-1)sum _{i=-1}^{s-1}f_i(Delta ).) In addition, if the size of every member of (mathscr {F}) belongs to the set (K:={k_1,k_2,ldots ,k_r}) with (min K>s-r,) then (mleqslant (k-1)sum _{i=s-r}^{s-1}f_i(Delta ).) (ii) Let (L={l_1,l_2,ldots ,l_s}) and (K={k_1,k_2,ldots ,k_r}) be two disjoint subsets of ({0,1,ldots ,p-1}) such that (min K>s-2r+1.) Assume that (Delta ) is a simplicial complex with (nin V(Delta )) and (mathscr {F}={F_1, F_2,ldots , F_m}) is a family of faces of (Delta ) such that (|F_j|pmod {p}in K) for every j and (|F_{i_1}cap F_{i_2}cap cdots cap F_{i_k}|pmod {p}in L) for any collection of k distinct sets from (mathscr {F}.) Then (mleqslant (k-1)sum _{i=s-2r}^{s-1}h_i(Delta ).) (iii) Let (L={l_1,l_2,ldots ,l_s}) be a subset of ({0,1,ldots ,p-1}.) Assume that (Delta ) is a near-cone with apex vertex v and (mathscr {F}={F_1, F_2,ldots , F_m}) is a family of faces of (Delta ) such that (|F_j|pmod {p}not in L) for every j and (|F_{i_1}cap F_{i_2}cap cdots cap F_{i_k}|pmod {p}in L) for any collection of k distinct sets from (mathscr {F}.) Then ( mleqslant (k-1)sum _{i=-1}^{s-1}f_i(text {link}_Delta (v)).)

This manuscript demonstrates robust optimality conditions, Wolfe and Mond–Weir type robust dual models for a robust mathematical programming problem involving vanishing constraints (RMPVC). Further, the theorems of duality are examined based on the concept of generalized higher order invexity and strict invexity that establish relations between the primal and the Wolfe type robust dual problems. In addition, the duality results for a Mond–Weir type robust dual problem based on the concept of generalized higher order pseudoinvex, strict pseudoinvex and quasiinvex functions are also studied. Furthermore, numerical examples are provided to validate robust optimality conditions and duality theorems of Wolfe and Mond–Weir type dual problems.

The smallest M-eigenvalue (tau _M ({mathcal {A}})) of a fourth-order partial symmetric tensor ({mathcal {A}}) plays an important role in judging the strong ellipticity condition (abbr. SE-condition) in elastic mechanics. Specifically, if (tau _M ({mathcal {A}})>0), then the SE-condition of ({mathcal {A}}) holds. In this paper, we establish lower and upper bounds of (tau _M ({mathcal {A}})) via extreme eigenvalues of symmetric matrices and tensors constructed by the entries of ({mathcal {A}}). In addition, when ({mathcal {A}}) is an elasticity Z-tensor, we establish lower bounds for (tau _M ({mathcal {A}})) via the extreme C-eigenvalues of piezoelectric-type tensors. Finally, numerical examples show the efficiency of our proposed bounds in judging the SE-condition of ({mathcal {A}}).

For (1le p<infty ), let (L_p({mathcal {M}},tau )) be the non-commutative (L_p)-space associated with a von Neumann algebra ({mathcal {M}}), where ({mathcal {M}}) admits a normal semifinite faithful trace (tau ). Using the trace (tau ), Banach duality formula and Gâteaux derivative, this paper characterizes an element (ain L_p({mathcal {M}},tau )) such that

where ({mathcal {B}}_p) is a closed linear subspace of (L_p({mathcal {M}},tau )) and (Vert cdot Vert _p) is the norm on (L_p({mathcal {M}},tau )). Such an a is called ({mathcal {B}}_p)-minimal. In particular, minimal elements related to the finite-diagonal-block type closed linear subspaces

(converging with respect to (Vert cdot Vert _p)) are considered, where ({e_i}_{i=1}^{infty }) is a sequence of mutually orthogonal and (tau )-finite projections in a (sigma )-finite von Neumann algebra ({mathcal {M}}), and ({mathcal {S}}) is the set of elements in ({mathcal {M}}) with (tau )-finite supports.