A family of inequalities originating from coding of messages

AbstractThis paper presents 96 new inequalities with common structure, all elementary to state but many not elementary to prove. For example, if n is a positive integer and a=ł(a1,…,an) and b=(b1,…,bn) are arbitrary vectors in R+n=[0,∞)n, and ρ(mi,j) is the spectral radius of an n×n matrix with elements mi,j, then∑i,jmin((aiaj),(bibj))⩽∑i,jmin((aibj),(biaj)),[2mm]∑i,jmax((ai+aj),(bi+bj))⩾∑i,jmax((ai+bj),(bi+aj)),ρ(min((aiaj),(bibj)))⩽ρ(min((aibj),(biaj))),[-4mm]∑i,jmin((aiaj),(bibj))xixj⩽∑i,jmin((aibj),(biaj))xixj,forallrealxi,i=1,…,n,∫∫log[(f(x)+f(y))(g(x)+g(y))]dμ(x)dμ(y)⩽∫∫log[(f(x)+g(y))(g(x)+f(y))]dμ(x)dμ(y).The second inequality is obtained from the first inequality (which is due to G. Zbăganu [A new inequality with applications in measure and information theories, in: Proceedings of the Romanian Academy, Series A1 (1), 2000, pp. 15–19]) by replacing min with max, and × with +, and by reversing the direction of the inequality. The third inequality is obtained from the first by replacing the summation by the spectral radius. The fourth inequality is obtained from the first by taking each summand as a coefficient in a quadratic form. The fifth inequality is obtained from the first by replacing both outer summations by products, min by ×, × by +, and the non-negative vectors a, b by non-negative measurable functions f, g. The proofs of these inequalities are mysteriously diverse.A nice generalization of the first inequality is proved: Let ∗ be one of the four operations +, ×, min and max on an appropriate interval J of R. Let a,b∈Jn. Denote by a∗a the n×n matrix ai,j=ai∗aj. Then the matrix a∗a is more different from b∗b than a∗b is from b∗a. Precisely, if ∣A∣=∑1⩽i,j⩽n∣ai,j∣, then ∥a∗a−b∗b∥⩾∥a∗b−b∗a∥

Existence of Probability Measures With Given Marginals

We show that if f is a probability density on Rn wrt Lebesgue measure (or any absolutely continuous measure) and 0 ≤ f ≤ 1, then there is another density g with only the values 0 and 1 and with the same (n−1)-dimensional marginals in any finite number of directions. This sharpens, unifies and extends the results of Lorentz and of Kellerer. Given a pair of independent random variables 0 ≤ X, Y ≤ 1, we further study functions 0 ≤ ϕ ≤ 1 such that Z = ϕ (X,Y) satisfies E(Z|X) = X and E(Z|Y) = Y. If there is a solution then there also is a nondecreasing solution ϕ(x,y). These results are applied to tomography and baseball

Limitations, improvements and alternatives of the silt density index

Reverse osmosis (RO) membrane systems are widely used in the desalination of water. However, flux decline due to fouling phenomena in RO remains a challenge. To minimize fouling, a reliable index is necessary to predict the fouling potential of the RO feed water. The ASTM introduced the silt density index (SDI) as a standard fouling index to measure the fouling potential due to colloidal and suspended particles. For decades, the SDI is worldwide accepted and applied. There are growing doubts about the predictive value of this parameter. In addition there are several deficiencies observed, affecting the accuracy and reproducibility e.g. no correction factor for temperature, nor for variations in membrane resistance, and no linear correlation with the concentration of colloidal/suspended particles. This paper gives an overview of our work on limitations, improvements and alternatives for the SDI. Firstly, the influence of the applied 0.45 μm test membrane on the SDI will be investigated. Variations in SDI values can be attributed to differences in properties of these membranes. In order to quantify the influence of pressure, temperature and membrane resistance on the SDI a mathematical relation was developed between the SDI and the MFI0.45, assuming cake filtration. In addition, also other fouling mechanisms were incorporated in the model using the well-known blocking laws. Based on a cake filtration fouling mechanism and assuming 100% particle retention, the models were used to normalize the experimental SDI values for temperature, pressure and membrane resistance to the SDI+. By applying this normalization, the results of SDI tests carried out under different conditions and/or with different membranes can be compared easily as was proven experimentally in the lab and at a seawater desalination plant. Finally, an alternative filtration index will be introduced, the volume-based SDI_v. The SDI_v compares the initial flow rate to the flow rate after filtering a standard volume of feed water using MF membranes with an average pore size of 0.45 μm. Our experimental results show that SDI_v is independent of the membrane resistance. In that way, it eliminates most of the disadvantages of the SDI and has great potential to replace the SDI in the fiel

