Give a Counterexample to Show That This is False if We Only Require F to Be Continuous

Counterexample

Given a counterexample of such a statement in any category A we may easily find a small subcategory A′ in which the counterexample remains such.

From: North-Holland Mathematical Library , 1990

The Linear Time - Branching Time Spectrum I.* The Semantics of Concrete, Sequential Processes

R.J. van Glabbeek , in Handbook of Process Algebra, 2001

14. Possible worlds semantics

In Veglioni and De Nicola [49], a nondeterministic process is viewed as a set of deterministic ones: its possible worlds. Two processes are said to be possible worlds equivalent iff they have the same possible worlds. Two different approaches by which a nondeterministic process can be resolved into a set of deterministic ones need to be distinguished; I call them the state-based and the path-based approach. In the state-based approach a deterministic process h is obtained out of a nondeterministic process g ∈ $\mathcal{G}$ by choosing, for every state s of g and every action a ∈ I (s) a single edge s $\underset{⟶}{a}$ s′. Now h is the reachable part of the subgraph of g consisting of the chosen edges. In the path-based approach on the other hand, one chooses for every path π ∈ PATHS(g) and every action a ∈ I (end (π)) a single edge end (π) $\underset{⟶}{a}$ s′ to continue with. The chosen edges may now be different for different paths ending in the same state. The difference between the two approaches is illustrated in Counterexample 16. In the state-based approach, the process in the middle has two possible worlds, depending on which of the two b-edges is chosen. These worlds are essentially abc and ab ^∞. In the path-based approach, the process in the middle has countably many possible worlds, namely abⁿc for n ≥ 1 and ab ^∞.

Counterexample 16.. State-based versus path-based possible worlds equivalence.

In [49], Veglioni and De Nicola take the state-based approach: "once we have resolved the underspecification present in a state s by saying, for example, s $\underset{⟶}{a}$ s, then, we cannot choose s $\underset{⟶}{a}$ 0 in the same possible world". However, they provide a denotational characterization of possible worlds semantics on finite processes, namely by inductively allocating sets of deterministic trees to BCCSP expressions (cf. Section 17), which can be regarded as path-based. In addition, they give an operational characterization of possible world semantics, essentially following the state-based approach outlined above. They claim that both characterizations agree. This, however, is not the case, as Counterexample 17 reveals a difference between the two approaches even on finite processes. In the path-based approach the process displayed has a possible world acd + bce (i.e., a process with branches acd and bce), which it has not in the state-based approach. As it turns out, the complete axiomatization they provide w.r.t. BCCSP is correct for the path-based, denotational characterization, but is unsound for the state-based, operational characterization. To be precise: their operational semantics fails to be compositional w.r.t. BCCSP.

Counterexample 17.. State-based versus path-based possible worlds equivalence for finite processes.

Counterexample 16 shows that a suitable formulation ⁸ of the state-based approach to possible worlds semantics is incomparable with any of the semantics encountered so far. The processes left and middle are state-based possible worlds equivalent, yet abbc ∈ T (middle) − T (left). Furthermore, the processes right and middle are tree equivalent, yet in the state-based approach one has abbc ∈ PW (right) − PW (middle).

Below I propose a formalization of the path-based approach to possible worlds semantics that, on finite processes, agrees with the denotational characterization of [49].

DEFINITION 14

A process p is a possible world of a process q if p is deterministic and p ⊑_RS q. Let PW (q) denote the class of possible worlds of q. Two processes q and r are possible worlds equivalent, notation q =_PW r, if PW (q) = PW (r). In possible worlds semantics (PW) two processes are identified iff they are possible worlds equivalent. Write q ⊑_PW r iff PW (q) ⊆ PW (r).

It can be argued that the philosophy underlying possible worlds semantics is incompatible with the view on labelled transition systems taken in this paper. The informal explanation of the action relations in Section 1.1 implies for instance that the right-hand process graph of Counterexample 8 has a state in which a has happened already and both bc and bd are possible continuations. In the possible worlds philosophy on the other hand, this process graph is just a compact representation of the set of deterministic processes {abc, abd}. None of the two processes in this set has such a state.

This could be a reason not to treat possible worlds semantics on the same footing as the other semantics of this paper. However, one can give up on thinking of non-deterministic processes as sets of deterministic ones, and justify possible worlds semantics – at least the path-based version of Definition 14 – by an appropriate testing scenario. This makes it fit in the present paper.

