FIXED-POINT PROBLEMS IN DISCRETE TOMOGRAPHY: A CASE OF SQUARE WINDOWS

We describe an iterative construction of Lagrangian tori in the complex Grassmannian \operatorname{Gr}(k,n) , based on the cluster algebra structure of the coordinate ring of a mirror Landau–Ginzburg model proposed by Marsh and Rietsch (2020). Each torus comes with a Laurent polynomial, and local systems controlled by the k -variables Schur polynomials at the n -th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over \mathbb{C} .

In a recent article, the coordinate ring of the nodal cubic was given the structure of a quantum homogeneous space. Here the corresponding coalgebra Galois extension is expressed in terms of quantum groups at roots of unity, and is shown to be cleft. Furthermore, the minimal quotient extensions are determined.

The Ehrenfeucht Conjecture [5] states that if Μ is a finitely generated free monoid with nonempty subset S, then there is a finite subset T⊂S (a “test set”) such that given two endomorphisms f and g on Μ, f and g agree on S if and only if they agree on T. In[4], the authors prove that the above conjecture is equivalent to the following conjecture: a system of equations in a finite number of unknowns in Μ is equivalent to a finite subsystem. Since a finitely generated free monoid embeds naturally into the free group with the same number of generators, it is natural to ask whether a free group of finite rank has the above property on systems of equations. A restatement of the question motivates the following.

We define the $\textit{Divisor Divisibility Sequence}$ associated to a Laurent polynomial $f\in\mathbb{Z}[X_1^{\pm1},\ldots,X_N^{\pm1}]$ to be the sequence $W_n(f)=\prod f(\zeta_1,\ldots,\zeta_N)$, where $\zeta_1,\ldots,\zeta_N$ range over all $n$'th roots of unity with $f(\zeta_1,\ldots,\zeta_N)\ne0$. More generally, we define $W_\Lambda(f)$ analogously for any finite subgroup $\Lambda\subset(\mathbb C^*)^N$. We investigate divisibility, factorization, and growth properties of $W_\Lambda(f)$ as a function of $\Lambda$. In particular, we give the complete factorization of $W_\Lambda(f)$ when $f$ has generic coefficients, and we prove an analytic estimate showing that the rank-of-apparition sets for $W_\Lambda(f)$ are not too large.

The general theory of quaternion q-difference equations is completely different from traditional q-difference equations in the complex space for the special algebraic structure of the quaternion space, such as the non-commutativity of multiplication among elements. In this paper, some properties of basic quaternion q-discrete functions, such as the quaternion q-exponential function, are obtained. The existence, uniqueness, and extension theorems of the solution for higher-order linear quaternion q-difference equations (QQDCEs) are established through constructing quaternion q-stepwise approximation sequences, and the equivalent variable transform between higher-order linear QQDCEs and quaternion linear q-difference equations is given. Moreover, some basic results, such as the Wronskian formula, Liouville formula, and general solution structure theorems of higher-order linear QQDCEs with constant and variable coefficients, are established by applying the quaternion characteristic polynomial and the quaternion determinant algorithm. In addition, some particular and general solution formulas of non-homogeneous QQDCEs are obtained under the non-commutative condition of quaternion elements. Finally, several examples are provided to illustrate the feasibility of our obtained results.

On integral representations of twisted group rings

Let ${\cal P}_f({\bf N})$ be the set of finite nonempty subsets of ${\bf N}$ and for $F,G\in{\cal P}_f({\bf N})$ write $F < G$ when $\max F < \min G$. Let $X=\{(F,G):F,G\in{\cal P}_f({\bf N})$ and $F < G\}$. A triangle in $X$ is a set of the form $\{(F\cup H,G),(F,G),(F,H\cup G)\}$ where $F < H < G$. Motivated by a question of Erdős, Nešetríl, and Rödl regarding three term arithmetic progressions, we show that any finite subset $Y$ of $X$ contains a relatively large triangle free subset. Exact values are obtained for the largest triangle free sets which can be guaranteed to exist in any set $Y\subseteq X$ with $n$ elements for all $n\leq 14$.

