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The first two chapters of this book offer a modern, self-contained exposition of the elementary theory of triangulated categories and their quotients. The simple, elegant presentation of these known results makes these chapters eminently suitable as a text for graduate students. The remainder of the book is devoted to new research, providing, among other material, some remarkable improvements on Brown's classical representability theorem. In addition, the author introduces a class of triangulated categories\"--the \"well generated triangulated categories\"--and studies their properties. This exercise is particularly worthwhile in that many examples of triangulated categories are well generated, and the book proves several powerful theorems for this broad class. These chapters will interest researchers in the fields of algebra, algebraic geometry, homotopy theory, and mathematical physics.

A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.

It has been known that the fractional ODEs involving Riemann–Liouville fractional derivatives maintain many “analogies” of the classic elliptic-type properties, including maximum principles, Hopf’s Lemma, the existence of real eigenvalues, etc. However, in this short work, we construct a counterexample demonstrating that Caputo-type fractional operators (and their compositions) fail to uphold the maximum principle and Hopf’s lemma. This finding is somewhat unexpected and serves as a benchmark problem or counterexample, reminding us to remain cautious when developing parallel theories for fractional differential equations and applying them in modeling. Since the maximum principle and Hopf’s lemma are often used to construct convex cones in Banach spaces and guarantee the existence of a principal eigenvalue in classical elliptic theory, this result leads us to conjecture that fractional Sturm–Liouville problems involving the Caputo fractional derivative may lack a conventional principal eigenvalue. This conjecture is partially proved in a following-up work.

The incorrect notion that kurtosis somehow measures \"peakedness\" (flatness, pointiness, or modality) of a distribution is remarkably persistent, despite attempts by statisticians to set the record straight. This article puts the notion to rest once and for all. Kurtosis tells you virtually nothing about the shape of the peak-its only unambiguous interpretation is in terms of tail extremity, that is, either existing outliers (for the sample kurtosis) or propensity to produce outliers (for the kurtosis of a probability distribution). To clarify this point, relevant literature is reviewed, counterexample distributions are given, and it is shown that the proportion of the kurtosis that is determined by the central μ ± σ range is usually quite small.

The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n – d. That is, any two vertices of the polytope can be connected by a path of at most n – d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.

We consider a fractal refinement of the Carleson problem for the Schrödinger equation, that is, to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the α-Hausdorff measure (α-almost everywhere). We extend to the fractal setting (α < n) a recent counterexample of Bourgain [5], which is sharp in the Lebesque measure setting (α = n). In doing so, we recover the necessary condition from [24] for pointwise convergence α-almost everywhere, and we extend it to the range n/2 < α ≤ (3n + 1)/4.

This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpotent blocks for all quasi-simple groups.

This paper is dedicated to investigating the L^p -bounds of wave operators W_(H,^2) associated with fourth-order Schrödinger operators H=^2+V on R^3 with real potentials satisfying |V(x)| x^- for some >0 . A recent work by Goldberg and Green (2021) has demonstrated that wave operators W_(H,^2) are bounded on L^p(R^3) for all 19 and zero is a regular point of H . In the paper, we aim to further establish endpoint estimates for W_(H,^2) in two significant ways. First, we provide counterexamples to illustrate the unboundedness of W_(H,^2) on the endpoint spaces L^1(R^3) and L^ınfty(R^3) for non-zero compactly supported potentials V . Second, we establish weak (1,1) estimates for the wave operators W_(H,^2) and their dual operators W_(H,^2)^* in the case where zero is a regular point and >11 . These estimates depend critically on the singular integral theory of Calderón–Zygmund on a homogeneous space (X,d) with a doubling measure d .

The question whether non-isomorphic finite p -groups can have isomorphic modular group algebras was recently answered in the negative by García-Lucas, Margolis and del Río [J. Reine Angew. Math. 783 (2022), 269–274]. We embed these negative solutions in the class of two-generated finite 2 -groups with dihedral central quotient, and solve the original question for all groups within this class. As a result, we discover new negative solutions and simple algebra isomorphisms.