Testing scenario. A testing scenario for possible worlds semantics can be obtained by making one change in the reactive testing scenario of failure simulation semantics. Namely in each state only as many copies of the process can be made as there are actions in Act, and, for a ∈ Act, the first test on copy p_a of p is pressing the a-button. If it goes down, one goes on testing that copy, but is has already changed its state; if it does not go down, the test on p_a ends.

Modal characterization. On well-founded processes, a modal characterization of possible worlds semantics can be obtained out of the modal characterization of ready simulation semantics by changing the modality ${\wedge }_{i\in I}{\phi }_i$ into ${\wedge }_{a\in X}a{\phi }_a$ with X ⊆ Act. Possible worlds of a well-founded process p can be simply encoded as modal formulas in the resulting language. Probably, this modal characterization applies to image finite processes as well. For processes that are neither well-founded nor image finite this characterization is not exact, as it fails to distinguish the two processes of Counterexample 1.

Classification. RT ≺ PW ≺ RS. PW is independent of S, CS and PF.

PROOF. "PW ≺ RS" ⁹ follows by the transitivity of ⊑ _RS .

"RT ⪯ PW" holds as σ is a ready trace of p ∈ $\mathcal{P}$ iff it is a ready trace of a possible world of p.

"S ⪯̸ PW" (and hence "RS ⪯̸ PW") follows from Counterexample 8. There S (left) ≠ S (right), but PW (left) = PW (right) = {abc, abd}.

"PF ⪯̸ PW" follows since PF ⪯̸ RS.

"CS ⋡ PW" follows since CS ⋡ RT, and "PF ⋡ PW" since PF ⋡ RT.

Finally, "RT ≠ PW" follows from Counterexample 18, taken from [49]. There the first process denotes two possible worlds, whereas the second one denotes four.

Counterexample 18.. Ready trace equivalent, but not possible worlds equivalent.

Infinite processes. The version of possible worlds semantics defined above is the infinitary one. Note that RT ^∞ ≺ PW. Exactly as above one even establishes

$p{⊑}_{RS}q\Rightarrow p{⊑}_{PW}q\Rightarrow p{⊑}_{RT}^{\infty }q,$

i.e., RT ^∞ ⪯^* PW ⪯^* RS. Finitary versions could be defined by means of the modal characterization given above. I will not pursue this here.

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Special Kinds of Theorems

Antonella Cupillari , in The Nuts and Bolts of Proofs (Third Edition), 2005

EXERCISES

Use counterexamples to prove that the following statements are false.

1.

Let f be an increasing function and g be a decreasing function. Then the function f + g is constant. (See front material of the book for the definitions of nonincreasing function and f + g .)

2.

If t is an angle in the first quadrant, then 2 sin t = sin 2t.

3.

Consider the polynomial P(x) = −x ² + 2x − 3/4. If y = P(x), then y is always negative.

4.

The reciprocal of a real number x ≥ 1 is a number y such that 0<y<1.

5.

The number 2 ⁿ + 1 is prime for all counting numbers n.

6.

Let f, g, and h be three functions defined for all real numbers. If f ○ g = f ○ h, then g = h.

Discuss the truth of the following statements; that is, prove those that are true and provide counterexamples for those that are false.

7.

The sum of any five consecutive integers is divisible by 5.

8.

If f(x) = x ² and g(x) = x ⁴, then f(x) ≤ g(x) for all real numbers x ≥ 0.

9.

The sum of four consecutive counting numbers is divisible by 4 (see exercise 7).

10.

Let f and g be two odd functions defined for all real numbers. Their sum, f + g, is an even function defined for all real numbers. (See front material of the book for the definitions of even and odd functions and f + g .)

11.

Let f and g be two odd functions defined for all real numbers. Then their quotient function f/g is an even function defined for all real numbers. (See front material of the book for the definitions of even and odd functions.)

12.

A six-digit palindrome number is divisible by 11.

13.

The sum of two numbers is a rational number if and only if both numbers are rational.

14.

Let f be an odd function defined for all real numbers. The function g(x) = (f(x))² is even. (See front material of the book for the definitions of even and odd functions.)

15.

Let f be a positive function defined for all real numbers. The function g(x) = (f(x))³ is always increasing. (See front material of the book for the definitions of increasing function.)

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Unstable Operations in Generalized Cohomology

J. Michael Boardman , ... W. Stephen Wilson , in Handbook of Algebraic Topology, 1995

CONJECTURE

If x ≠ 0 in BP_n (X), where X is a space, then $v_n^{i}x\ne 0$ for all i>0.

