The kappa statistic as a function of sensitivity and specificity

This paper explores analytically the connections between three commonly used statistical measures of agreement: sensitivity, specificity, and Cohen's (unweighed) kappa, which are employed often in clinical studies. In particular, we show that if both sensitivity and specificity are below 7/8, then a kappa of 3/4 or greater (which is considered to indicate good to excellent agreement) can never be achieved even for raw agreement approaching 100%. In addition we derive some of the known properties of kappa for two-by-two tables which apparently are not readily available in the literature.     

Existence of a solution of a difference scheme for one variational problem

The problem of minimizing the functional (A) $${}_a\smallint ^b \varphi (x,y,y',y'')dx$$ under the conditions (B) $$y(a) = a_0 ,y'(a) = a_1 ,y(b) = b_0 ,y'(b) = b_1$$ is replaced by the problem of finding the vector (y₁,y₂,...,y_n?1) on which the sum (C) $$\sum\limits_{\kappa = 0}^n {C_\kappa \varphi (x_\kappa ,y_\kappa ,\left. {\frac{{y_{\kappa + 1} - y_\kappa }}{h},\frac{{y_{\kappa + 1} - 2y_\kappa + y_{\kappa + 1} )}}{{h^2 }}} \right)}$$ takes a minimal value. Under certain conditions on ? andC _k it is proved that a solution exists for the difference scheme constructed. The method of differentiation with respect to a parameter is used for the proof.     

Non-reflection of the bad set for \check{I}_{\theta}[\lambda] and pcf

We reconsider here the following related pcf questions and make some advances: (Q1) concerning the ideal $\check{I}_{\kappa}[\lambda]$ how much reflection do we have for the bad set $S^{{\mathrm {bd}}}_{\lambda,\kappa} \subseteqq \{{\delta < \lambda} : {\mathop {\rm cf}\nolimits \, (\delta) = \kappa}\}$ assuming it is well defined (for transparency only)? (Q2) are there somewhat free black boxes? The advances in (Q2) will be used in subsequent for constructions of Abelian groups and modules.     

笛卡尔乘积图的圈点连通度

图G的圈点连通度,记为κ_c(G),是所有圈点割中最小的数目,其中每个圈点割S满足G-S不连通且至少它的两个分支含圈.这篇文章中给出了两个连通图的笛卡尔乘积的圈点连通度:(1)如果G_1≌K_m且G_2≌K_n,则κ_c(G_1×G_2)=min{3m+n-6,m+3n-6},其中m+n≥8,m≥n+2,或n≥m+2,且κ_c(G_1×G_2)=2m+2n-8,其中m+n≥8,m=n,或n=m+1,或m=n+11;(2)如果G_1≌K_m(m≥3)且G_2■K_n,则min{3m+κ(G_2)-4,m+3κ(G_2)-3,2m+2κ(G_2)-4}≤κ_c(G_1×G_2)≤mκ(G2);(3)如果G_1■K_m,K_(1,m-1)且G_2■K_n,K_(1,n-1),其中m≥4,n≥4,则min{3κ(G_1)+κ(G_2)-1,κ(G_1)+3κ(G_2)-1,2_κ(G_1)+2_κ(G_2)-2}≤κ_c(G_1×G_2)≤min{mκ(G_2),nκ(G_1),2m+2n-8}.     

Asymptotic behavior of sums of the type\sum\limits_{\kappa = m}^{n - 1} {\exp } (i\omega \sqrt {n\kappa } ) as n,m → ∞

The asymptotic behavior asn, m → ∞ of the sum $$\sum\limits_{\kappa ,\ell = m}^{n - 1} {\exp \left[ {i\omega \sqrt n \left( {\sqrt \kappa + \sqrt \ell } \right)} \right]} \Phi \left( {1 - \frac{{\left| {\sqrt \kappa - \sqrt \ell } \right|}}{\Delta }} \right)$$ is studied where π(t)=0 for t?0 and φ(t)=t for t > 0.     

Modulation space estimates for Schrödinger type equations with time-dependent potentials

We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian \({( - \Delta )^{\kappa /2}}\) with 1 ? κ ? 2. Consequently the classical Hamiltonian corresponding to the previous Schrödinger type equations is studied. As applications, a series of new boundedness results for the corresponding propagator are obtained in the framework of modulation spaces. The main results of the present article include the case of wave equations.     

