How to find derivative of a parametric function?

When a particle moves on the plane,it can be described uncounted points corresponding to coordinates in a Cartesian coordinate system with the same unit on the each axis.However,not all the processes are convenient to express y in terms x.At times a relationship between x and y in terms of a third variable which is a various time t,is mexessary.     

How does the method of implicit differentiation use?

The functions are described by expressing one variable explicitly in terms of another variable.For example y=2x~3+x or y= sinx,or,in general,y=f(x).In other words,an explicit relation between x and     

How does the core sit inside the mantle?

The k‐core, defined as the maximal subgraph of minimum degree at least k, of the random graph has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold d_k for the appearance of an extensive k‐core. The aim of the present paper is to describe how the k‐core is “embedded” into the random graph in the following sense. Let and fix . Colour each vertex that belongs to the k‐core of in black and all remaining vertices in white. Here we derive a multi‐type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k‐core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2‐core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k‐core. Based on this the study of the k‐core reduces to the analysis of Warning Propagation on a suitable Galton‐Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 459–482, 2017     

How to estimate the sum of a series?

<正>Now we’ll learn how to estimate the sum of a series?We have been able to use the integral test to show that a series ∞∑(n=1)a_n is convergent or divergent.But we also don’t know the best way to find the sum of the series.It is difficult to compute the sum of any series.Here is the best way to find an approximation to the sum of the series.You know     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

What is the limit of a function in precise way?

The precise definition of a limit is represented in graphically,numerically,analytically and verbally by using algebraic way.The intuitive definition of a limit is given as the limit of f(x),as x approaches a,equals L ,and we write lim x→af(x)=L.But the more questions will be asked if you think it broadly.Is the definition of the limit of function vague? How should be x close to a presented algebraically? How should be f(x) gets closer and closer to L presented algebraically? How close to a does x to     

Correction and strengthening of “How large are the level sets of the Takagi function?”

The purpose of this note is to correct an error in an earlier paper by the author about the level sets of the Takagi function (Monatsh Math 167:311–331, 2012), and to prove a stronger form of one of the main results of that paper.     

How small are the increments of a Wiener process?

Let $\text{γ}{}_{\mathrm{т}}\text{=(8(}\text{logTa}{}^{-1}{}_{\mathrm{T}}\text{+log log T)}\text{π}{}^2\text{a}{}_{\mathrm{T}}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$, 0<a_T?T<∞, and {W(t);0?t<∞} be a standard Wiener process. This exposition studies the almost sure behaviour of

, under varying conditions on a_T and T/a_T. The following analogue of Lévy's modulus of continuity of a Wiener Process is also given:

$\text{lim}{}_{\mathrm{h\to 0}}\text{inf}{}_{\mathrm{0?t?1}}\text{sup}{}_{\mathrm{0?s?h}}\text{(}\text{8 log h}{}^{-1}\text{π}{}^2\text{h}\text{)}\text{1}\text{2}\text{|W(t+s)?W(t)| =}{}^{\mathrm{a.s.}}\text{1.}$

and this may be viewed as the exact “modulus of non-differentiability” of a Wiener Process.     

How big can the circuits of a bridge of a maximal circuit be?

IfCE(G) is a maximum cardinality cocircuit of a 2-connected graphG, then no other maximum cocircuit is contained in one and the same block ofG-C. The analogous conjecture for real representable matroids would have important applications to classifying convex bodies with a certain Helly type property.     

How Big Are the Increments of a Two-Parameter Gaussian Process?

Some limit theorems on the increments of a two-parameter Gaussian process are obtained via estimating large deviation probability inequalities on the suprema of the Gaussian process which is a generalization of a two-parameter Lévy Brownian motion.     

How big are the increments of a multi-parameter Wiener process?

Summary Let _T = (2a _T(log Ta _T ^–1 +log log T))^–1/2, 0< a _T T< and let R^* be the set of sub-rectangles of the square [0, T ^1/2]x[0, T^1/2], having an area a _T . This paper studies the almost sure limiting behaviour of as T, where W is a two-time parameter Wiener process. With a _T =T, our results give the well-known law of iterated logarithm and a generalization of the latter is also attained. The multi-time parameter analogues of our twotime parameter Wiener process results are also stated in the text.Research partially supported by a Canadian NRC grant and a Canada Council Leave Fellowship     

When does the top homology of a random simplicial complex vanish?

Several years ago Linial and Meshulam (Combinatorica 26 (2006) 457–487) introduced a model called of random n‐vertex d‐dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability at which the d‐dimensional homology of such a random d‐complex is, almost surely, nonzero? Here we derive an upper bound on this threshold. Computer experiments that we have conducted suggest that this bound may coincide with the actual threshold, but this remains an open question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 26–35, 2015     

How to find derivative of a parametric function?(2)

Example 2:At time t≥0,aparticle moving in the xy-plane has vector given by V(t)=[X(t),Y(t)]=[t2,5t].What is the acceleration vector of the particle at timet=3? Solution:From the condition is given,we know X(t)=t2 and Y(t)=5t.     

How many bits of information does an independent variable yield in a multiple regression?

A simple measure from statistical information theory is proposed to measure the contribution of each independent variable in a multiple regression.