No counter-examples are known, although examples exist [13], [14], [24] where v_jx = 0 for all j < n. It holds if x reduces nontrivially to homology, therefore for n < 2p. We hoped to circumvent our lack of knowledge of unstable homology operations by working instead with the rather better understood unstable BP-cohomology operations and using the (not at all unstable) duality spectral sequence

${\text{Ext}}_{BP*}^{**}\left(B{P}^{*}\left(X\right),B{P}^{*}\right)\Rightarrow B{P}_{*}\left(X\right)$

of Adams [1] (see also [12]). The reason for optimism is that if we substitute Σ ^k (BP ^*/I_n ) for BP ^*(X), a standard calculation shows that the only surviving Ext group is Ext ⁿ = Σ ^m (BP ^*/I_n ), with m = f(n) – k – n; so that k ≥ f(n) – 1 implies –m ≥ n – 1, almost what we want. If we confine ourselves to additive operations, we obtain –m ≥ n – 2, off by one more. We can hope to work our way up from Σ ^k (BP ^*/I_n ) to a general BP ^*(X) by extension and the filtration (1.2).

This is all grounds for our suspicion that for a geometric unstable algebra, i.e. M = BP ^*(X) for some space X, the bounds in Theorem 1.5 should be one better (thus giving us −m ≥ n in the above discussion). Again, there are no known counter-examples, although spaces are known which have deg(x_i ) = f(n_i ), thus showing that the bounds cannot be improved by more than one.

Recently, with the help of Mike Hopkins, a new approach to the Johnson Question has been developed. It requires a much better understanding of the unstable splittings of BP. Now that we have so much explicit information on these splittings, this method of attack seems promising.

Outline. There are two main threads running through this work: the theory of additive unstable operations, which closely resembles the stable theory of [8], and the theory ofall unstable operations, which is radically different. The comonad tent is big enough to accommodate both, as well as the stable theory. We have kept the additive material in separate sections so that it can be read independently.

In Section 2, we discuss several classes of cohomology operation. In Sections 3 and 4, we study the E-(co)homology of group objects, in preparation for Sections 5 and 7, where we study modules and algebras from the additive point of view. In Section 6, we consider additive operations as linear functionals. In Sections 12 and 14, we study suspensions and complex orientation. In Section 16, we present the additive structure for each of our five examples E.

It turns out that much of the stable machinery does not extend to all unstable operations, because it relies too heavily on the bilinearity of tensor products. However, the approach in terms of comonads does work, and in Section 8 we develop the requisite comonad U. We also show in Section 9 that the corresponding comonad for unstable modules does not exist and compare the various stable and unstable structures. In Section 10, we convert the categorical elegance into machinery we can use; specifically, cohomology operations become linear functionals on Hopf rings. In Theorem 10.47, we display in full detail the definition of an unstable algebra from this point of view.

In Sections 11, 13, and 15, we revisit the cohomology of a point, sphere, and complex projective space $ℂP^{\infty }$ from this new Hopf ring point of view. These spaces alone yield almost enough generators and relations to specify the Hopf rings for our five examples E, as we discuss in detail in Section 17. The case E = KU is used to determine the structure of KU _*(KU, o), as quoted in [8, §14]. From a sufficiently elevated perspective, the results of Section 17, the additive results of Section 16, and the stable results of [8] all fit into a grand master plan.

In Section 20, we restrict attention to the case E = BP and use the additive operations to recover Quillen's theorem and prove Theorem 20.11. This relies on the relations developed in Section 18. In Section 21, we use nonadditive operations to improve Theorem 20.11 by one dimension to Theorem 21.12, which is Theorem 1.5.

In Section 22, we construct additive idempotent operations θ _n which yield the desired factorizations (1.7) in all except the top degree. In Section 23, we finish off Theorems 1.12 and 1.16 by constructing nonadditive idempotent operations. To do this, it is necessary in Section 19 to develop the notion of a Hopf ring ideal.

An index of symbols is included at the end.

This work is also notable for what it does not contain. There are no spectral sequences, except implicitly in the references. There are no explicit Steenrod operations, except in a few examples; in our wholesale approach, most individual operations never even acquire names. There are no formal indeterminates anywhere; the elements that are sometimes treated as such are really Chern classes x; but when xⁱ = 0, we can no longer take the coefficients of xⁱ .