Measurability in C({2^\kappa }) } and Kunen cardinals

A cardinal κ is called a Kunen cardinal if the σ-algebra on κ × κ generated by all products A×B, where A,B ? κ, coincides with the power set of κ×κ. For any cardinal κ, let $C({2^\kappa })$ be the Banach space of all continuous real-valued functions on the Cantor cube $C({2^\kappa })$ . We prove that κ is a Kunen cardinal if and only if the Baire σ-algebra on $C({2^\kappa })$ for the pointwise convergence topology coincides with the Borel σ-algebra on $C({2^\kappa })$ for the norm topology. Some other links between Kunen cardinals and measurability in Banach spaces are also given.     

Covering pairs by quintuples with index λ  0 (mod 4)

A (v, k. λ) covering design of order v, block size k, and index λ is a collection of k-element subsets, called blocks, of a set V such that every 2-subset of V occurs in at least λ blocks. The covering problem is to determine the minimum number of blocks, α(v, k, λ), in a covering design. It is well known that $ \alpha \left({\nu,\kappa,\lambda } \right) \ge \left\lceil {\frac{\nu}{\kappa}\left\lceil {\frac{{\nu - 1}}{{\kappa - 1}}\lambda} \right\rceil} \right\rceil = \phi \left({\nu,\kappa,\lambda} \right) $, where [χ] is the smallest integer satisfying χ ≤ χ. It is shown here that α (v, 5, λ) = ?(v, 5, λ) + ? where λ ≡ 0 (mod 4) and e= 1 if λ (v?1)≡ 0(mod 4) and λv (v?1)/4 ≡ ?1 (mod 5) and e= 0 otherwise With the possible exception of (v,λ) = (28, 4). © 1993 John Wiley & Sons, Inc.     

On an equation involving weighted quasi-arithmetic means

Let I ? ? be an interval and κ, λ ∈ ? / {0, 1}, µ, ν ∈ (0, 1). We find all pairs (φ, ψ) of continuous and strictly monotonic functions mapping I into ? and satisfying the functional equation $$ \kappa x + (1 - \kappa )y = \lambda \phi ^{ - 1} (\mu \phi (x) + (1 - \mu )\phi (y)) + (1 - \lambda )\psi ^{ - 1} (\nu \psi (x) + (1 - \nu )\psi (y)) $$ which generalizes the Matkowski-Sutô equation. The paper completes a research stemming in the theory of invariant means.     

On Linear Fractional Transformations Associated with Generalized J-Inner Matrix Functions

A class U_k1 (J){\mathcal{U}}_{\kappa 1} (J) of generalized J-inner mvf’s (matrix valued functions) W(λ) which appear as resolvent matrices for bitangential interpolation problems in the generalized Schur class of p ×q  mvf￠s S_k^{p ×q}p \times q \, {\rm mvf's}\, {\mathcal{S}}_{\kappa}^{p \times q} and some associated reproducing kernel Pontryagin spaces are studied. These spaces are used to describe the range of the linear fractional transformation T_W based on W and applied to S_k2^{p ×q}{\mathcal{S}}_{\kappa 2}^{p \times q}. Factorization formulas for mvf’s W in a subclass U^°_k1 (J) of U_k1(J){\mathcal{U}^{\circ}_{\kappa 1}} (J)\, {\rm of}\, {\mathcal{U}}_{\kappa 1}(J) found and then used to parametrize the set S_k1+k2^{p ×q} ?T_W [ S_k2^{p ×q} ]{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} \cap T_{W} \left[ {\mathcal{S}}_{\kappa 2}^{p \times q} \right]. Applications to bitangential interpolation problems in the class S_k1+k2^{p ×q}{\mathcal{S}}_{{\kappa 1}+{\kappa 2}}^{p \times q} will be presented elsewhere.     

The use of ROC for defining the validity of the prognostic index in censored data

The validity of a diagnostic marker can be summarised by using statistical measures either for the goodness of the fit like the deviance, measures of the explained variation like R² or the misclassification rate. Other intuitive measures are sensitivity and specificity in the case of binary response. In the absence of censored data the calculation of these measures is widely used. In the presence of censoring the estimation of time-dependent sensitivity and specificity is not well known. In this article we propose a new method for calculating ROC curves with censored data using the observed number of events and calculating the additional number of expected events for censored observations. The new method is illustrated with data for predicting mortality in patients surviving a myocardial infarction.     