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weightA, denoted by DA(G), is defined to be the least positive integer t such that, for every sequence (x1,..., xt) with xi ∈ G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and ai ∈ A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, EA(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some ai ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n − 1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote EA(G) and DA(G) by EA(n) and DA(n) respectively.

We introduce the notion of constructive suprema and of constructively directed sets. The Axiom of Choice turns out to be equivalent to the postulate that every supremum is constructive, but also to the hypothesis that every directed set admits a function assigning to each finite subset an upper bound. The Axiom of Multiple Choice (which is known to be weaker than the full Axiom of Choice in set theory without foundation) implies a simple set-theoretical induction principle (SIP), stating that any system of sets that is closed under unions of well-ordered subsystems and contains all finite subsets of a given set must also contain that set itself. This is not provable without choice principles but equivalent to the statement that the existence of joins for constructively directed subsets of a poset follows from the existence of joins for nonempty well-ordered subsets. Moreover, we establish the equivalence of SIP with several other fundamental statements concerning inductivity, compactness, algebraic closure systems, and the exchange between chains and directed sets.

On hyperspaces of generalized metric spaces

Green’s matrices

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in mathbb {K}[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(alpha )=f(beta ) for alpha ,beta in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

Eigenvalues are special sets of scalars associated with a given matrix. In other words for a given matrix A, if there exist a non-zero vector V such that, <i>AV= &lambda;V</i> for some scalar &lambda;, then &lambda; is called the eigenvalue of matrix A with corresponding eigenvector V. The set of all nxm matrices over a field F is denoted by M_nm (F). If m = n, then the matrices are square, and which is denoted by Mn (F). We omit the field F = C and in this case we simply write M_nm or M_n as appropriate. Each square matrix AϵM_nm has a value in R associated with it and it is called its determinant which is use full for solving a system of linear equation and it is denoted by det (A). Consider a square matrix AϵM_n with eigenvalues <i>&lambda;</i>, and then by definition the eigenvectors of A satisfy the equation, <i>AV = &lambda;V</i>, where <i>v={v₁, v₂, v₃…………v_n}</i>. That is, AV=<i>&lambda;</i>V is equivalent to the homogeneous system of linear equation (A-<i>&lambda;I</i>) v=0. This homogeneous system can be written compactly as (A-<i>&lambda;</i>I) V = 0 and from Cramer’s rule, we know that a linear system of equation has a non-trivial solution if and only if its determinant is zero, so the solution <i>&lambda;</i> is given by det (A-<i>&lambda;</i>I) =0. This is called the characteristic equation of matrix A and the left hand side of the characteristic equation is the characteristic polynomial whose roots are equals to <i>&lambda;</i>.

In the first part of the present paper, we shall make some simple observations on the ideal class groups of algebraic number fields, following the group-theoretical method of Tschebotarew. The applications on cyclotomic fields (Theorems 5, 6) may be of some interest. In the last section, we shall give a proof to a theorem of Kummer on the ideal class group of a cyclotomic field.

We propose a general recurrent neural-network (RNN) model for nonlinear optimization over a nonempty compact convex subset which includes the bound subset and spheroid subset as special cases. It is shown that the compact convex subset is a positive invariant and attractive set of the RNN system and that all the network trajectories starting from the compact convex subset converge to the equilibrium set of the RNN system. The above equilibrium set of the RNN system coincides with the optimum set of the minimization problem over the compact convex subset when the objective function is convex. The analysis of these qualitative properties for the RNN model is conducted by employing the properties of the projection operator of Euclidean space onto the general nonempty closed convex subset. A numerical simulation example is also given to illustrate the qualitative properties of the proposed general RNN model for solving an optimization problem over various compact convex subsets.