Extension of results of Kostant and Han

The Ramanujan Journal - Kostant proved that the mth coefficient of the $$(r^2-1)$$ th power of the Dedekind eta function is non-vanishing for $$r \ge \mathsf {max} \{4,m \}$$ and r an...     

Packing under convex quadratic constraints

Mathematical Programming - We consider a general class of binary packing problems with a convex quadratic knapsack constraint. We prove that these problems are $$\mathsf {APX}$$ -hard to...     

Do there exist complete sets for promise classes?

In this paper we investigate the following two questions:

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The computational complexity of evolutionarily stable strategies

The concept of evolutionarily stable strategies (ESS) has been central to applications of game theory in evolutionary biology, and it has also had an influence on the modern development of game theory. A regular ESS is an important refinement the ESS concept. Although there is a substantial literature on computing evolutionarily stable strategies, the precise computational complexity of determining the existence of an ESS in a symmetric two-player strategic form game has remained open, though it has been speculated that the problem is -hard. In this paper we show that determining the existence of an ESS is both -hard and -hard, and that the problem is contained in , the second level of the polynomial time hierarchy. We also show that determining the existence of a regular ESS is indeed -complete. Our upper bounds also yield algorithms for computing a (regular) ESS, if one exists, with the same complexities.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

Strange expectations and simultaneous cores

Let \(\gcd (a,b)=1\). J. Olsson and D. Stanton proved that the maximum number of boxes in a simultaneous (a, b)-core is

$$\begin{aligned} \max _{\lambda \in {\mathrm {core}}(a,b)} (\mathsf{size}(\lambda )) = \frac{(a^2-1)(b^2-1)}{24} \end{aligned}$$

and that this maximum is achieved by a unique core. P. Johnson combined Ehrhart theory with the polynomial method to prove D. Armstrong’s conjecture that the expected number of boxes in a simultaneous (a, b)-core is

$$\begin{aligned} \mathop {\mathbb {E}}\limits _{\lambda \in {\mathrm {core}}(a,b)}\left( \mathsf{size}(\lambda )\right) = \frac{(a-1)(b-1)(a+b+1)}{24}. \end{aligned}$$

We extend Johnson’s method to compute the variance to be

$$\begin{aligned} \mathop {\mathbb {V}}\limits _{\lambda \in {\mathrm {core}}(a,b)}\left( \mathsf{size}(\lambda )\right) = \frac{ab(a-1)(b-1)(a+b)(a+b+1)}{1440}, \end{aligned}$$

and also prove polynomiality of all moments. By extending the definitions of “simultaneous cores” and “number of boxes” to affine Weyl groups, we give uniform generalizations of all three formulae above to simply laced affine types. We further explain the appearance of the number 24 using the “strange formula” of H. Freudenthal and H. de Vries.

     

Absolute continuity and singularity of Palm measures of the Ginibre point process

We prove a dichotomy between absolute continuity and singularity of the Ginibre point process \(\mathsf {G}\) and its reduced Palm measures \(\{\mathsf {G}_{\mathbf {x}}, \mathbf {x} \in \mathbb {C}^{\ell }, \ell = 0,1,2\ldots \}\), namely, reduced Palm measures \(\mathsf {G}_{\mathbf {x}}\) and \(\mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x} \in \mathbb {C}^{\ell }\) and \(\mathbf {y} \in \mathbb {C}^{n}\) are mutually absolutely continuous if and only if \(\ell = n\); they are singular each other if and only if \(\ell \not = n\). Furthermore, we give an explicit expression of the Radon–Nikodym density \(d\mathsf {G}_{\mathbf {x}}/d \mathsf {G}_{\mathbf {y}}\) for \(\mathbf {x}, \mathbf {y} \in \mathbb {C}^{\ell }\).     

On effective σ‐boundedness and σ‐compactness

We prove several dichotomy theorems which extend some known results on σ‐bounded and σ‐compact pointsets. In particular we show that, given a finite number of $\Delta ^{1}_{1}$ equivalence relations $\mathrel {\mathsf {F}}_1,\dots ,\mathrel {\mathsf {F}}_n$, any $\Sigma ^{1}_{1}$ set A of the Baire space either is covered by compact $\Delta ^{1}_{1}$ sets and lightface $\Delta ^{1}_{1}$ equivalence classes of the relations $\mathrel {\mathsf {F}}_i$, or A contains a superperfect subset which is pairwise $\mathrel {\mathsf {F}}_i$‐inequivalent for all i = 1, …, n. Further generalizations to $\Sigma ^{1}_{2}$ sets A are obtained.     

Degrees of difficulty of generalized r.e. separating classes

Important examples of classes of functions are the classes of sets (elements of ^ω 2) which separate a given pair of disjoint r.e. sets: . A wider class consists of the classes of functions f ∈ ^ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: . We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let denote the Medvedev degrees of those such that no m + 1 sets among A ₀,...,A _k-1 have a nonempty intersection. It is shown that each is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions is the greatest element of . If 2 ≤ l < k, then 0 _M is the only degree in which is below a member of . Each is densely ordered and has the splitting property and the same holds for the lattice it generates. The elements of are exactly the joins of elements of for . Supported by National Science Foundation grants DMS 0554841, 0532644 and 0652732.     