О коэффициентах рядо в Фурье по множествам

The following statement is proved: Theorem.Let f(x), 0≦x≦2π, possess the Fourier expansion $$\mathop \sum \limits_{\kappa = - \infty }^\infty c_\kappa e^{in} \kappa ^x with \bar c_\kappa = c_{ - \kappa } , n_\kappa = - \bar n_{ - \kappa }$$ where {n _k } is a Sidon sequence. Then in order to have $$\mathop \sum \limits_{\kappa = - \infty }^\infty |c_\kappa |^p< \infty$$ for a given p, 1 $$\mathop \sum \limits_{k = 1}^\infty \left( {\frac{{\left\| f \right\|L^k (0,2\pi )}}{k}} \right)^p< \infty$$ . An analogous statement holds true for series with respect to the Rademacher system.     

Bemerkungen zu einem Satz von Fejér

Последовательность {ita_k} _(n) ^k =1/∞ вещественных ч исел называется дважды мо нотонной, еслиa _k -2a _k+1 +a _k+2 ≧0 дляk≧1. В работе доказываютс я следующие утвержде ния, являющиеся обобщени ем двух теорем Фейера:

1. Если {ita_k — дважды моно тонная последовател ьность, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^n {a_\kappa z^\kappa } > 1/2$$ дляи≧ 1.
2. Если О≦β<1 и последова тельность (k+1-2β)ak} дважд ы монотонна, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {ka_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } > \beta $$ , то есть $$\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } \varepsilon S_\beta ^\kappa $$ . При помощи 2) получены о бобщения и уточнения теорем из работы [1] о линейных комбинациях некотор ых однолистных функц ий.

     

On weakening tightness to weak tightness

The weak tightness wt(X) of a space X was introduced in Carlson (Topol Appl 249:103–111, 2018) with the property $$wt(X)\le t(X)$$. We investigate several well-known results concerning t(X) and consider whether they extend to the weak tightness setting. First we give an example of a non-sequential compactum X such that $$wt(X)=\aleph _0<t(X)$$ under $$2^{\aleph _0}=2^{\aleph _1}$$. In particular, this demonstrates the celebrated Balogh’s (Proc Am Math Soc 105(3):755–764, 1989) Theorem does not hold in general if countably tight is replaced with weakly countably tight. Second, we introduce the notion of an S-free sequence and show that if X is a homogeneous compactum then $$|X|\le 2^{wt(X)\pi \chi (X)}$$. This refines a theorem of de la Vega (Topol Appl 153:2118–2123, 2006). In the case where the cardinal invariants involved are countable, this also represents a variation of a theorem of Juhász and van Mill (Proc Am Math Soc 146(1):429–437, 2018). In this connection we also show $$w(X)\le 2^{wt(X)}$$ for a homogeneous compactum. Third, we show that if X is a $$T_1$$ space, $$wt(X)\le \kappa $$, X is $$\kappa ^+$$-compact, and $$\psi (\overline{D},X)\le 2^\kappa $$ for any $$D\subseteq X$$ satisfying $$|D|\le 2^\kappa $$, then (a) $$d(X)\le 2^\kappa $$ and (b) X has at most $$2^\kappa $$-many $$G_\kappa $$-points. This is a variation of another theorem of Balogh (Topol Proc 27:9–14, 2003). Finally, we show that if X is a regular space, $$\kappa =L(X)wt(X)$$, and $$\lambda $$ is a caliber of X satisfying $$\kappa <\lambda \le \left( 2^{\kappa }\right) ^+$$, then $$d(X)\le 2^{\kappa }$$. This extends of theorem of Arhangel$$'$$skiĭ (Topol Appl 104:13–26, 2000).     

$P_*(\kappa)$线性互补问题的Mehrotra型预估-校正算法

Mehrotra-type predictor-corrector algorithm,as one of most efficient interior point methods,has become the backbones of most optimization packages.Salahi et al.proposed a cut strategy based algorithm for linear optimization that enjoyed polynomial complexity and maintained its efficiency in practice.We extend their algorithm to P_*（κ）linear complementarity problems.The way of choosing corrector direction for our algorithm is different from theirs. The new algorithm has been proved to have an ο(（1+4κ）（17+19κ） √（1+2κn）^3/2log[（x⁰）^Ts⁰/ε] worst case iteration complexity bound.An numerical experiment verifies the feasibility of the new algorithm.     

How quadratic are the natural numbers?

