Bulletin of The Iranian Mathematical Society最新文献

Let (text {S}) be a finite set, each element of which receives a color. A rainbow t-set of (text {S}) is a t-subset of (text {S}) in which different elements receive different colors. Let (left( {begin{array}{c}text {S} tend{array}}right) ) denote the set of all rainbow t-sets of (text {S}), let (left( {begin{array}{c}text {S} le tend{array}}right) ) represent the union of (left( {begin{array}{c}text {S} iend{array}}right) ) for (i=0,ldots , t), and let (2^text {S}) stand for the set of all rainbow subsets of (text {S}). The rainbow inclusion matrix (mathcal {W}^{text {S}}) is the (2^text {S}times 2^{text {S}}) (0, 1) matrix whose (T, K)-entry is one if and only if (Tsubseteq K). We write (mathcal {W}_{t,k}^{text {S}}) and (mathcal {W}_{le t,k}^{text {S}}) for the (left( {begin{array}{c}text {S} tend{array}}right) times left( {begin{array}{c}text {S} kend{array}}right) ) submatrix and the (left( {begin{array}{c}text {S} le tend{array}}right) times left( {begin{array}{c}text {S} kend{array}}right) ) submatrix of (mathcal {W}^{text {S}}), respectively, and so on. We determine the diagonal forms and the ranks of (mathcal {W}_{t,k}^{text {S}}) and (mathcal {W}_{le t,k}^{text {S}}). We further calculate the singular values of (mathcal {W}_{t,k}^{text {S}}) and construct accordingly a complete system of ((0,pm 1)) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let (mathcal {D}^{text {S}}_{t,k}) denote the integral lattice orthogonal to the rows of (mathcal {W}_{le t,k}^{text {S}}) and let (overline{mathcal {D}}^{text {S}}_{t,k}) denote the orthogonal lattice of (mathcal {D}^{text {S}}_{t,k}). We make use of Frankl rank to present a ((0,pm 1)) basis of (mathcal {D}^{text {S}}_{t,k}) and a (0, 1) basis of (overline{mathcal {D}}^{text {S}}_{t,k}). For any commutative ring R, those nonzero functions (fin R^{2^{text {S}}}) satisfying (mathcal {W}_{t,ge 0}^{text {S}}f=0) are called null t-designs over R, while those satisfying (mathcal {W}_{le t,ge 0}^{text {S}}f=0) are called null ((le t))-designs over R. We report some observations on the distributions of the support sizes of null designs as well as the structure of null designs with extremal support sizes.

The primary aim of this paper is to focus on the stability analysis of an advanced neural stochastic functional differential equation with finite delay driven by a fractional Brownian motion in a Hilbert space. We examine the existence and uniqueness of mild solution of ( {textrm{d}}left[ {x}_{a}(s) + {mathfrak {g}}(s, {x}_{a}(s - omega (s)))right] =left[ {mathfrak {I}}{x}_a(s) + {mathfrak {f}}(s, {x}_a(s -varrho (s)))right] {textrm{d}}s + varsigma (s){textrm{d}}varpi ^{{mathbb {H}}}(s),) (0le sle {mathcal {T}}), ({x}_a(s) = zeta (s), -rho le sle 0. ) The main goal of this paper is to investigate the Ulam–Hyers stability of the considered equation. We have also provided numerical examples to illustrate the obtained results. This article also discusses the Euler–Maruyama numerical method through two examples.

Inclusion properties are studied for balls of the triangular ratio metric, the hyperbolic metric, the (j^*)-metric, and the distance ratio metric defined in the unit ball domain. Several sharp results are proven and a conjecture about the relation between triangular ratio metric balls and hyperbolic balls is given. An algorithm is also built for drawing triangular ratio circles or three-dimensional spheres.

In this paper, we show that for all (vequiv 0,1) (mod 5) and (vge 15), there exists a super-simple (v, 5, 2) directed design. Moreover, for these parameters, there exists a super-simple (v, 5, 2) directed design such that its smallest defining sets contain at least half of its blocks. Also, we show that these designs are useful in constructing parity-check matrices of LDPC codes.

In this paper, we consider a relativistic Abelian Chern–Simons equation

on a connected finite graph (G=(V, E)), where (lambda >0) is a constant; (a>b>0); (N_{1}) and (N_{2}) are positive integers; (p_{1}, p_{2}, ldots , p_{N_{1}}) and (q_{1}, q_{2}, ldots , q_{N_{2}}) denote distinct vertices of V. Additionally, (delta _{p_{j}}) and (delta _{q_{j}}) represent the Dirac delta masses located at vertices (p_{j}) and (q_{j}). By employing the method of constrained minimization, we prove that there exists a critical value (lambda _{0}), such that the above equation admits a solution when (lambda ge lambda _{0}). Furthermore, we employ the mountain pass theorem developed by Ambrosetti–Rabinowitz to establish that the equation has at least two solutions when (lambda >lambda _{0}).

In this paper, we present the complete list of (U_{q}(sl_{2}))-symmetries on quantum polynomial algebra (k_q[x^{pm 1},y]) in the case that the action of the generator K of (U_{q}(sl_{2})) is a non-toric automorphism. The conditions for the isomorphism of such structures are explored as well.

Let (mathcal {C}) be an abelian monoidal category. It is proved that the nilpotent category ({text {Nil}}(mathcal {C})) of (mathcal {C}) admits almost monoidal structure except the unit axiom. As an application, it is proved that Hom and Tensor functors exist over ({text {Nil}}(mathcal {C})) and tensor–hom adjunction remains true over the nilpotent category of the category of finite-dimensional vector spaces, which develops some recent results on this topic.

How to accurately model and effectively suppress the spread of network viruses has been a major concern in the field of complex networks and cybersecurity. Most existing work often considers the transmission between the infected and uninfected nodes (i.e., horizontal transmission) and assumes that all new nodes connected to the Internet are susceptible, but the nodes might have been implanted with a backdoor or virus by attackers or infected nodes before connecting to the Internet. This vertical transmission also provides an important route for virus propagation. In this paper, we investigate the propagation of network viruses under the combined influence of network topology and hybrid transmission (i.e., horizontal and vertical transmissions). Through rigorous qualitative analysis, we identify the propagation threshold (R_0) which determines whether viruses in the network tend to become extinct or persist, and explore the impacts of vertical transmission on the viral spread. Furthermore, we consider the problem of how to dynamically contain the hybrid spread of network viruses with limited resources. By utilizing optimal control theory, we prove the existence of an optimal control strategy. Finally, a group of representative simulation experiments verify the validity of the theoretical findings. Specifically, the simulation results show that the optimal control strategy proposed in this paper reduces the value of the target generic function J by 67.69% compared with no control.

For graphs G and H, the Ramsey number r(G, H) is the smallest number N, such that any red/blue edge-coloring of (K_N) contains either a red copy of G or a blue copy of H. Let (F_n=K_1+nK_2) be a fan and (W_4=K_1+C_4) be a wheel of order five. In this paper, we show that the Ramsey number (r(W_4,F_n)=4n+1) for all sufficiently large n. Moreover, this implies that a large fan (F_n) is (W_4)-good.

In this paper we will make use of Brouwer’s fixed point theorem to obtain collectively coincidence point results for multivalued maps belonging to similar classes.