On a problem posed by Pauli

A problem posed in a remark by Pauli is discussed: Is it possible to recover the state vector of a quantum system from the distribution functions of the physical observables of this system?     

On a problem posed by Murray and Tent

In this short note, we give a positive answer to a problem posed by Murray and Tent (J. Algebra 398:318–328, 2014).     

On a problem of J. P. Williams

Let be the algebra of all bounded operators on a Hilbert space . Let be a continuous function on the closed disk and let

for some 0,$"> for all and all with . Then is differentiable on . The paper shows that the function may have a discontinuous derivative.

     

On a problem of S.L. Sobolev

In his famous works of 1930 [1,2] Sergey L. Sobolev has proposed a construction of the solution of the Cauchy problem for the hyperbolic equation of the second order with variable coefficients in Rş. Although Sobolev did not construct the fundamental solution, his construction was modified later by Romanov (2002) and Smirnov (1964) to obtain the fundamental solution. However, these works impose a restrictive assumption of the regularity of geodesic lines in a large domain. In addition, it is unclear how to realize those methods numerically. In this paper a simple construction of a function, which is associated in a clear way with the fundamental solution of the acoustic equation with the variable speed in 3-d, is proposed. Conditions on geodesic lines are not imposed. An important feature of this construction is that it lends itself to effective computations.     

On a problem of B. Jónsson concerning subalgebras and endomorphisms

Presented by the R. W. Quackenbush.     

On a problem of W. J. LeVeque concerning metric diophantine approximation

We consider the diophantine approximation problem

where is a fixed function satisfying suitable assumptions. Suppose that is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdos). Here, we are going to extend and solve LeVeque's problem.

     

On a problem posed by Orey and Pruitt related to the range of the N-parameter wiener process in R d

Let W ^(N,d) be the N-parameter Wiener process with values in R ^d .It is shown that the range of W^(N,d) is all of R^d when d<2N, answering a question raised earlier by Orey and Pruitt.     

A note on a problem by H. S. Shapiro

В РАБОтЕ ДОкАжАНО, ЧтО limk _a *f(x)=f(x) пОЧтИ ВсУДУ, гДЕk _a(t)=a^?n k(a^?1t), t?Rⁿ, Для Дль ДОВОльНО шИРОкОг О клАссА ФУНкцИИk(t). ДАНы УслОВИь, пРИ кОтО Рых пОлУЧЕННыИ РЕжУл ьтАт РАспРОстРАНьЕтсь НА ФУНкцИУ $$k(x,y) = \gamma \frac{1}{{1 + |x|^\alpha }} \cdot \frac{1}{{1 + |y|^\beta }},$$ гДЕ α, β>1, А γ — НОРМИРУУЩ ИИ МНОжИтЕль тАкОИ, Чт О∫∫k(x, y) dx dy=1.     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

On a problem by V. A. Toponogov

We give the positive solution of a problem formulated by V. A. Toponogov and discuss some of its natural generalizations.     

On a ramsey-type problem of J. A. Bondy and P. Erdös. I

Let m → (C_k, C_n) signify the truth of the following statement: Let {V(G); ≥ m; if G contains no C_k, then $\text{G}$ contains a C_n. Bondy and Erdös [1] proved that for n > 3 2n ? 1 → (C_n, C_n). They conjectured that 2n ? 1 → (C_n, C_k) for all n > 3 and all k < n and could prove it only for $\text{k < (2n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. In this paper we prove this for all n > 4 and for all k < n.