In this paper we define a space σ(X) for approximate systems of compact spaces. The construction is due to H. Freudenthal for usual inverse sequences [4, p. 153–156]. We stablish the following properties of this space: (1) The space σ(X) is a paracompact space, (2) Moreover, if X is an approximate sequence of compact (metric) spaces, then σ(X) is a compact (metric) space (Lemma 2.4). We give th...

(1) sup λ>λ0, ε>0 ∥(−△+ V − (λ + iε) )−1∥∥ L2,σ(Rd)→L2,−σ(Rd) < ∞ provided that λ0 > 0, (1 + |x|)1+|V (x)| ∈ L∞ and σ > 12 . Here L(R) = {(1 + |x|)−σ f : f ∈ L(R)} is the usual weighted L2. The bound (1) is obtained from the same estimate for V = 0 by means of the resolvent identity. This bound for the free resolvent is related to the so called trace lemma, which refers to the statement that fo...

(1) sup λ>λ0, ε>0 ∥−4 + V − (λ + iε) )−1∥∥ L2,σ(Rd)→L2,−σ(Rd) <∞ provided that λ0 > 0, (1 + |x|)1+|V (x)| ∈ L∞ and σ > 12 . Here L(R) = {(1 + |x|)−σ f : f ∈ L(R)} is the usual weighted L2. The bound (1) is obtained from the same estimate for V = 0 by means of the resolvent identity. This bound for the free resolvent is related to the so called trace lemma, which refers to the statement that for...

Lemma 1. 1. I is a σ-ideal, 2. I ⊆ (s)0, 3. I 6 = (s)0 (in ZFC). Notice that such a σ – ideal was defined and investigated in several papers, see for example [4]. Since strongly meager sets and strong measure zero sets are (s)0 it makes sense to ask if they are in I. It is well-known that SN ⊆ I. In fact, if F : 2 −→ 2 is a continuous function and X ∈ SN then F”(X) ∈ SN . The purpose of this pa...

We have ∀X.A′ ≺ ∀X.B, so A′ ≺ B. Also, we have Σ ` V X : A′, so the RHS has type B = A. • Case (V : A′ p =⇒ ∀X.B) X 7−→ V : A′ p =⇒ B : We have A′ ≺ ∀X.B, so A′ ≺ B. Thus, the RHS has type B = A. • Case V : ∀X.A′ p =⇒ B 7−→ (V ?) : A′[X:=?] p =⇒ B : We have ∀X.A′ ≺ B, so A′[X:=?] ≺ B by Lemma 1. Thus, the RHS has type B. Definition 3. Well-typed contexts, written Σ B E : B ⇒ A, are defined in t...

We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ > 1. The proof relies on a geometric lemma of independent interest. Namely, let K ⊂ IR2 be a compact, simply connected set with smooth bo...

We establish a Julia–Carathéodory theorem and a boundary Schwarz– Wolff lemma for hyperbolically monotone mappings in the open unit ball of a complex Hilbert space. Let B be the open unit ball of a complex Hilbert space H with inner product 〈·, ·〉 and norm ‖ · ‖, and let ρ : B ×B 7→ R be the hyperbolic metric on B ([8], p. 98), i.e., ρ(x, y) = tanh √ 1− σ(x, y), (1) where σ(x, y) = (1 − ‖x‖)(1 ...

Remark 12.2. If Fs is N(μs, σ 2 s) and Fb ( 6= Fs) is N(μb, σ b ) then it can be easily shown that the problem is identifiable if and only if σs ≤ σb. When σs > σb, the model is not identifiable, an application of Lemma 2.4 gives α0 = α [ 1−(σb/σs) exp ( −σsσb(μb−μs)/2 )] . Thus, α0 increases to α as |μs − μb| tends to infinity. It should be noted that the problem is actually identifiable if we...

Digital signature is one of the basic primitives in cryptography. A common paradigm of obtaining signatures, known as the Fiat-Shamir (FS) paradigm, is to collapse any Σ-protocol (which is 3-round public-coin honest-verifier zero-knowledge) into a non-interactive scheme with hash functions that are modeled to be random oracles (RO). The Digital Signature Standard (DSS) and Schnorr’s signature s...

For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,Σ) = min{f(S) : S is of length n, over alphabet Σ}. Here, it is shown that 2f(n, {0, 1}) = n− o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a rem...