Case Report: Robotically Assisted Excision of Cystic Tumor Located in a Difficult to Access Area in the Liver

Introduction: Cystic liver lesions may be benign cysts, parasitic infestations, or malignant tumors requiring surgical resection. Hilar location and relation to major vasculature present challenges in conventional surgical access and resection.Materials and Methods: We describe totally robotic excision of a cystadenoma in a 55-year-old woman without complication. Time points in the accompanying video (https://youtu.be/Tn_QPgpSHA4) are embedded within the text.Results: Advantages of the robotic technique lie in overcoming the natural restriction of conventional laparoscopic instruments, easier repair, and control of intraoperative vascular injuries using EndoWrist® instruments, ergonomic dissection close to major vasculature and reduced intraoperative blood loss as dissection is easier.Discussion: Indications for robotic surgery included the large size of the cystic lesion, its intrahepatic location, and compression of the inferior vena cava (IVC) and right and middle hepatic veins. Had robotic removal of the lesion not been feasible, the entire lobe of the liver would have required resection.

Approval-Based Apportionment

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.

Learning to Crawl

Web crawling is the problem of keeping a cache of webpages fresh, i.e., having the most recent copy available when a page is requested. This problem is usually coupled with the natural restriction that the bandwidth available to the web crawler is limited. The corresponding optimization problem was solved optimally by Azar et al. (2018) under the assumption that, for each webpage, both the elapsed time between two changes and the elapsed time between two requests follows a Poisson distribution with known parameters. In this paper, we study the same control problem but under the assumption that the change rates are unknown a priori, and thus we need to estimate them in an online fashion using only partial observations (i.e., single-bit signals indicating whether the page has changed since the last refresh). As a point of departure, we characterise the conditions under which one can solve the problem with such partial observability. Next, we propose a practical estimator and compute confidence intervals for it in terms of the elapsed time between the observations. Finally, we show that the explore-and-commit algorithm achieves an O(√T) regret with a carefully chosen exploration horizon. Our simulation study shows that our online policy scales well and achieves close to optimal performance for a wide range of parameters.

Uniquely $K_r^{(k)}$-Saturated Hypergraphs

In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\leqslant k <r<n$, a $k$-uniform hypergraph $H$ with $n$ vertices is uniquely $K_r^{(k)}$-saturated if $H$ does not contain $K_r^{(k)}$ but adding to $H$ any $k$-set that is not a hyperedge of $H$ results in exactly one copy of $K_r^{(k)}$. Among uniquely $K_r^{(k)}$-saturated hypergraphs, the interesting ones are the primitive ones that do not have a dominating vertex—a vertex belonging to all possible ${n-1\choose k-1}$ edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of $\tau$-critical hypergraphs: a hypergraph $H$ is uniquely $\tau$-critical if for every edge $e$, $\tau(H-e)=\tau(H)-1$ and $H-e$ has a unique transversal of size $\tau(H)-1$.We have two constructions for primitive uniquely $K_r^{(k)}$-saturated hypergraphs. One shows that for $k$ and $r$ where $4\leqslant k<r\leqslant 2k-3$, there exists such a hypergraph for every $n>r$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\leqslant n \leqslant {(k+2)^2\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.