Notation. We make heavy use of the notation and machinery developed in [8]. Topologically, we generally work in the homotopy category Ho of unbased spaces. For compatibility with the unstable notation, the E-cohomology and E-homology of a spectrum X are written E ^*(X, o) and E _*(X, o). Algebraically, our most important categories are the categories FMod and FAlg of filtered E ^*-modules and algebras. These and the other categories we need were introduced in [8, §6]. We make frequent use of Yoneda's Lemma. All tensor products are taken over E ^* unless otherwise stated.

For reasons discussed in [8], we always give cohomology E ^*(X) the profinite topology [8, Definition 4.9], and complete it as in [8, Definition 4.11] to E ^*(X)^∧ as necessary. In contrast, the homology E _*(X) is always discrete. Because we emphasize cohomology, we invariably assign the degree i to elements of Eⁱ (X); this forces elements of E_i (X) to have degree –i.

One theorem provides all the duality and Künneth isomorphisms we need.

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Foundations of Complex Analysis in Non Locally Convex Spaces

Aboubakr Bayoumi , in North-Holland Mathematics Studies, 2003

Holomorphic Extensions of Dense Subspaces

The following theorem gives a counterexample to holomorphic extension problem. More precisely, it supplies us with a dense subspace which cannot be extended to the whole space all holomorphic functions.

Theorem 106

[17](1989)

Not every holomorphic function on

$l_o={\cap }_{p>0}{l}_p$

with the sup-topology, that is, the one defined by the F-norm

$\parallel x\parallel =\underset{p>0}{\sup }{\displaystyle \sum _1^{\infty }\left|x_n\right|{}^p}$

can be extended to a holomorphic function on l_p (1 > p >   0).

Proof

l_o is a dense subspace of each l_p (1 > p >   0), by Stile[[201],p.ll7]. Hence every bounding subsets of l_o is bounding as a subset of l_p. Since bounding subsets of l_o and of l_p are relatively compact, by Theorem 90, p. 186, it suffices to pick up a non compact subset of l_o which is compact as a subset of l_p. For example the set

(9.84) $D=\left\{0,x^{\left(1\right)},x^{\left(2\right)},\dots \right\},\mathit{where}$

(9.85) $x^{\left(n\right)}=\left(0,\dots ,0,{\left(\frac{1}{n}\right)}^{1-1/p},0,\dots \right),n\in N$

is compact in l_p if (1 > p >   0), but it is not compact in l_o. Notice that

$\begin{array}{l}\parallel x^{\left(n\right)}{\parallel }_o=\underset{p>0}{\sup }{\displaystyle \sum _1^{\infty }|}{x}_n|{}^p\\ =\underset{p>0}{\sup {\left(\frac{1}{n}\right)}^{\left(1-1l_p\right)p}=n^{1-p}\to \infty }\end{array}$

as n → ∞. This completes the proof of the theorem. ■

As another counter example where the holomorphic extension fails to occur from a non locally convex subspace to a locally convex one we obtain the following result.

Theorem 107

If f is a holomorphic function on l_p (1 > p >   0), then ƒ may not be extended to a holomorphic function on l ₁ . That is, not every holomorphic function on l_p can be extended holomorphically to its Banach-envelope l ₁.

Proof

The set

$A=\left\{x^{\left(n\right)};x^{\left(n\right)}=\left({\left(\frac{1}{n}\right)}^{1/p},\dots ,{\left(\frac{1}{n}\right)}^{1/p},0,0,\dots \right)\right\}$

is not bounding as a subset of l_p, for it is not relatively compact. However it is bounding as a subset of l ₁. See also proof of Theorem 95. This completes the proof of the theorem. ■

Remark 37

It follows from the preceding theorem and the fact that the linear completion l _p ^{^}  = l ₁ that

$l_p\underset{\ne }{\subset }{l}_p^{^}\text{;}$

also that

${\left(l_p\right)}_{\partial }=l_p\text{.}$

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Waves in Random Media

A.Z. Genack , ... B.A. van Tiggelen , in Encyclopedia of Condensed Matter Physics, 2005

Coherent Backscattering

Coherent backscattering is a pertinent counter example of the statement that multiple scattering destroys all interference effects, and an illustration of how mesoscopic wave–particle duality can be employed. Consider a typical random walk of a wave along an arbitrary sequence of scatterers before being reflected back toward the light source. In the radiative transfer picture, all interference effects are assumed to be washed out upon averaging over all configurations of the particles, and the intensity due to waves following two paths would be given by ${\left|{\psi }_1\right|}^2+{\left|{\psi }_2\right|}^2$ . However, in the wave picture, two oppositely propagating waves will always interfere constructively at the source. The reflection of energy is then given by