The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic

In this paper, we show within ${\mathsf{RCA}_0}In this paper, we show within that both the Jordan curve theorem and the Sch?nflies theorem are equivalent to weak K?nig’s lemma. Within , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).      

Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups

The partition algebra \(\mathsf {P}_k(n)\) and the symmetric group \(\mathsf {S}_n\) are in Schur–Weyl duality on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\), so there is a surjection \(\mathsf {P}_k(n) \rightarrow \mathsf {Z}_k(n) := \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is an isomorphism when \(n \ge 2k\). We prove a dimension formula for the irreducible modules of the centralizer algebra \(\mathsf {Z}_k(n)\) in terms of Stirling numbers of the second kind. Via Schur–Weyl duality, these dimensions equal the multiplicities of the irreducible \(\mathsf {S}_n\)-modules in \(\mathsf {M}_n^{\otimes k}\). Our dimension expressions hold for any \(n \ge 1\) and \(k\ge 0\). Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on \(\mathsf {M}_n^{\otimes k}\) and the quasi-partition algebra corresponding to tensor powers of the reflection representation of \(\mathsf {S}_n\).     

Exact inequalities for sums of asymmetric random variables, with applications

Let be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter . Let if and . Let . Let f be such a function that f and f′′ are nondecreasing and convex. Then it is proved that for all nonnegative numbers one has the inequality where . The lower bound on m is exact for each . Moreover, is Schur-concave in . A number of corollaries are obtained, including upper bounds on generalized moments and tail probabilities of (super)martingales with differences of bounded asymmetry, and also upper bounds on the maximal function of such (super)martingales. Applications to generalized self-normalized sums and t-statistics are given.      

Linear extensions of partial orders and reverse mathematics

We introduce the notion of τ‐like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω‐like means that every element has finitely many predecessors, while being ζ‐like means that every interval is finite. We consider statements of the form “any τ‐like partial order has a τ‐like linear extension” and “any τ‐like partial order is embeddable into τ” (when τ is ζ this result appears to be new). Working in the framework of reverse mathematics, we show that these statements are equivalent either to $\mathsf {B}{\Sigma }^{0}_{2}$ or to $\mathsf {ACA}_0$ over the usual base system $\mathsf {RCA}_0$.     

On a positive set theory with inequality

We introduce a quite natural Frege‐style set theory, which we call Strong‐Frege‐2 $(\mathsf {SF}_2)$, a sort of simplification of the theory considered in 13 (under the name Strong‐Frege‐3) and 1 (under the name F2). We give a model of a weaker variant of $\mathsf {SF}_2$, called $\mathsf {SF}_2\mathsf {AC}$, where atoms and coatoms are allowed. To construct the model we use an enumeration “almost without repetitions” of the Π¹₁ sets of natural numbers; such an enumeration can be obtained via a classical priority argument much in the style of 5 and 15 . © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

Lawvere Completion and Separation Via Closure

For a quantale V\mathsf{V}, first a closure-theoretic approach to completeness and separation in V\text-categories\mathsf{V}\text{-categories} is presented. This approach is then generalized to \mathbbT \mathbb{T} and a compatible \mathbbT\text-algebra \mathbb{T}\text{-algebra} structure on V\mathsf{V}.     

Gaussian Integral Means of Entire Functions

For an entire function \(f:\mathbb C\mapsto \mathbb C\) and a triple \((p,\alpha , r)\in (0,\infty )\times (-\infty ,\infty )\times (0,\infty ]\) , the Gaussian integral mean of \(f\) (with respect to the area measure \(dA\) ) is defined by $$\begin{aligned} {\mathsf M}_{p,\alpha }(f,r)=\left( \,\, {\int \limits _{|z| Via deriving a maximum principle for \({\mathsf M}_{p,\alpha }(f,r)\) , we establish not only Fock–Sobolev trace inequalities associated with \({\mathsf M}_{p,p/2}(z^m f(z),\infty )\) (as \(m=0,1,2,\ldots \) ), but also convexities of \(r\mapsto \ln {\mathsf M}_{p,\alpha }(z^m,r)\) and \(r\mapsto {\mathsf M}_{2,\alpha <0}(f,r)\) in \(\ln r\) with \(0 .     

-summand vectors in Banach spaces

The aim of this paper is to study the set of all -summand vectors of a real Banach space . We provide a characterization of -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an -sum of a Hilbert space and a Banach space without nontrivial -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces.

     

Coends of Higher Arity

Applied Categorical Structures - We specialise a recently introduced notion of generalised dinaturality for functors $$T : (\mathcal {C}^\mathsf {op})^p \times \mathcal {C}^q \rightarrow \mathcal...