About Quotient Orders and Ordering Sequences

Summary In preparation for the formalization in Mizar [4] of lotteries as given in [14], this article closes some gaps in the Mizar Mathematical Library (MML) regarding relational structures. The quotient order is introduced by the equivalence relation identifying two elements x, y of a preorder as equivalent if x ⩽ y and y ⩽ x. This concept is known (see e.g. chapter 5 of [19]) and was first introduced into the MML in [13] and that work is incorporated here. Furthermore given a set A, partition D of A and a finite-support function f : A → ℝ, a function Σf : D → ℝ, Σf (X)= ∑x∈X f(x) can be defined as some kind of natural “restriction” from f to D. The first main result of this article can then be formulated as: $$\sum\limits_{x \in A} {f(x)} = \sum\limits_{X \in D} {\Sigma _f (X)\left( { = \sum\limits_{X \in D} {\sum\limits_{x \in X} {f(x)} } } \right)} $$ After that (weakly) ascending/descending finite sequences (based on [3]) are introduced, in analogous notation to their infinite counterparts introduced in [18] and [13]. The second main result is that any finite subset of any transitive connected relational structure can be sorted as a ascending or descending finite sequence, thus generalizing the results from [16], where finite sequence of real numbers were sorted. The third main result of the article is that any weakly ascending/weakly descending finite sequence on elements of a preorder induces a weakly ascending/weakly descending finite sequence on the projection of these elements into the quotient order. Furthermore, weakly ascending finite sequences can be interpreted as directed walks in a directed graph, when the set of edges is described by ordered pairs of vertices, which is quite common (see e.g. [10]). Additionally, some auxiliary theorems are provided, e.g. two schemes to find the smallest or the largest element in a finite subset of a connected transitive relational structure with a given property and a lemma I found rather useful: Given two finite one-to-one sequences s, t on a set X, such that rng t ⊆ rng s, and a function f : X → ℝ such that f is zero for every x ∈ rng s \ rng t, we have ∑ f o s = ∑ f o t.

A Survey on Decidable Equivalence Problems for Tree Transducers

The decidability of equivalence for three important classes of tree transducers is discussed. Each class can be obtained as a natural restriction of deterministic macro tree transducers (MTTs): (1) no context parameters, i.e., top-down tree transducers, (2) linear size increase, i.e., MSO definable tree transducers, and (3) monadic input and output ranked alphabets. For the full class of mtts, decidability of equivalence remains a long-standing open problem.

UNIVERSALLY RATIONAL BELIEF HIERARCHIES

In a recent paper, Tsakas [2013 Rational belief hierarchies, Journal of Mathematical Economics, Maastricht University] introduced the notion of rational beliefs. These are Borel probability measures that assign a rational probability to every Borel event. Then, he constructed the corresponding Harsanyi type space model that represents the rational belief hierarchies. As he showed, there are rational types that are associated with a non-rational probability measure over the product of the underlying space of uncertainty and the opponent's types. In this paper, we define the universally rational belief hierarchies, as those that do not exhibit this property. Then, we characterize them in terms of a natural restriction imposed directly on the belief hierarchies.

ON A PROBLEM OF BERNIK, KLEINBOCK AND MARGULIS

In this paper, the Khintchine-type theorems of Beresnevich (Acta Arith.90(1999), 97) and Bernik (Acta Arith.53(1989), 17) for polynomials are generalised to incorporate a natural restriction on derivatives. This represents the first attempt to solve a problem posed by Bernik, Kleinbock and Margulis (Int. Math. Res. Notices2001(9) (2001), 453). More specifically, the main result provides a probabilistic criterion for the solvability of the system of inequalities |P(x)| < Ψ1(H) and |P′(x)| < Ψ2(H) in integral polynomialsPof degree ≤nand heightH, where Ψ1and Ψ2are fairly general error functions. The proof builds upon Sprindzuk's method of essential and inessential domains and the recent ideas of Beresnevich, Bernik and Götze (Compositio Math.146(2010), 1165) concerning the distribution of algebraic numbers.

ON BOUNDED ARTINIAN-FINITARY MODULES

Let G be a group, R be a ring and A be an RG-module. We say that A is an Artinian-finitary module overRG if for every element g ∈ G, the factor-module A/CA(g) is an Artinian R-module. The study of these modules was initiated by Wehrfritz. If D is a Dedekind domain and U is an Artinian D-module, then we can associate with U some numerical invariants. If V is the maximal divisible submodule of U, then V is a direct sum of finitely many indecomposable submodules. The number bd(U) of these direct summands is an invariant of U. The composition length bF(U/V) of U/V is another invariant of U. We consider the following special case of Artinian-finitary modules. Let D be a Dedekind domain and G be a group. The DG-module A is said to be bounded Artinian-finitary, if A is Artinian-finitary and there are the numbers bF(A) = b, bd(A) = d ∈ ℕ and a finite subset bσ(A) = τ Spec (D) such that lF(A/CA(g)) ≤ b, ld(A/CA(g)) ≤ d and Ass D(A/CA(g)) ⊆ τ for every element g ∈ G. In the article, the bounded Artinian-finitary modules under some natural restriction are studied.