[12] $\begin{array}{c}R(\theta =\pi )=|{\psi }_1+{\psi }_2{|}^2=|{\psi }_1{|}^2+|\psi {|}^2+2Re({\psi }_1{\psi }_2^{*})\\ =2(|\psi {|}^2+|\psi {|}^2)\end{array}$

which is twice the answer obtained from radiative transfer. Coherent backscattering is a direct consequence of the reciprocity principle. The angular line profile of coherent backscattering is the Fourier transform of the distribution of distances between the entry and exit of reflected photons, and as such a very sensitive way of probing the path length distribution of reflected photons. The algebraic long-range nature of this distribution generates a triangular cusp which is seen in Figure 8. It is noticed that the enhancement factor of 2 is lowered when the $\text{ℓ}/\lambda$ ratio approaches unity. This has been attributed to loop-type photon paths in the random medium, which then become numerous and give a flat contribution to the background.

Figure 8. Coherent backscattering of light measured, with two different values for the transport mean free path $\text{ℓ}$ . The typical angular width varies as $\lambda /\text{ℓ}$ . Narrow cone: a sample of BaSO₄ powder with $\text{ℓ}/\lambda =4$ , Broad cone: TiO₂ sample with $\text{ℓ}/\lambda =1$ . The inset confirms the triangular cusp predicted by diffusion theory, and also shows that the maximum enhancement factor is lowered for the sample with small $\text{ℓ}/\lambda$ . (Wiersma et al. (1995) Physical Review Letters 74: 4193.)

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Coherent Systems

Karl Schelechta , in Studies in Logic and Practical Reasoning, 2004

3.5.1 The formal results

S. Kraus, D. Lehmann, and M. Magidor have shown that for any logic |~   for $\mathrm{ℒ}$ , which is supraclassical, cumulative, and distributive, there is a D –smooth preferential model $\mathrm{ℳ}$ , s.t. for all finite $T\subseteq \mathrm{ℒ}{T}^{\mathrm{ℳ}}=\overline{\overline{T}}$ — where $T^{\mathrm{ℳ}}$ is the logic defined by the structure. ([KLM90], see also [Mak94], Observation 3.4.7.) We show that the restriction to finite T is necessary, by providing a counterexample for the infinite case. We start by quoting a lemma by D. Makinson.

Both Lemma 3.5.1 and our counterexample Example 3.5.1 have appeared in [Mak94], Section 3.4 (Lemma 3.4.9, Observation 3.4.10). The reader less familiar with transfinite ordinals can find there a more algebraic proof that our counterexample satisfies the logical properties claimed. Our technique of constructing a logic inductively by a mixed iteration of suitable length has, however, proved useful in other situations as well (see [Sch91-2]), moreover, it is very fast and straightforward: once you have the necessary ingredients, the machinery will run almost by itself.

Lemma 3.5.1

Let a logic |~   on $\mathrm{ℒ}$ be representable by a classical preferential model structure. Then, for all $A\subseteq \mathrm{ℒ},x\in \mathrm{ℒ},x\notin \overline{\overline{A}}$ there is a maximal consistent (under ⊢) $\Delta \subseteq \mathrm{ℒ}$ s.t. $\overline{\overline{A}}\subseteq \Delta ,x\notin \Delta$ , and $\overline{\overline{\Delta }}\ne \mathrm{ℒ}$ .

Proof:

Let $\mathrm{ℳ}=\left(\mathcal{X},\prec \right)$ be a representation of |~, i.e. $\overline{\overline{A}}=A^{\mathrm{ℳ}}$ for all $A\subseteq \mathrm{ℒ}$ . Let $A\subseteq \mathrm{ℒ}$ , $x\in \mathrm{ℒ}$ , and $x\notin \overline{\overline{A}}$ Then there is   < m, i  >   minimal in $\mathcal{X}\lceil M_A$ , with m ⊭ x. Note that by minimality, $m\models \overline{\overline{A}}.\Delta :=\left\{y\in \mathrm{ℒ}:m\models y\right\}$ is maximal consistent, $x\notin \Delta ,\overline{\overline{A}}\subseteq \Delta$ , and   < m, i  >   is also minimal in $\mathcal{X}\lceil M_{\Delta }$ , by M _Δ  ⊆ M_A . Thus, $m\models \overline{\overline{\Delta }}$ , and by classicality of the models, $\overline{\overline{\Delta }}\ne \mathrm{ℒ}$ .

