This supplement consists of several parts that refer directly to specific topics in the paper: A Proof of Equation (2) B Proof of Lemma 3.1 (Symmetric biproportional fit) C Technical details on why ”local affinity” is sufficient in Section 4.1 D Proof of Theorem 4.2 (Convergence of PSIPF) E Proof of Lemma 4.4 (L1-monotony) F Proof of Lemma 4.5 (Volume bounds) G Proof of Lemma 4.6 (Limit points)...

Lemma 4.4 in [E. Soubies, L. Blanc-Féraud and G. Aubert, SIAM J. Imaging Sci., 8 (2015), pp. 1607–1639] is wrong for local minimizers of the continuous exact `0 (CEL0) functional. The argument used to conclude the proof of this lemma is not sufficient in the case of local minimizers. In this note, we supply a revision of this lemma where new results are established for local minimizers. Theorem...

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

We present a new method for the constraint-based synthesis of termination arguments for linear loop programs based on linear ranking templates. Linear ranking templates are parametrized, well-founded relations such that an assignment to the parameters gives rise to a ranking function. This approach generalizes existing methods and enables us to use templates for many different ranking functions...

We obtain a global extension of the classical weak Harnack inequality which extends and quantifies Hopf–Oleinik boundary-point lemma, for uniformly elliptic equations in divergence form. Among consequences is boundary gradient estimate, due to Krylov well-studied non-divergence form equations, but completely novel framework. Another consequence new more general version lemma.

We show that Fan’s 1952 lemma on labelled triangulations of the n-sphere with n + 1 labels is equivalent to the Borsuk–Ulam theorem. Moreover, unlike other Borsuk–Ulam equivalents, we show that this lemma directly implies Sperner’s Lemma, so this proof may be regarded as a combinatorial version of the fact that the Borsuk–Ulam theorem implies the Brouwer fixed-point theorem, or that the Lustern...

This manuscript presents Hyers-Ulam stability and Hyers--Ulam--Rassias stability results of non-linear Volterra integro--delay dynamic system on time scales with fractional integrable impulses. Picard fixed point theorem  is used for obtaining  existence and uniqueness of solutions. By means of   abstract Gr"{o}nwall lemma, Gr"{o}nwall's inequality on time scales, we establish  Hyers-Ulam stabi...

The Szemerédi Regularity Lemma states that any sufficiently large graph G can be partitioned into a bounded (independent of the size of the graph) number of regular, or “random-looking,” components. The resulting partition can be viewed as a regularity graph R. The Key Lemma shows that under certain conditions, the existence of a subgraph H in R implies its existence in G. We prove the Regulari...

Dedicated to Alan George on the occasion of his 60th birthday Abstract. The splitting lemma is one of the two main tools of support theory, a framework for bounding the condition number of definite and semidefinite preconditioned linear systems. The splitting lemma allows the analysis of a complicated system to be partitioned into analyses of simpler systems. The other tool is the symmetric-pro...

S f(s)μ(ds) = 1, ∫ S f(s)μn(ds) = 0, and (1.3) does not hold. Theorem 1.1 presents Fatou’s lemma for weakly converging measures μn and nonnegative functions fn. This fact is useful for the analysis of Markov decision processes and stochastic games. Serfozo [7, Lemma 3.2] establishes inequality (1.4) for a vaguely convergent sequence of measures on a locally compact metric space S and for nonneg...