On the critical pair theory in abelian groups : Beyond Chowla's Theorem

We obtain critical pair theorems for subsets S and T of an abelian group such that |S+T| < |S|+|T|+1. We generalize some results of Chowla, Vosper, Kemperman and a more recent result due to Rodseth and one of the authors.Comment: Submitted to Combinatorica, 23 pages, revised versio

The impact of loco-regional recurrences on metastatic progression in early-stage breast cancer: a multistate model

To study whether the effects of prognostic factors associated with the occurrence of distant metastases (DM) at primary diagnosis change after the incidence of loco-regional recurrences (LRR) among women treated for invasive stage I or II breast cancer. The study population consisted of 3,601 women, enrolled in EORTC trials 10801, 10854, or 10902 treated for early-stage breast cancer. Data were analysed in a multivariate, multistate model by using multivariate Cox regression models, including a state-dependent covariate. The presence of a LRR in itself is a significant prognostic risk factor (HR: 3.64; 95%-CI: 2.02-6.5) for the occurrence of DM. Main prognostic risk factors for a DM are young age at diagnosis (</=40: HR: 1.79; 95%-CI: 1.28-2.51), larger tumour size (HR: 1.58; 95%-CI: 1.35-1.84) and node positivity (HR: 2.00; 95%-CI: 1.74-2.30). Adjuvant chemotherapy is protective for a DM (HR: 0.66; 95%-CI: 0.55-0.80). After the occurrence of a LRR the latter protective effect has disappeared (P = 0.009). The presence of LRR in itself is a significant risk factor for DM. For patients who are at risk of developing LRR, effective local control should be the main target of therapy

Dynamic phase diagram of the REM

By studying the two-time overlap correlation function, we give a comprehensive analysis of the phase diagram of the Random Hopping Dynamics of the Random Energy Model (REM) on time-scales that are exponential in the volume. These results are derived from the convergence properties of the clock process associated to the dynamics and fine properties of the simple random walk in the $n$-dimensional discrete cube.Comment: This paper is in large part based on the unpublished work arXiv:1008.3849. In particular, the analysis of the overlap correlation function is new as well as the study of the high temperature and short time-scale transition line between aging and stationarit

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43923/1/10_2005_Article_BF02188016.pd

Prognostic biomarker soluble ST2 exhibits diurnal variation in chronic heart failure patients

Aim: Soluble suppression of tumorigenicity-2 (sST2) is a strong prognostic biomarker in heart failure. The emerging understanding of circadian biology in cardiovascular disease may lead to novel applications in prognosis and diagnosis and may provide insight into mechanistic aspects of the disease–biomarker interaction. So far, it is unknown whether sST2 exhibits a diurnal rhythm. Repeated measurements of sST2 may aid in clinical decision making. The goal of this study was to investigate whether sST2 exhibits diurnal variation in patients with heart failure with reduced ejection fraction (HFrEF) and in control subjects, thereby enhancing its diagnostic and prognostic values. Methods and results: The study comprised 32 subjects: 16 HFrEF patients and 16 controls. Blood was collected at seven subsequent time points during a 24 h time period. sST2, N-terminal pro-B-type natriuretic peptide (NT-proBNP), melatonin, and cortisol were measured from serum. Peak values of sST2 clustered at daytime (modal value: 5 p.m.) in 87.6% of all subjects (81.3% of patients, P = 0.021; 93.8% of controls, P = 0.001), and minimum concentrations at night-time (modal value: 5 a.m.) in 84.4% (87.5% of patients, P = 0.004 81.3% of controls, P = 0.021). A cosinor analysis of mean normalized sST2 values revealed significant cosine shaped 24 h oscillations of patients (P = 0.026) and controls (P = 0.037). NT-proBNP in contrast did not show a diurnal rhythm, while melatonin and cortisol patterns were intact in all subjects. Conclusions: sST2 exhibits a diurnal rhythm with lower values in the morning than in the late afternoon. This new insight could lead to refinement of its diagnostic and prognostic values through specified and consistent sampling times with repeated measurements. For example, by measuring sST2 during the afternoon, when levels are at their highest, false negatives on prognosis prediction could be avoided