We now construct a supraclassical, cumulative, distributive logic, and show that the logic so defined fails to satisfy the condition of Lemma 3.5.1, and is thus not representable by a preferential structure.

Example 3.5.1

Let $\upsilon \left(\mathrm{ℒ}\right)$ contain the propositional variables p_i : i ∈ ω, r. (Note that we do not require $\mathrm{ℒ}$ to be countable, we leave plenty of room for modifications of the construction!) We shall violate compactness badly "in both directions" by adding the rules (infinitely many p_i ) |~ r and (infinitely many ﹁p_i ,) |~ r. To account for distributivity, we shall add for all $\varphi \in \mathrm{ℒ}$ (infinitely many p_i ∨ ϕ) |~ r ∨ ϕ and (infinitely many ﹁p_i ∨ ϕ) |~ r ∨ ϕ. Closing under |~   and classical logic ω ₁ many times to take care of the countably infinite rules will give the result.

The details:

We define the logic |~ by a mixed iteration: For $B\subseteq \mathrm{ℒ}$ define I ⁺ _{Β, ϕ} := {i  < ω: p_i ∨ ϕ ∈ B}, Ι ⁻ _{Β, ϕ} := {i  < ω: ﹁p_i ∨ ϕ ∈ B}.

Define now inductively

A ₀ := A

for successor ordinals (α a limit or 0, i ∈ ω):

${A_{\alpha +2i+}}_1:=\overline{{A_{\alpha }}_{+2i}}$

A _{α  +   2i  +   2} := A _{α  + 2i  +   1} ∪ {r ∨ ϕ : Ι ⁺ _{Aα  +   2i  +   1,ϕ} is infinite or Ι ⁻ _{Aα  +   2i  +   1,ϕ} is infinite

}

for limit λ:

$A_{\lambda }:=\cup \left\{A_i:i<\lambda \right\}$

$\overline{\overline{A}}:=A_{{\omega }_1}\text{.}$

We show |~   is as desired. Note that the defined logic is monotone.

1)

$\overline{A}\subseteq \overline{\overline{A}}$ is trivial.

2)

$A\subseteq B\subseteq \overline{\overline{A}}\to \overline{\overline{A}}=\overline{\overline{B}}$ :

2.1)

$\overline{\overline{A}}\subseteq \overline{\overline{B}}$ by monotony

2.2)

$\overline{\overline{B}}\subseteq \overline{\overline{A}}$ : Let $\varphi \in \overline{\overline{B}}$ . In deriving ϕ in $\overline{\overline{B}}$ , we have used only countably many elements from B. This is seen as follows. Let β be minimal such that ϕ ∈ Β_β . ϕ can be derived from at most countably many ϕ_i ∈ Β _{β  −   1} (β has to be a successor ordinal). Arguing backwards, and using ω.ω  = ω (cardinal multiplication), we see what we wanted. (This is, of course, the outline for an inductive proof.) As $B\subseteq \overline{\overline{A}}$ , using regularity of ω ₁, we see that there is some α  < ω ₁ s.t. all ϕ_j used in the derivation of ϕ from B are in A_α . But then ϕ ∈ Α _{α  + β} .

3)

Distributivity: We show by induction on the derivation of a, b that $a\in \overline{\overline{A}}$ , $b\in \overline{\overline{B}}\to a\vee b\in \overline{\overline{A\cap B}}$ . To get started, use $A_0\subseteq A_1=\overline{A}$ , and $a\in \overline{A}$ , $b\in \overline{B}\to a\vee b\in \overline{A}\cap \overline{B}$ . By symmetry, it suffices to consider the cases for a. Let a ₁,…,a_n ⊢ a by classical inference. By induction hypothesis, $a_1\vee b,\dots ,a_n\vee b\in \overline{\overline{\overline{A}\cap \overline{B}}}$ , but then $a\vee b\in \overline{\overline{\overline{A}\cap \overline{B}}}$ , as the latter is closed under ⊢. Assume now a  = r ∨ ϕ ∈ A_α has been derived from infinitely many p_i ∨ ϕ (i ∈ I) in A _{α  −   1}. By induction hypothesis, $p_i\vee \varphi \vee b\in \overline{\overline{\overline{A}\cap \overline{B}}}$ . So $p_i\vee \varphi \vee b\in {\left(\overline{A}\cap \overline{B}\right)}_{{\beta }_i}$ for β_i   < ω ₁. Again by regularity of ω ₁, all $p_i\vee \varphi \vee b\in {\left(\overline{A}\cap \overline{B}\right)}_{\beta }\left(i\in I\right)$ for some β  < ω ₁. But then $r\vee \varphi \vee b=a\vee b\in {\left(\overline{A}\cap \overline{B}\right)}_{\beta +2}$ · The case ﹁p_i ∨ ϕ is similar.