Abstract. For natural numbers n we inspect all factorizations n = ab of n with a ￡ ba \le b in \Bbb N\Bbb N and denote by n=a_n b_nn=a_n b_n the most quadratic one, i.e. such that b_n - a_nb_n - a_n is minimal. Then the quotient k(n) : = a_n/b_n\kappa (n) := a_n/b_n is a measure for the quadraticity of n. The best general estimate for k(n)\kappa (n) is of course very poor: 1/n ￡ k(n) ￡ 11/n \le \kappa (n)\le 1. But a Theorem of Hall and Tenenbaum [1, p. 29], implies(logn)^-d-e ￡ k(n) ￡ (logn)^-d(\log n)^{-\delta -\varepsilon } \le \kappa (n) \le (\log n)^{-\delta } on average, with d = 1 - (1+log₂  2)/log2=0,08607 ?\delta = 1 - (1+\log _2 \,2)/\log 2=0,08607 \ldots and for every e > 0\varepsilon >0. Hence the natural numbers are fairly quadratic.¶k(n)\kappa (n) characterizes a specific optimal factorization of n. A quadraticity measure, which is more global with respect to the prime factorization of n, is k^*(n): = ?_{1 ￡ a ￡ b, ab=n} a/b\kappa ^*(n):= \textstyle\sum\limits \limits _{1\le a \le b, ab=n} a/b. We show k^*(n) ~ \frac 12\kappa ^*(n) \sim \frac {1}{2} on average, and k^*(n)=W(2^{\frac 12(1-e) log n/log ₂n})\kappa ^*(n)=\Omega (2^{\frac {1}{2}(1-\varepsilon ) {\log}\, n/{\log} _2n})for every e > 0\varepsilon>0.     

Cohen’s linearly weighted kappa is a weighted average

An n × n agreement table F?=?{f _ij } with n ?? 3 ordered categories can for fixed m?(2??? m??? n ? 1) be collapsed into ${\binom{n-1}{m-1}}$ distinct m × m tables by combining adjacent categories. It is shown that the components (observed and expected agreement) of Cohen??s weighted kappa with linear weights can be obtained from the m × m subtables. A consequence is that weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to the m?× m tables, where the weights are the denominators of the kappas. Moreover, weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to all nontrivial subtables.     

Cardinal Invariants and Eventually Different Functions

In this paper we study several kinds of maximal almost disjointfamilies. In the main result of this paper we show that forsuccessor cardinals , there is an unexpected connection betweeninvariants a_e(), b() and a certain cardinal invariant m_d(⁺)on ⁺. As a corollary we get for example the following result.For a successor cardinal , even assuming that ^< = and 2= ⁺, the following is not provable in Zermelo–Fraenkelset theory. There is a ⁺-cc poset which does not collapse andwhich forces a() = ⁺ < a_e() = ⁺⁺ = 2. We also apply the ideasfrom the proofs of these results to study a = a() and non(M).2000 Mathematics Subject Classification 03E17 (primary), 03E05(secondary).     

A Theory of Stationary Trees and the Balanced Baumgartner–Hajnal–Todorcevic Theorem for Trees

Building on early work by Stevo Todorcevic, we develop a theory of stationary subtrees of trees of successor-cardinal height. We define the diagonal union of subsets of a tree, as well as normal ideals on a tree, and we characterize arbitrary subsets of a non-special tree as being either stationary or non-stationary. We then use this theory to prove the following partition relation for trees: Main Theorem. Let \({\kappa}\) be any infinite regular cardinal, let ξ be any ordinal such that \({2^{|\xi|} < \kappa}\) , and let k be any natural number. Then $$non-(2^{<\kappa})-special\, tree \rightarrow (\kappa + \xi)^{2}_k.$$ This is a generalization to trees of the Balanced Baumgartner–Hajnal–Todorcevic Theorem, which we recover by applying the above to the cardinal \({(2^{< \kappa})^{+}}\) , the simplest example of a non- \({(2^{< \kappa})}\) -special tree. As a corollary, we obtain a general result for partially ordered sets: Theorem. Let \({\kappa}\) be any infinite regular cardinal, let ξ be any ordinal such that \({2^{|\xi|} < \kappa}\) , and let k be any natural number. Let P be a partially ordered set such that \({P \rightarrow (2^{< \kappa})^{1}_{2^{< \kappa}} }\) . Then $$P \rightarrow (\kappa + \xi)^{2}_{k}.$$      

The Halton sequence and its discrepancy in the Cantor expansion

Let \(\kappa \) be an infinite cardinal, and \(2^\kappa <\lambda \le 2^{\kappa ^+}\). We prove that if there is a weak diamond on \(\kappa ^+\) then every \(\{C_\alpha :\alpha <\lambda \}\subseteq \mathcal {D}_{\kappa ^+}\) satisfies Galvin’s property. On the other hand, Galvin’s property is consistent with the failure of the weak diamond (and even with Martin’s axiom in the case of \(\aleph _1\)). We derive some consequences about weakly inaccessible cardinals. We also prove that the negation of a similar property follows from the proper forcing axiom.