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Partly from lack of effective conventional therapeutics, patients with irritable bowel syndrome (IBS) turn to complementary and alternative approaches, including Traditional Chinese Medicine (TCM). Key to TCM's approach to IBS is individualized therapies targeted at subgroups. Subgroups represent distinct patterns of dysregulation (e.g. "excess" or "deficiency") identified by both intestinal and extra-intestinal symptoms. Our objective was to identify operational criteria supporting the existence of TCM-based subgroups in IBS and to assess reliability and validity of these criteria. Using TCM principles, items were selected on face validity from conventional questionnaires. TCM practitioners evaluated items for content and face validity. Symptom items and a set of patient cases with item responses were validated by examining patient's pattern of response to items and assessing the consistency with which practitioners diagnosed patients on the spectrum of an "excess" or "deficiency" syndrome. Standard correlation analysis revealed 33 intestinal and extra-intestinal symptom items. There was high degree of practitioner agreement in assessing individual items to particular patterns. External validation by practitioners of cases showed high internal consistency among practitioners (Cronbach's alpha coefficients of 0.91 and 0.87 for excess and deficiency, respectively) and high correlation of average practitioner rating to original questionnaire generated scores (Pearson correlation coefficients of 0.94 and 0.92 for excess and deficiency, respectively). This pilot study provides preliminary support for a methodology to identify novel subgroups of IBS patients related to the TCM classification, which may differ in underlying pathophysiology and treatment responses.

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J₁,…,J_m〉 of a finitely generated nonabelian free group F = F(x₁,…,x_n) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log*(N)), where N is the sum of the word lengths of the given generators of H.

Stallings showed that every finitely generated subgroup of a free group is canonically represented by a finite minimal immersion of a bouquet of circles. In terms of the theory of automata, this is a minimal finite inverse automaton. This allows for the deep algorithmic theory of finite automata and finite inverse monoids to be used to answer questions about finitely generated subgroups of free groups.

In this paper, we attempt to apply the same methods to other classes of groups. A fundamental new problem is that the Stallings folding algorithm must be modified to allow for "sewing" on relations of non-free groups. We look at the class of groups that are amalgams of finite groups. It is known that these groups are locally quasiconvex and thus, all finitely generated subgroups are represented by finite automata. We present an algorithm to compute such a finite automaton and use it to solve various algorithmic problems.

Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups.

In the present paper we employ the generalized Stallings' folding method, developed by the author, to introduce a procedure, which has given a subgroup H of a free product of finite groups reads off its Kurosh decomposition from the subgroup graph of H.

We show every locally solvable subgroup of $\mathrm{\text{PLo}}(I)$ is countable. A corollary is that an uncountable wreath product of copies of $\mathbb{Z}$ with itself does not embed into $\mathrm{\text{PLo}}(I)$.

In this paper, we characterize several classes of groups by the properties of their weak congruence lattices. Namely, we give necessary and sufficient conditions for the weak congruence lattice of a group, under which this group is a $T$-group, $T^{\ast }$-group, metacyclic, cocyclic, hyperabelian, polycyclic, $N$-group and $\tilde{N}$-group. We also discuss groups for which all subgroups are simple, like the Tarski monsters, and we show that they represent a class of non-Dedekind groups with a particular embedding property for lattices of normal subgroups. All the mentioned characterized groups are related to (different) series of subgroups, and we represent these series as chains of intervals in the weak congruence lattice. The corresponding conditions are formulated in a purely lattice-theoretic language.

We give a simple algorithm to solve the subgroup membership problem for virtually free groups given as a graph of finite groups. For a fixed virtually free group with a fixed generating set X, the subgroup membership problem is uniformly solvable in time $O(N{log}^{\ast }(N))$ where N is the sum of the word lengths of the inputs with respect to X. For practical purposes, this can be considered to be linear time. The algorithm is simple enough to work out small examples by hand and concrete examples are given to show how it can be used for computations in $\mathrm{\text{SL}}(2,\mathbb{Z})$ and $\mathrm{\text{GL}}(2,\mathbb{Z})$. We also give fast algorithm to decide whether a finitely generated subgroup of a virtually free group is free.

Kötter and Kschischang presented in 2008 a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this alphabet is the map d : (U, V) ↦ dim(U+V)-dim(U∩V). In this paper we generalize this model to arbitrary modular lattices, i.e. we consider codes, which are subsets of modular lattices. The used metric in this general case is the map d : (u, v) ↦ h(u ∨ v) - h(u ∧ v), where h is the height function of the lattice. We apply this model to submodule lattices. Moreover, we show a method to compute the size of spheres in certain modular lattices and present a sphere packing bound, a sphere covering bound, and a Singleton bound for codes, which are subsets of modular lattices.

Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are planar. In addition, we classify the finite groups whose permutability graphs of subgroups are one of outerplanar, path, cycle, unicyclic, claw-free or C₄-free. Also, we investigate the planarity of permutability graphs of subgroups of infinite groups.

The permutability graph of subgroups of a given group G, denoted by Γ(G), is a graph with vertex set consists of all the proper subgroups of G and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are one of bipartite, star graph, C₃-free, C₅-free, K₄-free, K₅-free, K_1,4-free, K_2,3-free or P_n-free (n = 2, 3, 4). We investigate the same for infinite groups also. Moreover, some results on the girth, completeness and regularity of the permutability graphs of subgroups of groups are obtained. Among the other results, we characterize groups Q₈, S₃ and A₄ by using their permutability graphs of subgroups.