We use the lemma to obtain the negative result, as the logic constructed above does not satisfy the lemma's condition:

Consider now A := Ø. Assume there is ϕ s.t. infinitely many $p_i\vee \varphi \in \overline{A}$ , thus there is ϕ s.t. infinitely many p_i ∨ ϕ are tautologies. But then ϕ has to be a tautology (consider (p_i ∨ ϕ) ↔ (﹁ϕ  → p_i ) and finiteness of ϕ!), thus ϕ and $\varphi \vee r\in \overline{A}$ . Likewise for ﹁p_i ∨ ϕ. So, the rules (infinitely many p_i ∨ ϕ) |~ r ∨ ϕ, etc. give nothing new, and $\overline{A}=\overline{\overline{A}}$ . In particular, $r\notin \overline{\overline{A}}$ .

Assume now $\Delta \subseteq \mathrm{ℒ}$ to be maximal consistent. So Δ decides all p_i : i ∈ ω. Thus either infinitely many or p_i , or ﹁p_i in Δ. Thus, $r\notin \overline{\overline{\Delta }}$ . Hence |~   is not representable by a preferential structure. □

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Reconstruction Theory and Nonlinear Time Series Analysis

Floris Takens , in Handbook of Dynamical Systems, 2010

5.3 Examples

We have seen that the existence of probability measures of relative frequencies, and hence the validity of stationarity, can be violated in two ways: one is that the elements of the sequence, or of the time series, under consideration, move off to infinity; the other is that limits of the form

$\underset{n\to \infty }{lim}\frac{\sum _{i=0}^{n-1}\psi \left(p_i\right)}{n}$

do not exist. If we restrict ourselves to time series generated by a dynamical system with compact state space, then nothing can move off to infinity, so then there is only the problem of existence of the limits of these averages. There is a general belief that for evolutions of generic dynamical systems with generic initial state, the probability measures of relative frequency are well defined. A mathematical justification of this belief has not yet been given. In my opinion this is one of the challenging problems in the ergodic theory of dynamical systems; see also[39,45].

There are however non-generic counterexamples. One is the map $\phi \left(x\right)=3x$ modulo 1, defined on the circle $\mathbb{R}$ modulo 1. If we take an appropriate initial point, then we get an evolution which is much like the 0-, 1-sequence which we gave as an example of a non-stationary time series. The construction is the following. As the initial state we take the point $x_0$ which has, in the ternary system, the form $0.{\alpha }_1{\alpha }_2\dots$ with

–

${\alpha }_i=0$ if the integer part of $ln\left(i\right)$ is even;

–

${\alpha }_i=1$ if the integer part of $ln\left(i\right)$ is odd.

If we consider the partition of $\mathbb{R}$ modulo $\mathbb{Z}$ , given by $I_i=[\frac{i}{3},\frac{i+1}{3})$ , $i=0,1,2$ , then ${\phi }^i\left(x_0\right)\in I_{{\alpha }_i}$ ; both $I_0$ and $I_1$ have a unique expanding fixed point of $\phi$ : 0 and 0.5 respectively; long sequences of 0's or 1's correspond to long sojourns very near 0 respectively 0.5. This implies that for any continuous function $\psi$ on the circle, which has different values in the two fixed points 0 and 0.5, the partial averages

$\frac{\sum _{i=0}^{n-1}\psi \left({\phi }^i\left(x_0\right)\right)}{n}$

do not converge for $n\to \infty$ .

For other examples see[43] and the references given in that paper.

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Survey of Regularity Results

Frank Morgan , in Geometric Measure Theory (Fifth Edition), 2016

Exercises

8.1

Try to come up with a counterexample to Theorems 8.1 and 8.4.

8.2

Try to draw an area-minimizing rectifiable current bounded by the trefoil knot. (Make sure your surface is orientable.)

8.3

Illustrate the sharpness of Theorem 8.3 by proving directly that the union of two orthogonal unit discs about 0 in R ⁴ is area minimizing.

Hint: First show that the area of any surface S is at least the sum of the areas of its projections into the x ₁-x ₂ and x ₃-x ₄ planes.

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Definitions and Basic Properties

Weiyang Ding , Yimin Wei , in Theory and Computation of Tensors, 2016

5.4.3 A Counter Example

We will give a 4^th-order counter example in this section to show that the set of all monotone $\mathcal{Z}$ -tensor is a proper subset of the set of all nonsingular $\mathcal{M}$ -tensor, when the order is larger than 2.

Recall the outer product X ∘ Y, where ${\left(X\circ Y\right)}_{i_1{i}_2{i}_3{i}_4}=x_{i_1{i}_2}\cdot y_{i_3{i}_4}$ . Let $\mathcal{J}=I_n\circ I_n$ , where I_n denotes the n  × n identity matrix. It is obvious that the spectral radius $\rho \left(\mathcal{J}\right)=n$ , since the sum of each row of $\mathcal{J}$ equals n. Take

$\mathcal{A}=s\mathrm{ℐ}-\mathcal{J}\left(s>n\right),\mathbf{x}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill ⋮\hfill \\ \hfill 1\hfill \\ \hfill -\delta \hfill \end{array}\right]\left(0<\delta <1\right)\text{.}$

Then $\mathcal{A}$ is a nonsingular $\mathcal{M}$ -tensor and

$\mathcal{A}{\mathbf{x}}^3=s{\mathbf{x}}^{\left[3\right]}-\left({\mathbf{x}}^{⊺}\mathbf{x}\right)\mathbf{x}=s\left[\begin{array}{c}\hfill 1\hfill \\ \hfill ⋮\hfill \\ \hfill 1\hfill \\ \hfill -{\delta }^3\hfill \end{array}\right]-\left(n-1+{\delta }^2\right)\left[\begin{array}{c}\hfill 1\hfill \\ \hfill ⋮\hfill \\ \hfill 1\hfill \\ \hfill -\delta \hfill \end{array}\right]=\left[\begin{array}{c}\hfill s-n+1-{\delta }^2\hfill \\ \hfill ⋮\hfill \\ \hfill s-n+1-{\delta }^2\hfill \\ \hfill \left(n-1+\left(1-s\right){\delta }^2\right)\delta \hfill \end{array}\right]\text{.}$

When $\delta \le \sqrt{\frac{n-1}{s-1}}$ , the vector $\mathcal{A}{\mathbf{x}}^3$ is nonnegative while x is not. Therefore $\mathcal{A}$ is not a monotone $\mathcal{Z}$ -tensor, although it is a nonsingular $\mathcal{M}$ -tensor.

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Eigenvalues in Riemannian Geometry

In Pure and Applied Mathematics, 1984

Remark 3

We note that Cheng's theorem is a counterexample to the metaphysical principle: Large spaces–small eigenvalues, and small spaces–large eigenvalues. We refer to the rubric as a metaphysical principle because (i) large spaces and small spaces are not well defined, (ii) the rubric is valid in sufficiently many situations, where there might be agreement as how to declare a space larger or smaller, as to serve as a guide to intuition in unknown situations, (iii) it therefore serves as a barometer of the striking character of a result serving as a counterexample to the principle, and therefore, (iv) a commitment to the validity of such a principle leads to either (a) a more penetrating study into the determination of geometric size or (b) declaring that large eigenvalues might serve—in appropriate situations—as a definition for a small space.

Many of the results described in this book reflect the validity of this principle. Compare the normalization of geometric data (Section XII.7), the Weyl formula (Section I.3), and especially the domain monotonicity of eigenvalues with vanishing Dirichlet boundary data (Section I.5), for early examples of this principle. The principle is also a very helpful guide to appreciating the results of Chapter IX. But here, the Rauch and Bishop theorems say that increasing the curvature decreases the size of a geodesic disk of fixed radius, and the Cheng theorems state that increasing the curvature also decreases the lowest Dirichlet eigenvalue (contrary to our expectation that it be increased).

We note that the Obata theorem in the next section states that if M is a compact Riemannian manifold of dimension n ⩾ 2, with Ricci curvatures bounded below by the constant (n − 1)K, K > 0, then λ₁(M) ⩾ n K = λ₁(M _K )—the result consistent with the metaphysical principle. Interestingly, the strong form of the theorem states that λ₁(M) = n K if and only if M is isometric to M _K —our proof uses Cheng's theorem, in violation of the principle.

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